Choosing between 3D printing and CNC machining can be pivotal for manufacturers, designers, and engineers. Each method has distinct advantages, depending on factors like production volume, material requirements, and the complexity of the design. 3D printing offers unparalleled flexibility. It allows for rapid prototyping and intricate geometries that are difficult to achieve with traditional methods.
On the other hand, custom CNC machining is known for its precision and ability to work with a wide range of materials, making it ideal for producing high-quality parts with tight tolerances. Understanding these differences is crucial for selecting the right technique for your project, whether looking to innovate quickly or produce durable, high-quality components.
This article will look at the key aspects of 3D printing and CNC machining. It will help you determine which method best suits your needs.
3D printing, or additive manufacturing, is a cutting-edge process to create three-dimensional objects from a digital model. This technique involves building the object layer by layer, which sets it apart from traditional subtractive manufacturing methods that remove material from a larger block. The digital model, typically created using Computer-Aided Design (CAD) software, serves as a blueprint for the object, guiding the printer through each layer's construction.
CNC machining, short for Computer Numerical Control machining, is a versatile manufacturing process involving automated machine tool control using computer programs. This subtractive manufacturing technique removes material from a solid block or workpiece to create a highly precise and accurate finished part.
Below are some of the key differences between 3D printing and CNC machining:
3D printing offers nearly limitless design flexibility, allowing for the creation of complex geometries and intricate details that would be difficult or impossible to achieve with traditional manufacturing methods. This freedom is due to the addictive nature of the process, which builds parts layer by layer without the need for specific tooling or support for internal features. As a result, designers can focus on optimizing the functionality and aesthetics of the part without being constrained by the manufacturing process itself.
On the other hand, CNC machining is limited by certain design constraints inherent to the subtractive process. These constraints include tool access, tool path, and the need to avoid undercuts and internal geometries that are difficult to reach with cutting tools. Additionally, CNC machining requires careful planning of tool paths. It may also involve multiple setups and repositioning of the workpiece to achieve the desired shape. This method can restrict the complexity of parts that can be efficiently produced.
The precision of 3D printing varies depending on the specific technology used. Generally, the resolution can range from 0.016 mm to over 1 mm, with typical consumer-grade 3D printers achieving around 0.2 mm precision. While this is sufficient for many applications, achieving high precision consistently across different geometries and materials can be challenging.
CNC machining produces parts with high precision and tight tolerances. It can achieve precision levels as fine as 0.005 mm by employing slow feeds, new cutters, and shallow cuts. This makes CNC machining ideal for applications with critical dimensional accuracy and surface finish, such as aerospace and medical device manufacturing.
Operating a 3D printer generally requires less specialized skill compared to CNC machining. The process involves preparing a digital file, selecting orientation, and adding necessary support. Once the setup is complete, the printer performs the build with minimal human intervention, making it accessible to users with basic technical knowledge.
CNC machining demands a higher level of operator skill due to the complexity of setting up the machine, programming tool paths, and selecting appropriate cutting tools. Operators need to have a deep understanding of the machining process, material properties, and the capabilities of the equipment to consistently produce high-quality parts. This expertise is critical for achieving the desired precision and surface finish.
3D printing typically has a low setup time, but the actual build time can be lengthy. It often takes several hours or even days to complete, depending on the size and complexity of the part. This makes it suitable for prototyping and low-volume production, where quick iteration and design flexibility are prioritized.
CNC machining can have high setup and programming times, particularly for complex parts. However, once the setup is complete, the cutting stages are very fast, allowing for rapid part production. This makes CNC machining well-suited for high-volume production runs where speed and efficiency are essential.
The surface finish of 3D-printed parts can vary widely based on the technology used. Common issues include graininess, rough textures, and visible layer lines. Post-processing methods such as sanding, polishing, and media blasting can improve the finish. However, achieving a smooth, high-quality surface can be challenging without additional work.
CNC machining can produce excellent surface-quality parts, particularly when using longer cut times and finer cutting tools. The process can achieve a highly uniform and precise finish, making it ideal for applications where aesthetic and functional surface properties are important. Various post-processing techniques, such as anodizing and powder coating, can further enhance the surface finish of machined parts.
The strength of 3D-printed parts is often lower than that of their machined counterparts. Depending on the printing process and material, the strength can range from 10% to 20% of the native material's properties. This is due to the layer-by-layer construction, which can introduce weaknesses and inconsistencies in the final part.
CNC machining produces parts with the full strength of the native material, as the process involves removing material from a solid block. It results in parts with superior mechanical properties and durability, making CNC machining the preferred choice for applications where strength and reliability are critical.
Choosing between 3D printing and CNC machining is influenced by your specific needs. 3D printing is ideal for rapid prototyping, complex geometries, and low-volume production with diverse material options like plastics and metals. It offers customization and reduced waste but may require post-processing for a smooth finish.
CNC machining excels in high precision, consistency, and the ability to work with a wide range of materials, including metals and composites. It's best for producing parts with tight tolerances and in larger volumes. Ultimately, the choice depends on the project's complexity, material requirements, and production scale.
The critical value serves as a boundary that defines a specific range where the test statistic acquired during hypothesis testing, is improbable to lie within. The critical value is a benchmark against which the obtained test statistic is compared during hypothesis testing. This comparison helps in deciding whether to reject the null hypothesis or not.
On a graph, the critical value explains the boundary between the acceptance and rejection areas in hypothesis testing. It aids in judging the statistical importance or significance of a test statistic. This article will explain the following basics of critical value:
What is the critical value?
Types of Critical Value.
How to Calculate Critical Value?
Examples of Critical Value.
Summary.
A critical value is a specific test statistic value that establishes a confidence interval’s limits (upper and lower). It also sets the threshold for determining statistical significance in a statistical test. It indicates the distance away from the average (mean) of the distribution needed to comprehend a particular portion of the overall variation in the data (Such as 90%, 95%, or 99%).
There are various types of critical values used in statistical analysis which depend on the nature of the test and the distribution being considered. Some of the common types include:
Z-Score Critical Values
T-Score Critical Values
χ² (Chi-Square) Critical Values
F-Statistic Critical Values
Used in hypothesis testing for population means when the population standard deviation is known. Z-scores correspond to specific percentiles of the standard normal distribution.
Specify the alpha level.
Compute 1 minus the α level to derive the adjusted value for a 2-tailed test. Deduct the alpha level from 0.5 in the case of a 1-tailed test.
Use the z distribution table to find the area and obtain the z critical value. In the case of a left-tailed test, include a negative sign to the critical value after obtaining it.
Used in hypothesis testing for population means when we don't know the population standard deviation and have a small sample size. T-scores come from the t-distribution and vary based on the degrees of freedom.
Used in chi-square tests for independence or goodness of fit. Chi-square critical values are based on the chi-square distribution and depend on the degrees of freedom and the chosen level of significance.
Calculate the degrees of freedom for the Chi-Square distribution.
Choose the significance level (α).
Refer to the Chi-Square distribution table.
Locate the critical value corresponding to the degrees of freedom and chosen significance level.
Utilized in ANOVA (Analysis of Variance) tests to compare variances between multiple groups. F-statistic critical values are taken from the F-distribution and depend on the degrees of freedom of the groups being compared.
Compute the alpha level.
Deduct one from the size of the initial sample to get the degree of freedom. Denoted as X.
Deduct one from the second sample to determine the 2nd Degree (df) of freedom. Labeled as y.
Consult the F distribution table, locating the value where the column representing x intersects with the row representing y. This intersection provides the F critical value necessary for the analysis.
To calcauate the different critical values first we need to select the test and use the related distribution table according to the test. There’s no universal formula and methods for finding the value of the critical values, it just depend on the test selection. Follow the below steps to find the critical value using different test:
Type of hypothesis test: Z-test, t-test, chi-square test, F-test, etc.
Level of significance (α): Typically, 0.05 or 0.01.
Tailed Ness of the test: One-tailed or two-tailed.
Degrees of freedom (df): Often needed for t-tests, chi-square tests, and F-tests.
The critical values for both 1-tailed and 2-tailed tests can be determined using the confidence interval. The process to calculate the critical value is as follows:
Deduct the confidence level from 100%.
Change this value into decimal form to obtain α (alpha).
If the test is 1-tailed, the alpha level remains the same as in step 2. However, for a 2-tailed test, the α level is divided by 2.
The critical value can be determined by consulting the relevant distribution table, based on the type of test and the alpha value.
Examples Related to Critical Value
In this section, we calculate the different critical values using the its respective test and formulas. For the better underestnfding of the calculations provide the detailed steps.
Example 1: Chi-Square Critical Value Calculation
Suppose you’re conducting a chi-square test to analyze the independence between two categorical variables in a survey. Your contingency table has 3 rows and 4 columns.
Solution:
Calculate Degrees of Freedom (df):
For a chi-square test of independence, degrees of freedom (df) are computed as:
df = (Number of rows - 1) * (Number of columns - 1)
df = (3 - 1) * (4 - 1)
df = 2 * 3 = 6
Choose Significance Level (α):
Suppose we are working with a significance level at α = 0.05, which is commonly used in hypothesis testing.
Refer to Chi-Square Distribution Table:
Consult the Chi-Square distribution table with 6 degrees of freedom and α = 0.05.
Locate Critical Value:
Find the critical value for 6 degrees of freedom at α = 0.05 in the Chi-Square distribution table.
Chi-square Distribution Table
Critical Value = χ² = 12.592 (df=6, α=0.05)
Alternatively, you can use the Critical Value Calculator to determine the critical value quickly, saving your time and efforts by manual calculations.
Example 2: Z – score Critical Value Calculation
Suppose we're conducting a hypothesis test to determine if the average IQ of a population is significantly different from a claimed mean IQ of 100, with a population standard deviation known to be 15. We'll perform a two-tailed test at a significance level (α) of 0.21.
Solution:
Specify the alpha level: α = 0.21 (significance level)
Compute 1 minus the alpha level for a two-tailed test: For a two-tailed test, the adjusted alpha level is
1 - α = 1 - 0.21 = 0.79.
Use the Z-distribution table to find the critical Z-value:
Therefore, the critical Z-score for a two-tailed test is approximately ± 0.81.
This article explored the essential concept of critical values in hypothesis testing. We understood their role in defining boundaries for the test statistic and judging its statistical significance. We delved into the formulas and steps for calculating critical values for various scenarios like Z-tests, T-tests, and Chi-Square tests. We examined different types of critical values and observed their application in practical examples.
PCB stands for printed circuit board. You will find PCBs in pretty much all electronic devices. It is usually green/blue in color. The PCB is a circuit in a board that permanently holds all the components of a circuit. It is the main part of an electronic device. This board controls and regulates the function of the whole device. A circuit may work perfectly in a breadboard. But breadboard circuits are not suitable for use. It will only be eligible to be used in a ready-made product if implemented in a PCB. This is why PCB designing should be done with utmost care.
It takes a lot of knowledge and expertise to manufacture good-quality PCBs. PCBway is a trusted PCB manufacturer. While their head office is located in China, they ship PCBs worldwide. The following image shows the home page of PCBway.
PCBway Fabrication House is the best PCB manufacturer for professionals and hobbyists. There you can not only print your PCBs. You can get consultancy regarding the whole manufacturing process. Together with a top-notch design and an expert manufacturer, you can produce a high-quality and durable PCB. The following pictures shows how the order page of PCBway looks.
PCB design should be an accurate process. It involves several critical steps. Different challenges may arise in each step of this process. It is important to detect and solve these problems at the early stage of manufacturing the product. Otherwise, we cannot guarantee a reliable electronic gadget. This article discusses the common problems faced in PCB designing and practical solutions to solve these.
This problem often occurs when the designer is a newbie. The wrong placement of components causes some problems. Due to this mistake, the size of the PCB increases unnecessarily. That costs unnecessary money. Soldering becomes difficult if the components are haphazard. Electromagnetic interference (EMI) may also occur and overall signal integrity may also be affected. Misplaced components also result in the following problems:
We should place components in a PCB wisely. The components should be arranged in such a way that the traces will be as short as possible. The components that are supposed to be directly connected, must be placed close to each other. The following image shows a decent arrangement of components in a PCB.
Making 90-degree traces is a big no-no for pro-level PCB designs. The sharp edge created by this type of tracing creates extra stress on the traces. These traces are more likely to crack or break. The life span of such badly designed PCBs is less than usual. The corners of a right-angled PCB have higher electric field density than standard ones.
Right-angle PCB traces affect signal integrity. The effect of a PCB trace's right-angled corner is the same as that of a transmission line coupled to a capacitive load. This is called parasitic capacitance. As a result, the transmission line signal's rising time is slowed down.
Designers should always avoid right-angled traces. In PCB designing software like Proteus, there are functions for making curved traces. We must use those to make our traces curved and not susceptible to cracks after long-term use. The following image shows the difference between a bad and a good trace.
PCB designers must ensure signal integrity. Signal transmitting across the PCB should not distort. Poor signal Integrity is caused mostly by wrongly designed traces, crosstalk, impedance mismatch etc. Signal distortion causes transmission errors.
Modern PCB designing platforms have many important tools that help PCB designers maintain good signal integrity. By knowing how to use them properly, you can avoid errors like uncontrolled line impedances, propagation delays and signal attenuation.
The following figure shows a relationship between coupling traces and SI(signal integrity parameters.)
EMI causes noise and signal interference. Noise degrades the performance of a PCB. EMI increases with frequency. This may cause many problems in high-frequency circuits and designs where components are congested.
EMI can be handled with a combination of design strategies. One method is to use ground planes. We have to place ground planes in such a way that they absorb and redirect electromagnetic emissions. It is also possible to reduce EMI by reducing the current loop area.
It is also important to Shield critical components and traces. Additionally, careful routing of high-frequency traces away from sensitive analog signals can minimize interference.
Heat is generated in PCBs in several ways. Some prime sources of heat generation in PCBs are the active devices or chips that generate heat. Another source is created when an RF power is applied to the circuit. In the case of a double-layered PCB, the copper has extremely high thermal conductivity. On the other hand, the substrate is a thermal insulator that has a very low conductivity. A good-quality PCB must Have a high heat flow. There must be sufficient heat sinks around active components. It is important for keeping the circuit cooler by more efficient heat transfer from the heat source to the heat sink.
The following picture shows some EMI shielding films.
Structurally a PCB can be perfectly alright. Still, it will be useless if the power supply is not adequate. Power distribution should be according to the requirements of each and every component. There may be different voltage and current requirements for different components of a PCB. For example, a PCB may consist of a microcontroller that operates at 5V, but there may be a motor driver that operates at a different voltage level. So, different amounts of voltage and current must be supplied to different parts of the circuit. It is necessary to design a reliable power distribution network. There are PDN analyzers that can detect anomalies in the PDN.
If you know how to use a PCB designing software, you already know this term. DRC stands for Design rule check. DRC error occurs when you do not maintain the minimum trace-to-trace distance defined by the software. For example, the minimum spacing between two traces of a 2-layered PCB is 6 mils (mil=1/1000 inch). If any two traces of your PCB layout are closer than this, the software will show a DRC error. The same error messages will appear also when-
Traces are overlapped with each other
The power plane and the GND plane touch one another
The minimum standard distance between a trace and an adjacent via is not maintained.
We should never ignore DRC errors. If we print a PCB without solving DRC errors, chances are high that it will blow away after powering up.
There should be no DRC error in the PCB layout. Each DRC error detected by the software should be corrected before printing the PCB. You have to edit your design to meet the requirements of the software. You may need to adjust the sizes of the traces and vias to comply with the rules. The following image shows the DRC tool of a PCB designing software.
You may design a flawless PCB, but manufacturing errors can still occur. You can see many short circuits and broken traces. Sometimes it becomes also impossible to read the texts written on the silkscreen. The following picture shows a broken PCB trace.
A good collaboration with the manufacturer may help you solve these manufacturing defects. The customer should provide clear and detailed documentation. It is essential to include the fabrication drawings and assembly instructions. It is helpful to perform a manufacturability (DFM) check to identify potential issues before production. Automated optical inspection (AOI) and in-circuit testing (ICT) during manufacturing can also find defects early.
A well-designed ground system is required for Modern high-speed electronics. PCBS need to operate at their best performance. If the PCB ground is not properly implemented, the circuit board may experience many different problems with noise and electromagnetic interference (EMI).
Sometimes the ground net in a PCB design can appear confusing. Yes, there are many connections, but since most designs will have one or more ground planes in their layer stacked up, you just add a via to the ground, and the work is done. Right? Theoretically, that is correct, practically, there are lots more that need to go into your PCB grounding technique to build a good power delivery network.
A single, continuous ground plane is typically the best approach for minimizing ground loops. The following image shows a PCB layout with properly designed POWER and GND planes.
Soldering issues such as cold joints, bridging, and insufficient solder can lead to unreliable connections and component failures.
The following image shows an example of an accidental short circuit.
Designing with manufacturability in mind can help prevent soldering issues. Ensuring appropriate pad sizes and clearances for components can facilitate proper soldering. Specifying the correct solder mask and paste layers in the design files is also important. Automated soldering processes, such as reflow soldering, should be used whenever possible to ensure consistent and reliable solder joints. Inspecting solder joints using AOI and X-ray inspection can catch defects before final assembly.
Inadequate clearance between traces, pads, and components can lead to shorts and an increased risk of crosstalk, affecting the PCB's reliability and performance.
Following the clearance guidelines provided by the PCB manufacturer is essential. Maintaining adequate spacing between traces and pads can prevent shorts and crosstalk. Using the DRC tool in the CAD software to check for clearance violations can help identify and rectify issues before fabrication. Also, consider the voltage levels and environmental factors like humidity and temperature. It can guide appropriate clearance settings.
Designing multilayer PCBs introduces complexity, such as ensuring proper layer stack-up, signal routing, and maintaining signal integrity across layers.
Planning the layer stack-up early in the design phase is critical for multilayer PCBs. Assigning specific layers for power, ground, and signal routing can help manage complexity. Using blind and buried vias can optimize space and routing options. Ensuring proper alignment of vias and traces across layers is essential for maintaining signal integrity. Simulation tools can assist in verifying the performance of multilayer designs and identifying potential issues.
Tolerance means the maximum deviation from the design at the time of the manufacturing process. There are always big differences between theory and practice. Your design may be perfect in your software, but you have to consider manufacturing tolerances in practice. If we do not take it into account, our PCB may fail.
It is best to check the manufacturing tolerances of the PCB manufacturing company. The following image represents PCBway’s manufacturing tolerance guideline. To learn more about PCBway's manufacturing tolerance policy, you can click here.
While designing a PCB, it is necessary to take environmental factors into account. You must consider the temperature, humidity, and atmospheric pressure of the environment where it is likely to be used. For example, a PCB designed for an industrial purpose should be more robust than a PCB of a home appliance. PCBs are likely to damage early if they are not compatible with their surroundings.
First of all, we have to select the components according to their specifications and operating temperatures. For industrial products, all components and the board itself should be industrial-graded. For better heat dissipation, use thermal vias, heat sinks etc.
Following is a chart of high TG materials used by PCBway.
PCB manufacturing is a process that needs a lot of scrutinization, time and dedication. PCBs are often printed on a trial-and-error basis. It should be our goal to save as much money and time as possible while not compromising the PCB quality. We should correct all DRC errors before printing a PCB. We need to provide a proper thermal management system, and proper shielding for removing signal interference. We should not tend to make the design on an ad-hoc basis. Rather, we must always try to make a durable PCB. We should choose a dependable manufacturer.
It can become challenging to make award ceremonies rewarding and fun, especially if you are in a technical business related to engineering. However, it is essential to have such celebratory moments to recognize your colleagues, coworkers, and team.
Moreover, such ceremonies are a chance to celebrate yourself and those around you, and to achieve that; the ceremony must be executed flawlessly. It must create an atmosphere of festivity and jubilation so that your team feels the spirit and recognition for their work and is, therefore, motivated to continue achieving milestones for the business.
Hosts bring life to the ceremony or party and must be chosen carefully for the occasion. We recommend choosing a host with some connection to your business’s industry. Still, it is also essential that they have an uplifting and witty personality capable of withholding engaging banter to keep the mood lighthearted and joyous and, therefore, automatically keep the team attentive throughout the ceremony.
On celebratory occasions, it is crucial to avoid choosing a host who tends to give boring and long speeches- those can be saved for conferences and in-office occasions.
Your management could also invite popular guest speakers or charming celebrities, which may increase attendance at the event and spark excitement among those present.
Choosing an exciting and alluring venue is always a must to create a celebratory atmosphere. For example, if you were to host the award ceremony in an auditorium, it would not spark any excitement among the employees, and they would perceive it as just another boring office event.
However, hosting the event in an exciting environment, such as near or on the beach, would make the atmosphere more festive and relaxed for employees, allowing it to feel like a celebratory moment.
It is vital to make your team and workers feel like they are part of a family, and the best way to do this is by proving that you know them personally and not just by their professional achievements.
Therefore, sometimes it is essential to gift them Custom Trophies and awards engraved with fun, quirky recognitions, such as ‘The detail-detective,’ ‘The Innovator,’ or ‘The Early Bird.’ Such recognitions give an insight into their personality and portray their achievements.
Furthermore, since teamwork in the workplace is essential for a business’s success, it is also essential to give out customized awards to teams, such as ‘The Financial Gurus’ or ‘The Creative Collective.’
You could take these customized awards and customize the entire trophy designs. For example, the trophy design could have wings to signify that the employee has reached new heights or grown. Similarly, the award could be key-shaped to symbolize that they have unlocked new doors in their career or have finally cracked the code to unlock their highest potential.
Such awards that resonate personally with the teams will no doubt stimulate rounds of laughter and prompt light-hearted conversations.
It is vital to keep the ceremony short and allocate a good proportion of time for colleagues to mingle and interact with one another. This opportunity may lead to many new bonds within the workspace, enhancing productivity and motivation levels.
One way to achieve this is by hosting live entertainment acts and creating a laid-back atmosphere. The most popular is to invite a band or singers who will create a memorable moment for your employees and even guide some of them onto the dance floor. Furthermore, your management could host well-known comedians who would make your employees erupt into rounds of laughter.
Moreover, you could set up an area for live painters who could perfectly capture the moment's essence. This painting could be set up in your office building and forever take your employees back to relive their memorable moments.
After a series of accomplishments, your employees more than deserve a weekend getaway to relax and unwind so they can come back to work more refreshed and ready than ever.
Therefore, if your budget allows for it, we recommend hosting a gala at a weekend getaway, followed by a range of leisurely activities to help your employees unwind and escape from the stress of deadlines and daily tasks.
Such leisurely activities could include golf, spa sessions, massages, swimming pools, and even a tour around whichever city or town you visit.
A weekend getaway will surely provide your employees with an experience of a lifetime that they will always cherish. Furthermore, it will also be an opportunity for your employees to get to know one another better, satisfying their social needs , especially when they return to the workplace.
Undoubtedly, a business’s most valuable asset is its employees. Therefore, it is essential to keep them motivated if you want to ensure higher returns for your business. Recognizing their work and managing your workforce more efficiently and effectively will save time and increase productivity for your employees and the business.
Hi readers, I hope you are all well. In this post, we can talk about the vectors briefly. Physical quantities can be defined through magnitudes but some physical quantities can be defined through both magnitudes and direction, these types of quantities defined through both magnitude and directional properties are known as vectors, and the quantities that can be explained through magnitude, not with direction are known as scalars. some vectors are force, velocity, displacement, and acceleration.
Vectors can explain the direction and magnitude of the quantity but they can't provide their position. It is an essential tool of mathematics that can be used in physics for knowing the direction or magnitude. It cant be used in the 18th century but can be used in the modern era of the 19th century late and it can be presented by the scientists Oliver Heavisde and Willard Gibbs they can develop the vector analysis which can be used to express the modern laws of electromagnetism which can be presented by the James clerk maxwell.
In different fields of physics like mechanics, and mathematics or in engineering, vectors can be used to explain the different qnatites in mathematical form with magnitude and direction. Now we can start are brief discussion about vectors their definition, mathematical representation, operations, types, and their application in different fields of physics.
Definition, mathematical representation, operations, and their application in different fields of physics in detail are given below:
It can be defined as:
"The quantity which can described through both direction and magnitude is known as vector quantity or vectors."
In mechanics or geometry firstly term vector can be used but in some articles, the word or term vector can also be used for tuples. because mostly in mechanics which is the branches of physics vector quantities are used for magnitude and direction. some examples of vector quantities are given there:
Representation of vectors in detail is given there:
The vector quantity can be represented through the bold letter. For instance, the normal letter is v but for the vector, it can be written in bold form like v. Another example is the normal letter is written as a,b, and c but if it can be used for vector it can be written as a, b and c.
The vectors can also be represented by putting the arrowhead over the letter. Some examples are given there:
Vectors can also be represented in graphical form through an arrow. In the graphical representation, the arrow points to the direction of the vector, and the length of an arrow can represent the magnitude of the vectors.
For instance, a vector can cover the displacement from point A to point B then the arrow length represents the magnitude and the arrow point represents the direction from point A to point B.
The components of the vectors can also be expressed in the coordinate axis. Their components can be expressed in the two-dimensional cartesian coordinate system or the three-dimensional cartesian coordinate system. In a dimensional coordinate system, there are two axes x and y so the vector A in the two-dimensional system can be broken and written as AX on the x component and Ay in the y component.
But in the three-dimensional cartesian coordinate system x, y, and z are the three components and the vector A is written as Ax on the x component, Ayon the y component, and Az in the z component.
In the two-dimensional coordinate system, the vector A is mathematically written as;
A = AXi + Ayj
In the three-dimensional coordinate system, the vector A is mathematically written as;
A = Axi + Ayj + Azk
There i, j, and k are the unit vectors in the direction of vector components x, y, and z.
Various types of vectors are used in physics or mechanics some types of vectors with their details are given there:
Null or zero vector
Equal vector
Position vector
Negative of a vector
Like or unlike vector
Unit vector
Displacement vector
Coplanar vector
Co initial vector
Collinear vector
Their description is given there:
A null vector is also termed a zero vector. In vector is referred to as a zero or null vector when its magnitude is zero and there is no specific direction where the arrow points. In null or zero vector the length of magnitude is also zero. Their starting and ending points are the same. For instance, the vector OQ has the line segment and the starting point is O and the end is at the same point Q so their magnitude is 0.
Two different vectors are termed equal vectors if they have equal magnitude and also their direction. Rather they may have different starting points but their magnitude and the direction are same. For instance, if the vector magnitude is equal to the magnitude of vector b and their arrow is pointed in the same direction.
The vector can represent the origin and the position at any point related to the initial or origin point. Position vector can describe the direction of a vector from the main origin point to the endpoint.
The vector is termed as negative a vector if the vector which is given has the same magnitude and direction but at a point, any vector can change its direction means they have the same magnitude but have opposite directions so this vector which has the same magnitude but the opposite direction is known as negative of the vector. For example, vector A and vector B have the same magnitude but they have opposite directions and are written as A = -B
The vectors are termed as like vectors if they have the same direction but if the vectors do have not the same direction then they are called unlike vectors. For example, if the vector AB has the same direction then they are like vectors but if vector AB does have not the same direction then they are unlike vectors.
The vectors are termed unit vectors if they have only one magnitude. A unit vector is slo referred as the direction vector. The formulas that are used for unit vectors are:
V = VV
V represents a unit vector, V represents a vector, and V represents the magnitude of a vector.
If the quantity can be a displacement from point A to point B then the displacement between the AB is termed a displacement vector. For example, if a quantity is moved from point A and reaches point B then the distance between these points is termed a displacement vector it can also be termed the position vector.
The vectors that are placed in three-dimensional space with the same plane then it can be termed as the coplanar vectors. All vectors are parallel to each other in the same plane.
Two vectors are termed collinear if they are parallel to each other and not dependent upon the magnitude or direction. Collinear vectors are also termed parallel vectors. For example, if the vector A and vector B are opposite in direction and both have different magnitudes but they are parallel to each other then it can be called a collinear or parallel vector.
When two or many vectors have the same origin or initial point on the same plane then it can termed as the co-initial vectors. For example, the vectors A, B and C can originate from the same point with the same plane then it can be called as co initial vector.
Different mathematical operations with vectors are given below:
Addition of vector
Subtraction of vector
Dot product ( Scalar product)
Scalar multiplication
Cross product (Vector product)
For addition and subtraction of the vector, we can use the tip-to-tail rule in which the tail of the second vector can be placed on the tip of the first vector and the first vector tail is placed on the tip of the second vector.
For the addition of the vector we can use the tip-to-tail rule their mathematical representation is given below;
If we have the vector A their component on x or y is AX and Ay and the other vector B then their component on x and y is Bx and By and their sum is represented by C then they can be added as:
C = A+ B
Then,
A+ B = (Ax + Bx, Ay+ By)
Like the addition of a vector, the subtraction of a vector can also follow the head-to-tail rule. so the component method and mathematically expressed equation for the subtraction of a vector are given below:
If we have the vector A their component on x or y is AX and Ay and the other vector B then their component on x and y is Bx and By and their subtraction is represented by C then they can be subtracted as:
C = A– B
Then,
A– B = (Ax – Bx, Ay– By)
The cross product is also known as the vector product. In the vector product, the vector A and the vector B are perpendicular to each other then their product in mathematical form can be written as:
A B = ( AyBz– AzBy) i + (AzBx– AxBz) j + ( AxBy – AyBx) k
The magnitude of the vector or cross product is,
A B = ABsinθ
In this, the sinθ is the angle between the vector A and the vector B.
In physics cross products are used to understand the rotational and the circular motion of an object also they can be used to calculate precise calculations like torque.
When we multiply the vector with the scalar which has only magnitude and these quantities do have not a direction but if the scalar is negative then it means that their direction can be reversed. Their mathematical form is given there,
If we have the vector A and their component on x and y axis is Ax or Ay and the scalar is K then they can be written as:
KA = ( KAX , KAy)
The vector A and the vector B, their dot product is the scalar quantity so that's why they are also termed as the scalar product. Their mathematical expression is given below:
A B = AxBx + AyBy + AzBz
Their product can express their magnitude, not their direction. So the magnitude of the scalar or the dot product is,
A B = AB cosθ
cosθ is the angle between the vector A and the vector B for expressing the magnitude.
We can use the scalar and dot product in physics to calculate or determine the angle between the two vectors if one vector is projecting to another vector.
Vector quantities can play a very essential role in the physics for calculating numerous calculations. Vectors can be used in various fields of physics because by busying the vector mathematical operations we can do very precise and accurate calculations. Now we can discuss some applications of vector quantites in physics. The field of physics in which the vectors are used is given there:
In Mechanics
Newton’s law of motion
Electromagnetism
Maxwell’s equation
Bernoullis equation
Fluid dynamics
Quantum mechanics
Velocity field
Schrodinger equation
Their detail is given there:
To describe the motion, forces, and displacement in mechanics we can use the vector quantities. Some vector quantities explanation is given there:
Acceleration: acceleration is the time rate change of velocity, and the vector can express the change of direction and magnitude along with experiencing the quick velocity changing
Displacement: the position which can be changed by the moving object can be expressed by vectors. And the direction with magnitude of these objects can be expressed through vectors.
Velocity: velocity is the rate of change of displacement, their magnitude, direction, and speed can be represented by the vectors.
The equation of motion which can be used in the physics of the vector quantities can also involved in them. For example, the formula for the uniform acceleration is
v = v0 + at
There v expresses the velocity vector, v0 indicates the initial velocity, a is for acceleration and t expresses the time in which the velocity can be changed.
In Newton, the law of motion all formulas of Newton's law can be expressed by using vectors. The law of motion with its formula which indicates the magnitude and direction is given there:
The first law of motion: the object always remains at rest unless the external force can act upon it. And through these external forces, the object can be moved.
The second law of motion: the force that can act on the moving object is equal to the mass of an object and also equal to the acceleration. It can be written as F = ma where F is the vector quantity force, m indicates mass and a indicates the acceleration.
The third law of motion: is that every reaction has an equal and opposite reaction it can indicate the magnitude and direction of every action and reaction.
The dynamics of the object, its direction, and magnitude can be expressed through vectors in these equations of motion.
The vector spaces that can be involved in quantum mechanics are also termed as the Hilbert spaces. The state of particles is represented by the vectors which are used in quantum mechanics and it is also called wavefunction and state vectors. These vectors can also indicate the direction and magnitude of the particles in quantum mechanics.
In the field of fluid dynamics, the vectors represent the flow of fluid along with their direction and they can also represent the properties of the fluid which can flow. Vectors that can be used in fluid dynamics can also represent their magnitude with properties.
Schrodinger equation is the time-independent equation. This equation can be described and indicate the quantum state of any physical system independent of time. The vectors which can be used in these equations are written there:
iћ∂∂tψ (r, t) = H ψ ( r, t)
there, ћ expressed the Planck constant, ψ (r,t)expressed the wavefunctions, and the H expressed the Hamilton operator.
In electromagnetism, the vector quantities can expressed and describe the electrical field and the magnetic field. These vectors can also explain the relationship and the interactions of current and the charges in both electrical and magnetic fields.
The vectors that can be used to express the electric field are;
E = Fq
Where E is the vector that can indicate the electrical field in the space at any point, F is the force that can be experienced by the unit positive charges in the electric field and the q expresses the charges which are present in the electrical field.
The vector that expresses the magnetic field and the Lorentz force law is written there:
F = q (v B)
The magnetic field can be expressed through vector B and this vector B can describe the magnetic forces which are present in the magnetic field. The force F experienced by the charges can be expressed by the vector q with the same velocity which can indicated by v.
Bernoulli's equation can be derived and dependent on the vector. The vectors which are related in the Bernoullis equation are height, pressure, and velocity in the flowing fluid. The Bernoullis equation formula in which vectors are used is given there:
P + 12 ρv2 + ρgh = constant
There P represents the pressure, ρ represents the fluid density, h represents the height of the fluid according to the reference point, and v represents the fluid velocity.
The electric and magnetic field construction and how these fields can change their current and charges are described through the Maxwell equation. The vectors which can represent these fields current and charges are given there:
The vectors that can represent the current and charges in the field are given there:
⛛ E = -∂B∂t
The vectors that represent the Gauss law formula for electricity are given there:
⛛ E = pe0
The vectors that can represent the amperes law formula are given there:
⛛ B = μ0J + μ0ε0∂E∂t
The vectors that can represent the formula of Gauss law for magnetism are given there:
⛛ B = 0
Fluid particles that are present in space at different points can be explained and described through vectors like v (r,t), and in this v is the vector for velocity, r for the position of the particle, and t is for a time in which the velocity of the particles can be changed and dependent upon the time.
In the modern era of science, vectors can be used in many advanced topics in physics some topics with descriptions are given there:
Tensors
Gradient
curl
Vector calculus
Divergence
The complex physical quantities are also represented and described by the vectors and these vectors which describe complex quantities are generalized, known as tensors. In the continuum mechanics or in the theory of relativity tensors mostly tensors are used.
The vector field gradient is represented by ⛛ and the scalar field gradient is Φ. When the rate of Φ increased then the magnitude rate also increased and this gradient mathematically can be written as:
⛛Φ = (∂Φ∂x , ∂Φ∂y, ∂Φ∂z )
The ability and the tendency of the field that can be moved or rotated in a point is termed the curl of the vector field A. Mathematically it can written as:
⛛ A = (∂Az∂y – ∂Ay∂z , ∂Ax∂z – ∂Az∂x, ∂Ay∂x – ∂Ax∂y)
The concept of vectors in different fields of physics and mathematics can be extended through vector calculus because we can do different mathematical operations with vectors like curl, gradient, and divergence.
By using the divergence formula we can measure the rate of flow of vectors in the vector field of A. The mathematical expression and the formula of divergence are given below:
⛛ A = ( ∂Ax∂x + ∂Ay∂y + ∂Az∂z )
In physics, vectors are used as an essential tool because they can provide comprehensive information about the quantity and can also analyze and provide the description of the magnitude and the direction of the quantity in a very efficient way. Mostly it can be used in the field of physics like fluid dynamics, mechanics, electromagnetism, quantum mechanics, and in mathematical operations to derive or express the formula. Vectors can play a very essential role in physics or mathematics. Vectors can become the backbone of calculations in physics or in mathematics because they can help in doing very crucial calculations.
Hi friends, I hope you are all well. Today we can talk about the dimensions of all physical quantities which include the base quantities and the derived quantities. Dimensions of physical quantities are the fundamental part that helps us understand the physical and natural properties of any physical quantity. In modern science and technology, in engineering, and in different fields of physics where physical quantities units are used, the dimensions of these units help to derive a new formula and are also used in derivations. Dimensions of physical quantities also convey the detail of the types of physical quantity.
Dimension of physical quantities also helps to check the correctness of the equation and the formula that we can derive or use to solve the problems. Dimension of the physical quantities systematically expressed the physical quantities units. For analysis of the formula and the physical quantities, we can also use the dimensions of these quantities. Now we can start our detailed discussion about the dimensions of physical quantities which include the base quantities dimensions, derived quantities dimensions, and the application and analysis of the physical quantities in physics.
Physical quantities are defined as:
“The quantities which can be measured and quantified by measurements are termed as physical quantities.” For example mass, length weight, electric current, and many other various quantities. These physical quantities can be expressed in the algebraic form and when measured we can use different units for their measurements. To know the correctness of these units we can use dimension analysis.
For example for the quantity length we can use the unit meter and their symbol is m so by using the dimension of the meter we can verify them.
There are two main types of physical quantities units which are given below:
Base quantities
Derived quantities
The units that can be used to describe these physical quantities are known as base quantity units and the derived quantity units.
Dimension of physical quantities is defined as:
"The fundamental quantities which can be expressed in the form of raised power to describe the physical quantities are termed as dimensions of physical quantities."
The unit and the dimension of the physical quantity are written in the square brackets. For example, the unit of length is m and the symbol is m but their dimension is L.
The dimensions of base physical quantities and the derived physical quantities in detail are given below:
There are seven basic base physical quantities units which are also known as the building blocks units from which other units are derived. The dimensions of these base physical quantities with their units and definitions in detail are given there:
Base quantity |
Symbol |
Unit |
SI unit symbol |
Description |
Dimension |
Length |
L |
meter |
m |
The unit meter is used for the length and the length describes the distance between the two objects and also describes the height and the width of an object. |
L |
Mass |
m |
kilogram |
kg |
The quantity mass is used to measure the amount of matter of the objects. The measurements that can be measured are expressed in the unit kilogram. |
M |
Time |
t |
second |
s |
The physical quantity of time is used to measure the duration of the process and events like the duration of waves and oscillations. These measurements can be expressed with the unit second. |
T |
Electric current |
I |
Ampere |
A |
The flow of the electric charge in the electric circuits can be measured by using the term electric current and this measurement is expressed with the unit ampere. |
I |
Amount of substance |
N |
mole |
mol |
To measure the number of atoms, molecules, and other entities in the compound or matter we can use the term amount of substance the unit which is used to express the amount of substance is mole. |
N |
Luminous intensity |
J |
candela |
cd |
The power of light can be measured in terms of luminous intensity and these measurements are expressed with the unit candela. |
J |
Thermodynamic temperature |
Ө |
kelvin |
k |
The temperature in the form of heat which may be endothermic or exothermic released from the thermodynamic system is calculated or, measured and these measurements are expressed with the unit kelvin. |
Ө |
Derived units are derived from the seven basic base quantities units such as area, volume, power, and many others. Some of these derived units with their symbols and dimensions are given below:
Derived Quantities |
Symbols |
Formula and relation with other quantities |
Dimensions |
Dimensions related to the formula |
SI units |
SI units symbol |
Force |
F |
mass acceleration |
m= M acceleration = LT-2 |
MLT-2 |
Newton |
N |
Area |
A |
length breadth |
L= L breadth=L |
M0L2T0 |
Meter square |
m2 |
Density |
ρ |
mass volume |
m= M volume = L3 |
ML-3T0 |
Kilogram cubic per meter |
kgm-3 |
stress |
σ |
Forcearea |
F= MLT-2 A= L2 |
ML-1T-2 |
Newton per meter square |
Nm-2 |
Surface energy |
σs |
energyarea |
energy= ML2T-2 Area = L2 |
ML0T-2 |
Candela meter per second square |
Jm-2 |
Impulse |
J, imp |
force time |
force= MLT-2 Time= T |
MLT-1 |
Newton per second |
Ns-1 |
Strain |
ε |
change in dimensionorginal dimension |
Dimensionless |
—- |
—--- |
—--- |
Hubble constant |
H0 |
velocity of recession distance |
velocity= LT-1 distance= L |
M0L0T-1 |
Per second |
s-1 |
Coefficient of elasticity |
δ |
strssstrain |
stress= ML-1T-2 strain=1 |
ML-1T-2 |
Newton per meter square |
Nm-2 |
Volume |
V |
Length height breadth |
length= L height= L breadth= L |
M0L3T0 |
Cubic meter |
m3 |
Thrust |
N |
Force |
force = MLT-2 |
MLT-2 |
Newton per meter square or Pascal |
Nm-2or Pa |
Linear acceleration |
a |
velocitytime |
velocity= LT-1 time= T |
M0LT-2 |
Meter per second square |
ms-2 |
Work |
W |
Force distance |
force= MLT-2 distance= L |
ML2T-2 |
joule |
J |
Specific volume |
v |
volume/mass |
volume= L3 mass= M |
M-1L3T0 |
Cubic meter per kilogram |
m3kg-1 |
Specific gravity |
s.g |
density of material density of water |
—--- |
Dimenonless |
||
Tension |
T |
Force |
force= MLT-2 |
MLT-2 |
Newton meter per square and Pascal |
Nm-2 or Pa |
Surface tension |
Y |
force/ length |
Force= ML0T-2 length = L |
ML0T-2 |
Newton per meter |
Nm-1 |
Radius of gyration |
k |
distance |
distance= L |
L |
meter |
m |
Angular velocity |
ω |
angle/ time |
angle= L time= T |
LT-1 |
Radian per second |
rs-1 |
Momentum |
p |
Mass velocity |
mass= M velocity= LT-1 |
MLT-1 |
Kilogram meter per second |
Kg ms-1 |
Rate flow |
Q |
volume/ time |
Volume =L3 Time= T |
M0L3 T-1 |
Cubic meter per second |
m3s-1 |
Frequency |
λ |
No vibrations/ time |
Time= T |
M0L0T-1 |
meter |
m |
Heat |
Q |
energy |
energy= M1L2T-2 |
M1L2T-2 |
joule |
J |
Buoyant force |
force |
force= M1L1T-2 |
M1L1T-2 |
Newton |
N |
|
Plancks constant |
h |
energy/ frequency |
energy= M1L2 frequency= T |
M1 L2T-1 |
Joule second |
Js |
To derive the formula and the relationship between the different numerous physical quantities we can use a method or a technique known as dimensional analysis. Dimensional analysis can also be used to identify and determine the correctness of the equation and the formula and also change the units from one system to another systems. Dimensional analysis is based on the following points which are given there:
Principles of dimensional analysis
Steps involved in the dimensional analysis
Details of these points are given below:
Principles of dimensional analysis include:
Derivation of formulas
Dimensional homogeneity
Conversion of units
We can convert the units with the help of dimensional analysis in some other units. But the units that can we convert are expressed in terms of the base units. Some examples of conversion of the units are given there:
The unit square per meter which we we used to express the velocity is converted into some other base unit like km/h through dimension analysis.
If we can measure the distance in the units meters then we can convert it into the other base units like meters per hour.
The formula or the equation that we may derive and use for solving the problems has equal and the same similar dimensions on both sides which proves that the equation is correct. if the dimensions on both sides are not the same it proves that the equation is not correct. So dimension analysis also helps to identify or determine whether the equation or formula is correct or not. Some examples are given there:
For example, the formula or area is A = Length breadth. The dimension of the area is L2 and on the other side, the dimension of length is L and breadth dimension is L so on both sides dimensions are the same proves that the formula is correct.
Another example is a force equal to F= ma and the dimension of force is MLT-2 and on the other side the dimension of m is M and the dimension of a is LT-2. so it proves that the formula is correct because the dimensions on both sides are the same.
We can derive the different authentic formulas by using numerous physical quantities with the help of dimensional analysis. Dimensional analysis helps to understand the property of quantity which we can use to derive the formula. Some examples are given there:
We can derive the period of the pendulum by using the dimension of time and distance.
We can also derive the formula for the force by using different physical quantities like mass, acceleration, and time and we can also check whether the equation is correct or not with the help of dimensional analysis.
The steps which are involved in the dimensional analysis are given there:
Set up the equation
Identify the base quantity
Write the dimensional formula
Solve for the unknowns
For dimensional analysis, it is essential to check or confirm that the equation or formula we are using has the same dimensions on both sides and they are equal to each other. Some examples are given below:
The formula which is used for work is W= force distance the dimension for work is the same and equal to the dimension of force and the distance.
Another example is the formula of force is F= ma and the dimension of force is equal to the dimension of mass and acceleration so that's why the dimension analysis helps to identify the equation is correct.
In dimensional analysis, it is essential to identify the physical quantities that we are using to derive the formula or to solve the problems. Some examples are given below:
Identify the mass, length, volume, and other physical quantities and then use their dimension for dimensional analysis.
To identify the unknown physical quantities we can use the dimensions. And by the help of dimensions we can identify and determine the right physical quantity. Some examples are given below:
We can identify the gravitational constant G by using the dimensions and also determine their dimensions with the help of dimensional analysis.
For another example, we can determine the pendulum period by using the different dimensional equations, and after determining we can also identify their dimension with the help of dimensional analysis.
In the formula that we derived or used to solve the problems, it is essential to write the dimension of all quantities that can be used in the formula. Some examples are given below:
The dimensional formula of the area is L2.
The dimensional formula of power is Ml2T-3.
In the era of modern sciences and technology, engineers and other scientists measure complex quantities. Dimensional analysis helps to measure these complex quantities and also helps to derive new complex formulas and check the correctness of the derived formula by using the dimensions. Now in the modern era of physics and engineering, dimensional analysis can be used in many different fields. Some applications of dimensional analysis are given below:
Biophysics
Relativity
Plancks constant
Engineering
In biophysics, we can study and understand how the substance in the body or biological tissues can be moved or flow. Because diffusion mechanism can be used and we can measure and express them in some units so that's why we can use dimension analysis in it also
Some examples are given below:
The coefficient of diffusion can be described as the flow of material from one to another place and its dimensional formula is L2T-1.
To understand the relationship between mass and energy we can use the dimensional analysis method. Because they help to determine or identify the quantities that can be used in their dimensions. some examples are given there:
The formula which can be derived by Einstein is based on the relationship between the mass and the speed of light the formula is F=mc2 and the dimensions of m is M and the dimension of c2 is LT-1.
In quantum mechanics, we can understand the energy levels in the atoms and with the help of dimensional analysis, we can check the correctness of equations and detail understand the the energy levels in the atoms. Examples of dimensional analysis in quantum mechanics are given there:
For dimensional analysis and understanding the energy level we can use the formula of Plancks constant which is E=hv in this h is for energy level v is for frequency and their dimension is ML2T-1.
In engineering dimensional analysis is used for stress and strain because they can design the new materials and check their capacity to bear the load. So that's why dimensional analysis is used to determine the dimensions of stress and strain. Their example is given there:
In engineering, the stress on the object at force per unit area and strain changes the length and their dimensional analysis ML-1T-2.
Dimensional analysis plays a vital role in the modern era of science and modern technology because in this era we can measure complex physical quantities and checking the correctness of dimensional analysis is essential. It can also help to derive the new formula and understand the natural properties of the quantities. By using dimensional analysis scientists and engineers can achieve highly precise and accurate measurements of complex physical quantities.
With time when the physical quantities become more complex and developed then the dimensional analysis techniques can also be developed and adopt different changes in them. The steps that can included in the dimensional analysis can be developed and more steps are included for the analysis of complex physical quantities which helps the scientist to understand the quantities and give the most precise and accurate result according to complexity.
Hi friends, I hope you are all well. In this article, we can discuss the uncertainty in the measurements which can be measured. In the era of modern science and technology or modern physics, scientists can measure complex quantities and these measurements are not precise and accurate somehow doubt is present in these measurements, these doubts are suspicious known as uncertainty in the measurements. In physics or other fields of technology and engineering measurement is essential to measure or understand the quantity of a material or an object. Because every measurement is correct there are always some doubts or doubtful digits and they are called uncertainty in the measurements.
Now in this article, we can explore the history, definition, quantifying methods, and different techniques that can be used to minimize uncertainty and also explore their applications and significance in different fields of engineering and physics.
All substantial National measurement institutes can research the uncertainty in the measurements and give detailed documents about the measurement which is known as GUM and stands for the “Guide of Uncertainty in the Measurements.” this document gives the details about the uncertainties in the measurements. In metrology measurements when we can take measurements of the object many times it is confirmed that somehow measurements are not correct and precise. Or the doubtful measurements are termed uncertainty. All the measurements are not always correct because the measurement results depend upon the instrument's efficiency and the skilled person who can take measurements That is why uncertainty comes into measurements due to many various factors that may depend upon the environmental factors also.
Measurements are essential to determine the quantity of the physical quantity or any objects. Measurements also play a vital role at the economic level. The quality of laboratories can also determined by their calibration results because the exact measurements help to understand the quantity of the object. So that's why the ASME which stands for the American Society of Mechanical Engineers can present different standards for the uncertainty in the measurements. According to their standards, the measurements are done and engineers and scientists in laboratories can also measure the quality of different measurements.
Uncertainty in the measurements is defined as:
"The measurement which can we measured have lack of certainty and they have a great difference between the true value and the measurement value which be measured."
In simple words, the measurements we can measure have some doubts and their results do not according to the expectations and lack sureness are termed as uncertainty in the measurements. Uncertainty in the measurements is common because not every measurement is accurate or precise. For instance, we can measure the length which is about 6.7cm with a meter ruler but the true value range is about 6.62 or 6.75cm so the uncertainty in this measurement is approximately 0.05cm. Another example is if we measure the height of an object and the measurement which we can measure is 5.5m but the true value range is 5.3m or 5.7m so in this measurement the uncertainty is approximately equal to 0.02m.
There are two major types of uncertainty in measurements which are given below:
Type A uncertainty
Type B uncertainty
Type A uncertainty is defined as:
"The uncertainty measurement which can be evaluated through the different methods of statical analysis are known as type A uncertainty measurements."
Generally, in type A uncertainty measurements we can measure or collect different data about the measurement and then observe the series of collected data, and then evaluate the uncertainty which are present in these measurements.
For example, we can take measurements of an object many times or maybe 20 times and then evaluate and observe the results of these measurements and then analyze the uncertainty in these measurements. from empirical data, we can directly identify or determine the type A uncertainty in the measurements. Another example is when we want precise and accurate measurements then we can measure the same measurements many times like if we can measure the length and the measurements range between 5.7cm to 14cm and in between different measurements occur after these measurements we can observe or estimate the average uncertainty.
Type A uncertainty can be measured by repeated measurements of the measuring object and evaluated through statistical methods or techniques. Some statistical methods that are included in the evaluation of type A uncertainty are given below:
Confidence intervals
Arithmetic means
Standard error of the mean
Degree of freedom
Standard deviation
Confidence interval is defined as:
"The true and standard values that can be measured through the measurement of quantity, and confidence interval convey the range of accuracy with a confidence level of the true measured value."
This formula can be used for a normal distribution which has a 95% confidence interval. The formula is given there:
μ z SEM
μ z SEM
There,
z is for the confidence level which can we desire and it is approximately equal to 1.96 for the normal distribution with a confidence level of 95%
Arithmetic means is defined as:
“The set of measured numbers, the all measured numbers are added and dived by the total numbers which are present in the set and the central number which is present in the set also added with the all measured numbers”.
To know an average of all measured measurements we can use the arithmetic formula because, through this statistical technique, we can evaluate the type A uncertainty in the measured sets.
The formula which is used to calculate the average of the measured set is given there:
x = 1n i=1nxi
The standard error of the mean is defined as:
“ the uncertainty which is present in the average that can be calculated from the set of measurements with no of measurements and the standard deviation, these estimate can be conveyed by the standard error of the mean.”
The formulas that are used for calculating the standard error of the mean are given there:
SEM= σN
there,
SEM = standard error of the mean
σ= standard deviation
N = no of measurements
Standard deviation can be defined as:
“The average can be measured from the dispersion of the set of collected measurements.”
Standard deviation can be used to measure the average variance which can be essential for evaluating the Type A uncertainty.
The formula that can be used for the standard deviation or measure the average variances is given there:
σ= 1n-1i=1n(xi-x)2
The degree of freedom can be defined as:
After the calculation of the standard deviation and the average the final numbers can be calculated freely for statistical analysis and the final values can help to understand the type A uncertainty and the degree of freedom.
The formula that is used for the degree of freedom is given there:
v= n - 1
The main causes and sources of the Type A uncertainty in the measurements are given there:
Environmental changes
Human factors
Instrumental fluctuations
The methods that can reduce the uncertainty in the measurements are given below:
Advanced measurement techniques
Repeated numbers of measurements
Control environmental factors
Improving measurements techniques
Type b uncertainty can be defined as:
"the uncertainty which can be evaluated by using different methods except the statistical analysis of measurements. Type B uncertainty can't be evaluated through statistical analysis they can be evaluated through calibration certificates, scientists' judgment, and through the publishers."
Type B uncertainty can be measured differently from Type A uncertainty in the measurements because it is mostly evaluated through the collected information and through the publishers. This type of error is also common the main sources and the causes of these uncertainties can be explained below.
The methods that can be used to evaluate the type B uncertainties in the measurements are given below:
Expert judgment
Manufacturer specification
Theoretical analysis
Reviews of calibration certificates
Reference material data
Chats of collected data information
When we can do measurements but do have not direct measurements and data then the uncertainty measurement is provided to the experts who have experienced and understand the limitations and uncertainties of the measurements. Then the judges understand the uncertainty and then identify the type of uncertainty and try to reduce these uncertainties. Some examples are given below:
The scientists who can do experiments and want to change into theory the council and the judges understand the experiment according to their experience and then allow them.
A well-experienced scientist or metrologist can measure the uncertainty in the measurements through their experience and knowledge.
The type B uncertainty can be evaluated by using the calibration certificate because the calibration certificates convey information and details about the accuracy and precision of the measuring instruments. Calibration certificates also provide information about the correction of uncertainty in the measurements.
The uncertainty of the voltmeter is 0.05 provided in the calibration certificate and how to recover this uncertainty information is also present in it.
The uncertain information and documents are provided through the reference material data. The information and values that are provided through these reference data help to improve the uncertainty in the measurements that can be calculated.
The certified uncertainty of the gas analyzer is approximately about 0.1% and the reference data is provided to minimize the uncertainty in the measurements.
The instrument's specification accuracy, precision, and limits can only defined by the manufacturers because they understand the nature of their instruments and they also determine or estimate the uncertainty that can produced by their instrument during the measurements.
For the dimensional measurements the instruments we can use a digital micrometer and their accuracy is about 0.002 it can also budget the uncertainty measurement in it.
There are different theoretical models are present that can convey detailed information about the uncertainties in the measurements. Because these models are based on the assumptions and the practical experiences. By using these models we can also estimate and identify the uncertainties in the measurements which can be measured.
We can estimate the uncertainty and the precision in the vacuum of the speed of light and we can measure these uncertainties that are based on precision and we can calculate them.
The sources which can use the uncertainty in the measurements are given below:
Environmental conditions
Previous measurements
Manufactures specification
Theoretical models
Instrumental calibration
Previous measurements
In the combination of different type b components, we can use the root sum square method to estimate and calculate the uncertainty in the measurements. The all components that are combined are independent but we can combine them to estimate the uncertainty in the measurements precisely.
uc(y) = i=1nciu(xi)2
We can reduce the type B uncertainty in the measurements if we can follow these given steps. Because it can help to reduce tp understanding of the uncertainties in the measurements. The methods and the steps are given below:
Used advanced measurement techniques
Improved calibration
Used high-quality reference data
Enhanced environmental controls.
Generally, uncertainty in the measurements occurs due to many sources but the major two are
Random error
Systematic error
The sources and details of these errors are given below:
Random errors are common because they can caused by many different sources and they may be reduced by doing repeated measurements and by estimating the main cause of error. Some major sources which can cause this error are given there:
Observer variability
Environmental noise
Instrumental fluctuations
This error occurs due to the imperfect instruments usage and the unskilled persons who can take measurements but the main sources and causes of systematic error are given there:
Methodological error
Instrumental error
Observer error
Environmental factors
The techniques and some methods that are used to reduce the uncertainty are given there:
Replicates and repeat measurements
Randomization
Calibration and standardization
Improved experimental designs
Control variables
To reduce the uncertainty in the measurements we can use many different advanced techniques some are given there:
Error-correcting algorithm
Automated data collection
High precision instruments
Skilled persons
In scientific research the measurement of uncertainty is essential and it is also essential to reduce it because the scientists try to make precise and accurate measurements according to the calibration certificates, the significance of uncertainty in scientific research is given there:
Reproducibility
Peer review
Validation
Transparent reporting
In the field of modern technology and engineering, in measurements uncertainty and error are common but by using different techniques we can reduce them. Some applications of uncertainty in the management are given there:
Medical and biological research
Pollution monitoring
Quality control
Climate modeling
Safety standards
Drug efficacy
Diagnostic accuracy
The uncertainty in the measurements is common but in modern science and technology or different fields of science and physics, we can reduce the uncertainty in the measurements using many different techniques because scientists and engineers want to measure the precise and accurate measured values. Because the experts can agree on the measurements which are according to the standard values of the calibration certificates. Because the national measuring institutes and the American Society of Measurements can present the standard suits for measurements that are used to reduce or estimate the uncertainty in the measurements.
Hello friends, I hope you are all doing well. In this article, we can talk about the precision and accuracy of the measurement. Both of these are used to analyze errors in the measurements which can be calculated. In the era of science and modern technology, accuracy and precision are essential to depict the quality of the measurements. To measure the error in the measurement precision and accuracy techniques can be used because both are used for comprehensive measurement analysis.
To calculate or describe the random errors precision of measurement can be used but if we want to describe the systematic error, accuracy of measurements can be used. Now the details of precision and accuracy, their definition, differences, examples, and their application in different fields are given below:
The main differences between precision and accuracy are given below:
| Precision | Accuracy |
Definition: The reproducibility which can be measured in the measurement is known as precision. On the other hand, precision can also be defined as the set of repeatability measurements and all measurements are close to each other but we precise the closest value. Because highly precise measurements can convey the best and very similar results. |
Definition: The value that we can measure is close to the true value of the measurement is known as accuracy. Another definition of accuracy is that the value that can be measured is approximately equal or close to the accepted and the true value of the measured object. Because accuracy provides the correct measurement of the object which can we measured. If the measurement that we can calculate is close to the standard value of measurement then it can be an accurate measured value. |
Accurate: The measurements that can be calculated are precise but they can't be accurate because high precision can cause low accuracy. |
Precision: The measurements that we can calculate are accurate but they are not precise because high accuracy affects the precision and causes low precision measurement.
|
Determined: To determine the precision we can calculate the object measurement many times and from a set of measured measurements we can determine the precise measurement value. |
Determined: Accuracy can be determined from a single measurement of an object which can be measured. |
Random error: Random error can affect the precise measurement which we can determine. |
Systematic error: The accuracy of the measurement can be affected by systematic error. |
Accurate: The measurements that we can measure are precise but not correct every time. |
Precision: The measurements that can be measured are accurate but these measurements are not correct anyway. |
Reproducibility: Precision can give a degree of reproducibility. |
Conformity: Accuracy provides the degree of conformity of measurements. |
The terms precision and accuracy are different from each other, but somehow they are related to each other. Because when we measure the object or any quantity measurement then it may be accurate but not to be precise but if we determine the periscred measurement then it not to be accurate so that why both terms are not present in the measurements at the same time.to understand the relation and the difference between the accuracy and measurement and to improve the measurement techniques some common points are given there:
When the measurements that we can calculate are close to the standard or true value but the measured values are not similar to each other then they have high accuracy but low precision because the measurements are not matched to each other. Some examples are given below:
The measurement of the field which we can calculate is 80 cm, 99.90 cm, and 100.02cm but the true or standard value is 100.10 cm so the values are accurate but they can’t be precise.
If we can calculate the length of an object and the measured value is 70.01m, 80.67m, and 90.01m, the true value is 80.67 then the all values are not precise but they are accurate.
The measured values that can be calculated through measurements are not close to the true value and also the set of measurements are not close to each other all the measurement values are different which means they have low precision and low accuracy. Some examples are given below:
The measurements of the yields are 75.00 cm, 100.00 cm, and 112.00cm but the true value is 120.00cm. These values are not close to each other and neither one value is close to the true value so that is why they have low accuracy and low precision.
For instance, if the measurement is 95.00 cm, 106.00 cm, and 101.00cm but the true value is 105.07cm then they have both low accuracy and low precision.
If the measured values that we can measure are close to the true and the standard value and the set of measurements that we can calculate are close to each other then we can say that the measurements which we can be calculated have high accuracy and also have high precision. Some examples of highly accurate and highly precise measurements are given below:
The measurement of the length of the cylinder is 101m, 102m, and 103m and the true or standard value or accuracy is 102m then these values are precise and close to each other and they are also accurate because the measurement is close to the true value.
The measurement of the yield is 80cm, 82cm, 83cm, and 85cm or the true value is 83cm then they have high accuracy and high precision because the values are close to the true value and all sets of measurements are close to each other.
The measured measurements are similar or close to each other but these measurements are not close to the true or standard value. some examples of these measurements which have high precision but low accuracy are given below:
The quantifying of accuracy and precision in detail are given below:
By using the numerous different statistical measures we can quantify the accuracy. The steps which are involved in quantifying the accuracy are given there:
Root mean squared error
Absolute error
Mean absolute error
Relative error
Their details are given there:
When we take the square roots of the average squared errors then it is known as root mean squared error. The formulas which are used for root mean squared error are given below:
RMSE= 1ni=1n(measured valuei-true values)2
Absolute errors are used to determine the difference between the true value and the measured value. The formulas which are used to determine the absolute error are given below:
Absolute error= measured value- true value
In the set of measurements to calculate the average error in all the measurements mean absolute error formulas are used. The mean absolute error formula is given there:
MAE= 1ni=1nmeasured value- true value
To express the true value percentage with the absolute error we can use the relative error formula. The relative error formula is given there:
Relative error= (absolute errortrue value) 100
By using the measure of variability we can quantify the precision. the steps which are included in the measure of variability are given there:
Coefficient of variation (CV)
Range
Variance
Standard deviation
Details of these points are given below:
Range is used to measure the difference between the maximum and the minimum values in the set of measured measurement values. The formula which can be used to measure the range is given below:
Range= Maximum value- minimum value
By using the mean values from the set of measurements to can measure the dispersion of the measurement set. The formulas which are used in standard deviation are given below:
Standard deviation: 1n-1i=1n(measured value-mean value)2
The square root of standard deviation is termed variance. The formula of variance is given there:
Variance= 1n-1i=1n (mmeasured value- mean value)2
To express the ratio of the standard deviation we can use the formula of coefficient of variation. The formula for the coefficient of variation is given there:
Coefficient of variation: standard deviationmean value 100
The classification of the accuracy and the precision are given below:
To measure the statistical measurements we can use accuracy and we can use accuracy in the binary classification to determine or identify the conditions of measurements. By using the accuracy to can determine the exact ratio of the true positive or the true negative.
The two main types of errors that affect the precision and accuracy but these two types of error have also types two main types of errors are given below:
Systematic error
Random error
Details of these errors with their types are given there:
Systematic error is defined as an error that occurs many times due to faulty equipment or may be due to an unskilled person's measurement techniques and many other reasons. But this type of error can directly affect the measurement which can be measured and give the inefficient or bad results of the measurements.
The sources that can cause the systematic error are given below:
Observational errors
Environmental errors
Theoretical errors
Instrumental errors
Observational errors occur due to human mistakes. When they can read the reading and observe the reading efficiently then observational errors occur some examples are given below:
The reading that can be measured by a human is approximately 3.4cm but it can't observe the certainty and uncertainty within it and also can’t read the exact digits of the reading.
When the unskilled person tries to observe the readings of very small quantities then the observational error occurs maximum because to measure the small quantities skilled person is used to observe the readings.
The error occurs due to environmental factors like temperature, humidity, and the fluctuation in the electromagnetic interference and also due to the airflow which can move instruments during the measurements. Some examples of environmental errors are given below:
When we use the thermometer to measure the temperature changing continuously in temperature can cause errors in the measurements.
To measure the current when we used the galvanometer the needle of the galvanometer moved again and again due to the magnetic field these fluctuations can cause errors in the measurements.
The error occurs due to the usage of imperfect or faulty instruments during measurements.
To measure the measurements without error firstly perfect instruments are essential because without this the measurement can't be correct when we measure anything. Some examples of instrumental errors are given below:
To measure the length of an object a ruler can be used but the readings on the ruler are visible clearly because it can be helped to measure the exact length.
A weight machine can be shown the weight of 2kg even if we can't put any object on it this is the instrumental error.
Theoretical errors occur when we assume the theory model but nothing can happen in reality according to these models. Theoretical misassumptions can cause different errors in the measurement and provide the un accurate or precise results. Some examples of theoretical errors are given there:
During physics quantities measurements the air resistance is considered negligible.
These errors occur due to various small unpredictable variables and these errors can’t be controlled because they occur in all measurements due to many different small or uncontrolled able variables. Some sources which can cause the random errors are given below:
Environmental variability
Instrumental vartibility
observer variability
Sample variability
The fluctuations that can be occurred in the environment at any time like fluctuations in the temperature. Some examples are given there:
The flow of air can distract the measuring instrument from its place and cause errors in the reading
Air currents can also cause errors in the measurements.
The changing that occurs in the instrument during the reading there are many different instruments present that can use the noise or changes in it during measurements. Some examples are given below:
During measuring the current the galvanometer changing the needle again and again can cause errors in the readings or measurements.
When the numerous observer measured the measurement at the same time their readings were different from each other due to sight issues and many other issues. Some examples are given below:
When the numerous observed reading through the stopwatch and stopwatch couldn't stop at the same time slime gave them vision issues.
When we can measure the same object but the different samples have changes naturally we can measure them. These natural changes can also cause errors in the measurements. Some examples are given below:
When taking different samples of the same thing or object naturally their width or thickness can be changed then different measurements come with errors.
The accuracy and the precision can be improved if we can follow the following steps which are given there:
For improving accuracy:
Environmental control
Calibration
Instrument maintenance
standardization
For improving precision:
Training
Environmental control
Reptation
Instrument quality
Application of precision and accuracy in different fields are given below:
Scientific research
In modern technology and sciences
In measurements
Healthcare fields
Monitoring environment
In mathematics calculations.
Precision and accuracy both play vital roles in measurements because in modern physics scientists can measure very small quantities and these quantities of measurement can be accurate without error due to the accuracy and precision. Both terms are the fundamental p[art of measurement in the analysis of comprehensive measurement.
Precise and accurate measurements provide the quality of the measurements because without these terms in measurements, too many errors occur and they are not precise and accurate.
It’s no secret that the tendrils of tech are tightly wrapped around the security equipment sector, and that’s good news for people and organizations with assets to protect. What you can’t afford to do is remain out of the loop on the developments that are impacting the market at the moment, so stick around and we’ll fill you in on the details that matter in 2024.
The combination of IoT with security systems is continuing to gain momentum this year. The idea is that interconnected devices can create smarter, more responsive security environments that are less reliant on manual monitoring. In this context it should be no surprise to find a whole host of network-enabled hardware when you explore security devices available today.
There are perks and pitfalls to IoT’s increased hold over security, so here’s a look at these so you can make an informed procurement decision:
Real-Time Monitoring: IoT sensors enable continuous surveillance, providing real-time alerts for unauthorized access or anomalies. With 25% of businesses suffering at least $1,500 in losses to theft each month of the last year, it’s clear action is needed here.
Enhanced Automation: Automated locks, lights, and alarms respond instantly to detected threats, boosting efficiency.
Remote Access: Manage security systems from anywhere using mobile apps, which is perfect for frequent travelers or remote site managers.
Data Analytics: Collect detailed data on usage patterns and potential vulnerabilities, which improves predictive maintenance.
Cybersecurity Risks: Interconnected devices can be vulnerable to hacking, and there was a 400% increase in IoT malware over the course of 2023. This means ensuring adequate encryption and secure protocols is crucial.
Interoperability Issues: Different brands and standards can cause compatibility problems, which makes selecting compatible hardware/software a must.
Complexity in Setup/Management: Installing an integrated system requires expertise, and professional installation often necessary.
Let’s say you run a commercial warehouse. Integrated IoT allows it to monitor inventory levels while simultaneously tracking foot traffic through connected cameras. This dual function not only tightens security but also optimizes operations by predicting stock shortages based on entry patterns.
Image Source: Microsoft Designer
Surveillance systems imbued with artificial intelligence (AI) are in the process of redefining how we monitor and secure spaces, in addition to its influence over data security . These systems use advanced algorithms to provide smarter, more precise monitoring. Here’s a look at the ins and outs of what to expect from this tech:
Object Recognition: AI can distinguish between people, animals, and objects, which reduces false alarms.
Behavior Analysis: Monitors and identifies suspicious activities based on movement patterns, and thus enhances threat detection.
Facial Recognition: Identifies individuals from databases quickly, which in turn streamlines access control.
Improved Accuracy: Drastically cuts down on human error in surveillance by automating routine tasks.
24/7 Vigilance: Never sleeps or gets tired, and instead constantly watches over premises without gaps in coverage.
Scalable Solutions: Easily scalable for different environments, from small businesses and even domestic premises to large campuses.
Privacy Concerns: Potential misuse of facial recognition data raises ethical issues, so balancing security with privacy is essential. Indeed a KPMG report found that 53% of people see AI as introducing privacy problems, meaning there’s lots of awareness around this aspect.
High Costs: Advanced AI systems require significant investment in both hardware and software initially. Even if you outsource this to a third party vendor, the price of being an early adopter will not be cheap.
Data Management Complexity: Handling vast amounts of video data relies on robust storage solutions and processing power.
If you’re responsible for security at a retail store, you could use an AI-powered system that not only spots shoplifting attempts but also tracks customer flow. The twofold benefits of loss prevention and gleaning insights into peak shopping times means that optimizing staff allocation is a much less complex affair.
Image Source: Microsoft Designer
Biometric authentication has been part of the security sector for some time, but it’s only in 2024 that we are seeing the true extent of what the latest developments in this field can offer. Basically, it's not just about fingerprints anymore, as today we're dealing with multi-modal biometrics, combining several types of identifiers. Let’s unpack this in more detail:
Facial Recognition: Scans and verifies facial features, and is enhanced with AI, as mentioned.
Iris Scanning: Uses unique patterns in the eye for identification.
Voice Recognition: Analyzes vocal characteristics.
Behavioral Biometrics: Monitors keystroke dynamics, mouse movements and other identifying behaviors for cybersecurity purposes.
High Security: Difficult to replicate, as it adds a robust layer of protection over traditional methods like passwords or PINs.
Convenience: Faster access with fewer errors, since users don’t need to remember complex passwords.
Scalability: Easily integrates into various environments, from smartphones to secure business facilities.
Privacy Concerns: Collection and storage of biometric data can be intrusive, so organizations must ensure proper safeguards are in place.
False Positives/Negatives: Environmental factors or physical changes can affect accuracy; continuous updates and calibrations needed.
Costly Implementation: Setting up advanced systems requires substantial investment initially.
Consider the case of a high-security government building. Using multi-modal biometrics, an employee's identity gets verified through both iris scanning at the entrance and voice recognition at internal checkpoints. This layered approach ensures only authorized personnel gain access while maintaining a seamless user experience.
Image Source: Microsoft Designer
From a practical, commercial perspective it makes sense for businesses to prioritize the adoption of security equipment that intertwined elements of IoT, AI and advanced biometrics. This is a way of keeping physical and digital assets protected, while also ensuring the safety of employees and customers in 2024.
For homeowners, it is not as vital to ramp up security spending at the moment. That said, if you are thinking of installing equipment or upgrading an existing setup, looking for connected devices that are also AI-enabled makes sense. Biometrics is less relevant here but could bring convenience to the table if you want to move on from the traditional lock-and-key approach to access.
Hi everyone, I hope you are doing well. Today in this post we can discuss significant figures. Significant figures can be evolved and traced a few hundred years ago and they can be developed from time to time according to the precise calculations that can be done in mathematics and modern science and technology.
The origin of significant figures can be traced to ancient times in Greece, Egypt, or many other countries where they can be used as Significant figures in calculations or mathematics. With time these significant figures can be used by astronomers, mathematicians, and scientists as well. To measure accurate and precise measurements they can use significant figures with scientific notation. In modern physics and mathematics, significant figures can be used essentially.
Now we can start our detailed discussion about significant figures and their definition, rules, examples, and problems in detail.
Details of significant figures are given below:
Significant figures can be defined as:
"In the number or calculated value the specific digits which provide precision and accuracy and also impart and convey the necessity of digits in the calculated measurement."
Significant figures include the zero or the nonzero digits. Significant figures are also known as significant digits and sig figs. When we make measurements or calculate the measurement such as when we measure length, pressure, and volume and the measured value is large then we can use significant or scientific notation method. or the most capable digits are estimated and called significant figures.
Such as if we measure the volume of a cylinder and the measurement is approximately equal to 3.97L and the uncertainty is approximately equals to 0.04L. The actual volume of this cylinder is approximately between 3.93L and 1.01L, but we can't know the certain digits in this measurement completely. But all of these three figures are called significant figures.
Additionally, the value 345.0 has only 3 significant figures 3,4, and 5 the o digit is not termed as a significant figure because it is only a placeholder.
Some digits are not to be considered as a significant figure. So the types of digits that can't be considered significant figures are given below:
Spurious
Leading zero
Trailing zero
The zero can only be present in the value or measurement as a placeholder. For example 1800 the last two zeros were only a placeholder but we can write the scientific notation we can raise the power of ten but in significant value, we only estimate and choose the figures that can provide accurate and precise measurement.
The zero which can be present at the start of measurement and the value is known as leading zero. For instance, 00367 the first two zeros (00) known as leading zero and it can’t be a significant figure in this value 3,6, and 7 are the significant figures. Another example is 0.00456089 in this value the leading zero is not a significant figure but the zero which is present in between the digits like 6,0,8 and 9 is referred to as a significant figure.
Spurious digits are those when we do calculations on the instruments then the high-resolution measurement results precisions and the accuracy digits are known as spurious digits.
Another example is if the zero is present in the value after the decimal it can be considered a significant figure like 12.00 then these two zeros are also significant figures and in this value, the total no of significant figures is 4 but if the decimal is not present like 12300 then the last two zero is not considered as significant figure and in this value, the total significant digits are 5. After the decimal point, the zero is also considered as the significant figure for example 1.000 the last trailing zeros are also the significant figures and the total significant figures in this value is 4. But if the decimal is not present like 1000 then this value has only one significant figure and trailing zeros are only a placeholder.
Some examples of significant figures are given below:
123790: All digits in this value are known as significant figures.
0.0056:in these digits the leading zero is not a significant figure the last two 5,6 are significant figures.
34986097: all digits are significant figures and the zero which is present inside them like 6,0,9,7 all are significant figures.
1001: all digits are the significant figures.
To identify the significant figures from the measurement and for the value, there are some rules. The rules that can help to identify the significant figures are given below:
Non-zero digits in the measurement are to be significant
Between the two significant digits, the zero is also a significant figure.
Starting, leading zeros are not a significant figure.
Zero after the decimal point on the right side can also be a significant figure.
Trailing zero without the decimal point is not significant.
All non-zero digits in the measurement or the value are significant figures some examples are given below:
9144 all the digits in this value are significant. So the total significant digits is 4.
1.5678 all the digits after decimal or before decimal are considered as significant figures and the total significant digits in it is 5.
568.9 these all digits are significant figures.
The digit zero is not to be considered to be significant but in between the two the significant digit zero also becomes a significant digit some examples are given below:
10054 in this value all digits including zero are referred to as significant figures.
102.60809 these all digits are considered significant figures the zeros after the decimal point are also considered significant figures.
101.101 in this value all digits are also considered as the significant digits or figures.
The starting or the leading zero is not considered to be a significant figure it can only considered as a placeholder some examples are given below:
000102 in this value last 3 digits 1,0and 2 are called significant but the starting three zeros are not considered to be significant but it is only placeholder or leading zeros.
0.065 the last two digits 6 and 5 are sig figs but the starting zero or zero after the decimal point is not considered a significant figure.
The trailing zero, the zero on the right side after the decimal point can considered a significant figure. Some examples are given below:
1.89900 these all digits are considered to be significant digits. In this value, the digits 1,8,9,9,0,0 all are significant figures including the trailing zeros.
0.1200 in this value 4 digits are called significant figure 1,2,0 and 0 are significant figures which are trailing zeros but the first leading zero is not considered as the significant figure.
The trailing zeros without the decimal point are not considered as a significant figure. some examples are given below:
19900 the first 3 digits 1,9 and 9 are considered as the significant digits but the last trailing zeros are not to be considered as the significant figure.
1096800 in this value the first 5 digits are considered to be significant but the last trailing zeros are not a significant digit or a figure.
The rounding-off technique can be used in mathematics in all calculations. In this technique the last digit if it is greater than 5 then it can be added to the previous number. for representing the rounding off number we can use n. To round the significant digit with n there are some rules which are given below:
In the measurement value if the last digit is greater or equal to the 5 then we can +1 inti the previous digit. For instance, 1.345 then after rounding the digit the new value is 1.35 and it has 3 significant figures now.
Another example is if we have 5 significant figures like 5.9867 then after rounding off the last digit the new value is 5.987 and they have only four digits in it.
Now if we have the last digit less than 5 then it can’t be added to the previous number some examples are given below:
1.563 the last digit is less than 5 so it can’t added 1 to the previous digit and the significant figure in this value is 4. Another example is that 67.91 has the 4 significant figures and they can’t be changed because the last digit value can’t be added.
To calculate the numerical value of significant figures for nonzero significant figure x and for the precision and accuracy p we can use the formula which is given below:
10n. Round (x10n)
Where the n is equal to,
n=[log 10(x)]+1-p
In the measurements precision and accuracy, both are essential. Generally, we can use precision for the stability of the measurements with repeated values and the accuracy used for the closeness to the actual and accurate measurement of the thing and the object. but with time in modern technology and science, both precision and accuracy are termed for trueness or the most closeness exact value of the measurements. In the scientific community, the accuracy and precise measurement of the object is essential because without accuracy the measurements and problems become more complicated That is why accuracy and precision both are essential for measurements.
In addition and subtraction, the result of measurements can calculated with the same decimal point before and after the addition and subtraction for example:
12.56+ 0.5=13.1 they have the 3 significant figures in it another example is that
67.9+9 =76.9 and the significant figure in it is 3.
The result that can be taken from measurement and the result after division and multiplication of values both have the same significant figures some examples are given below:
4.56 1.4 = 6.384 and we should rounded to 6.4 then they have 2 significant figures like before the division. another example is that 2.44 then it is equal to 9.6 they have also 2 significant digits like before the multiplication or division.
In the measurement, certainty means that we confidently know the significant digit in the measurements. This known digit conveys and provides the precision of measurement and is also considered reliable because accuracy and precision are very essential in the measurements.
For example, if we can measure the length of an object with a ruler that has a millimeter marking and the measurement is equal to 24.87 then the digits 2,4,8, and 7 are certain because measurement tools can provide precise and accurate measurements.
Rather uncertainty in the significant figures means that which measurements we can take are not precise and accurate and we have doubts about them. When an unskilled person or limited measurement tool is used for measurement then chances of uncertainty increase because a skilled person can take accurate and precise measurements of the object and the high precision measurement tool is also needed.
For example, if we can measure the length and the measurement is 24.56 then the last digit 6 creates uncertainty because it can be rounded off and the original length can't be measured precisely and efficiently. So that's if we want to remove uncertainty then we again measure the length and then determine the significant figures in it.
Some examples of significant figures are given below:
406.62 all digits in this are significant figures and the total significant figures is 5.
0.00034 the last two digits are signification figures and the starting zeros are leading zeros.
12090 all are significant figures and the last zero is only a placeholder.
74.0 the zero after decimal is also a significant figure the total significant figure is 3.
67.08 in this all digits are significant figures and the total is 4.
These are some common examples of significant figures.
Some practical problems to understand the significant figures are given below:
Identify the significant figures in the given problems:
7696
0.0074
690.00
60.09
74.0
Solutions to these problems are given below:
All 4 digits are significant figures.
Leading zeros are not significant digits but the last 2 digits 7 and 4 are significant.
All 5 digits including zero after decimal are also considered significant digits.
6,0,0 and 9 all digits are considered significant figures.
7 and 4 including trailing zero after the decimal are also considered significant figures.
The calculation to determine or calculate the significant figures some problems are given below:
26.7+ 8.1
62.4 0.04
2.4+ 9.2
58.0+ 4.5
0.056/0.0007
The solution to these problems is given below:
34.8 in this value 3, 4, and 8 are significant figures.
2.496 in this value 4 significant digits are present.
11.6 in this solution all digits are considered significant figures.
62.5 in this solution of a problem all digits are significant digits.
80 are the solution and both are significant figures.
To determine the significant figures, perform the following operations:
(2.7+ 4) 7.9
(5.6 8.9) + 8
Solutions to these problems are given below:
(6.7) x 7.9
=52.93 In this solution, all 4 digits are significant figures.
(49.84) + 8
= 57.84 In this solution, all 4 digits are significant figures.