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Dimensions of Physical Quantities

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.

What are physical quantities?

Definition:

   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.

Types:

 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.

Dimensions of physical quantities: 

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: 

Base physical dimensions and their units: 

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:

Derived physical units with their dimensions: 

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:

Dimensional analysis:

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: 

      Principles of dimensional analysis include:

•  Derivation of formulas 

• Dimensional homogeneity 

• Conversion of units 

Conversions 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.

 Dimensional homogeneity: 

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.

Derivation of formulas:

  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.

Steps which are involved in the 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 

Set up the equation: 

  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. 

Identify the base quantities: 

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.

Solve for the unknowns: 

              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.

Write the dimensional formula: 

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.

Applications of dimensional analysis:

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 

Biophysics:

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.

Relativity:

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.

Quantum mechanics:

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.

Engineering:

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.

Conclusion:

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.

Total Uncertainty in the Measurements

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.

History:

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. 

What is the uncertainty in the measurements: 

Definition: 

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.

Types of uncertainty in the measurements:

There are two major types of uncertainty in measurements which are given below: 

• Type A uncertainty 

• Type B uncertainty 

Type A uncertainty in measurements:

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 evaluating methods:

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 intervals:

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."

Formula: 

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 mean: 

           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. 

Formula: 

The formula which is used to calculate the average of the measured set is given there: 

                             x = 1n i=1nxi

Standard error of the mean: 

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.”

Formula: 

 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:

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.

Formula:

    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

Degree of freedom:

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.

Formula: 

       The formula that is used for the degree of freedom is given there:

                           v= n - 1

Sources of Type A uncertainty in the measurements: 

The main causes and sources of the Type A uncertainty in the measurements are given there:

• Environmental changes

• Human factors 

• Instrumental fluctuations

Methods used to reduce type A uncertainty:

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 in the measurements: 

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.

Type B uncertainty measurement evaluating methods: 

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

Expert judgment: 

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.

Calibration certificates:

         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. 

Example: 

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.

Reference material data:

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.

Example:

            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.

Manufactures specification:

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.

Example: 

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.

Theoretical analysis: 

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.

Example: 

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.

Sources which can cause type B uncertainty: 

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  

The combining component of Type B uncertainty: 

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. 

Formula:

               uc(y) = i=1nciu(xi)2

Methods to minimize the type B uncertainty: 

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.

Sources of uncertainty in the measurements:

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 error: 

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

Systematic error: 

              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 

Techniques and methods to reduce uncertainty: 

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

Advanced measurement techniques: 

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 significance of uncertainty: 

                 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 

Application of uncertainty in the management: 

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

Conclusion: 

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.