UNIT AND MEASURMENT

Physical Quantities

Physical quantity is the name of a characteristic or property or phenomenon in our nature which can be measured. Physical quantities can be classified into two groups – fundamental quantities and derived quantities.
Fundamental quantity
Fundamental quantity is the one which cannot be expressed in terms of any other quantity. There are basically 7 fundamental quantities −

• Mass

• Length

• Time

• Temperature

• Electric current

• Luminosity

• Number of particles

There 2 more quantities which were discovered later and added to the list of fundamental quantities −

• Plain angle

• Solid angle

Derived quantity
Derived quantity is that characteristic, which can be expressed in terms of many fundamental quantities.
Example − Area, volume, speed, force, work, etc
Unit of Measurement
To measure a physical quantity, we need some standard measurements of that quantity and this standard measurement is known as unit. All measurements of this quantity are multiples of unit. Measurement is simply a comparison of physical quantity with unit.
The measurement of a physical quantity is mentioned in two parts, the first part gives how many times of the standard unit and the second part gives the name of the unit. For example, we have to study for 2 hours. The numeric part 2 says that it is 2 times of the unit of time and the second part hour says that the unit chosen here is an hour.
Thus,Measurement = numerical value × unit
System of units
Commonly used systems of units are −
The FPS system (Foot, Pound and Seconds system) developed by British.
The CGS system (Centimetre, Gram and Seconds system) developed by French.
The SI system (System International). The SI system uses the MKS system (Metre, Kilogram and Second system).
Properties of Units
Each unit should have the following property −
Common size
The unit should have a size that is most commonly used while measuring a physical quantity.
If a common size is used, it makes the measurement easy and efficient.
Example − 1meter for length, 1kg for mass.
Well defined
The unit should be well defined so that it is independent of any other physical quantity and change in any other quantity should not affect the unit of measurement.
Hence, unit of measurement is always compared with speed of light so that it is well defined.
Invariable with time
The unit should not change with time i.e. unit of measurement for a physical quantity should be the same irrespective of time.
Possible to reproduce
The unit of measurement should be fixed in such a way that it should be easy to reproduce whenever required.
Name of Units
The following table gives the fundamental quantities and their units in SI −

• Base Quantity	Name	Symbol
  Length	meter	m
  Mass	kilogram	kg
  Time	second	s
  Electric Current	ampere	A
  Thermodynamic temperature	kelvin	K
  Amount of substance	mole	mol
  Luminous Intensity	candela	cd
  Plain Angle	radian	rad
  Solid Angle	stradian	strad

Measurement of Length (Direct Method)
For the measurement of length we have number of techniques, which are broadly divided into two main groups −

• Direct Method

• Indirect Method

Direct Method
When we can reach both the ends of length, the measurement is done by direct method. In direct method instruments used are meter scale, Vernier calliper, screw gauge / spterometer.
Meter scale is an instrument that has numeric calibrations on it to measure length. The precision / least count of meter scale is 1mm i.e. measurement with accuracy till 1 decimal point is only available.
Example − Meter scale can measure length of 21.6cm or 21.7cm, but it is unable to measure a length between the two.
Vernier calliper is a measuring device that has better least count i.e. it is more accurate that the meter scale. It has the least count of 0.1mm.
Example − Vernier calliper can measure length of 21.65cm or 21.66cm
Screw gauge is a measuring device that uses the concept of pitch and circular scale. Least count of screw gauge is given by −
Least count = pitchno. of divisions of circular scaleExample – Screw gauge can measure length of 21.654cm or 21.655cm
Measurement of Length (Indirect Method)
Indirect Method
When we cannot reach both the ends of length, the measurement is done by indirect method. The use of indirect method is again classified depending upon the distance −

• Large distance

• Small distance

For measurement of length of large distance we use the following indirect methods −
We calculate the length using a formula −
length = velocity × time
In this method, the phenomenon of reflection is used to calculate the length/distance.
Example
If a man wants to know the distance of a wall from his current position, then that man will create a sound signal that will hit the surface of the wall and return back, by measuring the time required for the wave to come back and the velocity of wave, distance can be calculated using the above equation.
This indirect method is also used by ships to measure depth and locate foreign objects in sea. Ships use ultra-sound to implement this method.
Another application of this method which is implemented using ultra-sound is for sonography i.e. medical use.
For the calculation of distance between planets radio waves are used, the process of measurement of distance is done using this indirect method itself.
Measurement of Length Parallax Method
Indirect Method
When we cannot reach both the ends of length, the measurement is done by indirect method. The use of indirect method is again classified depending upon the distance −

• Large distance

• Small distance

For measurement of length of large distance we use the following indirect methods −
We calculate the length using a formula −
length = velocity × time
Trigonometry / parallax method
Large distances such as the distance of a planet or a star from the earth cannot be measured directly with a metre scale. An important method in such cases is the parallax method.
To calculate large distance, the parallax method uses two observation points for the single object as shown in figure.

• 

Equation used to calculate the distance is given by −
r = xθWhere,
r = Distance to be measured.
x = Distance between observation points
θ = Parallax angle.
Suppose we need to calculate the distance of a middle range star from earth.

• 

Step 1 − The two observation points are decided depending upon the location of the earth. The star is observed with a difference of 6months in between the two observation points.
Step 2 − To measure the distance, the formula of parallax method is used −
r = xθWhere,
r = Distance to be measured.
x = Distance between observation points
θ = Parallax angle.
Here, the distance between the observation points is known i.e.
x = 2 aU
Calculating the parallax angel is a difficult task, so as to make it easy, the concept of “rays coming from a faraway star is always parallel” is used.
Step 3 − A faraway star is observed together with the middle range star from both the observation points, so as to calculate the parallax angle. From figure −
θ = θ₁ + θ ₂
Step 4 − As both ‘the distance between observation points’ and ‘parallax angle’ is known, the distance between earth and middle range star can be measured easily by −
r = xθ∴ r = 2 aUθ₁ + θ₂

• 
  

Measurement of Length Size of Molecule

• To measure a very small size like that of a molecule (10^-9 m), we have to adopt special methods.

  We cannot use a meter scale as it has a minimum count of only 1meter, we cannot use Vernier calliper with 10^-4 m minimum count, and screw gauge with minimum count 10^-5 m is also not useful. Even a microscope has certain limitations.

  To observe an object, we project light on it and the object acts as a secondary source and reflect the light. Light has a wavelength of 10^-8 m, any object smaller than the wavelength of light will not be visible as the wave will move across the object without any interaction.

  So to estimate the size of molecule we perform the following calculations −

  Volume of any substance is given by −

  V = L × B × W

  Where,

  L = Length

  B = Breadth/thickness

  W = Width

  ∴ B = VL × W

  As we know,

  Area = L × W

  ∴ B = VArea

  Now, for the measurement of size of molecule, we will consider the molecule of Oleic acid as it has large molecular size.

  The idea is to first form mono-molecular layer of oleic acid on water surface.

  We dilute the oleic acid by alcohol for this we dissolve 1 ml of oleic acid in alcohol to make a solution of 20 ml.

  Then we take 1 ml of this solution and dilute it again to 20 ml, using alcohol. So, the concentration of the solution is −

  120 × 20 oleic acid consentration per ml. of solution

  Next we sprinkle some lycopodium powder on the surface of water in a large trough and we put drops of the diluted solution with the help of burette in the water.

  The oleic acid drop spreads into a thin layer of molecular thickness on water surface.

  Let `V` be the volume of each drop and a total of `n` drops are poured on the water surface, then the total volume of the solution is −

  nV

  As soon as the solution comes in contact of air, the alcohol molecules evaporate and we obtain a layer purely formed by the molecules of oleic acid.

  The volume of acid layer is given by −

  Volume = nV × 1400

  Now to get the area of the layer, a graph paper is rolled over the surface of the layer which imprints the shape of the acid layer on to the paper. Using this imprint of acid layer its area is gained.

  Hence, thickness of oleic acid molecule is given by −

  Thickness = VolumeArea

  Thickness = nV20 × 20 × A

• Significant Figures - Introduction

• Number of figures which are important, their count (in numbers) is known as significant figures.

  For example,

  • If the distance between Delhi to London is specified as 5000km, then there is only one significant figure which is 5 and,
  • If the distance between Delhi to London is specified as 5430km, then there are three significant figure which are 5, 4 and 3. Similarly,
  • If the distance between Delhi to London is specified as 5432km, then there are four significant figure which are 5, 4, 3 and 2.

  In any number or measurement last digit of all important is uncertain while the rest are certain.

  For example,

  • In the measurement of 5432, the digits 5, 4 and 3 are certain while 2 is uncertain.
  • In the measurement of 5430, the digit 5 and 4 are certain while 3 is uncertain.
  • In the measurement of 5400, the digit 5 is certain while 4 is uncertain.

  Rules for significant figures – Whole number

  • All the non-zero digits are significant.
  • The trailing zero(s) in a number are not significant.
  • All the zeros between two non-zero digits are significant.

  Example

  Measurement	Significant figures
  5500	2
  5005	4
  5050	3
  0050	1

  Rules for significant figures – Decimal number

  • All the non-zero digits are significant.
  • All the zeros between two non-zero digits are significant.
  • The zero(s) on the right of decimal point but to the left of the first non-zero digit are not significant.
  • The trailing zero(s) in a number with a decimal point are significant.

  Example

  Measurement	Significant Figures
  6.2	2
  0.02	1
  6.02	3
  6.00200	6
  0.00260	3

  Significant Figures after Calculation Addition

In the operations of addition and subtraction of two numbers, the result will carry as many figures after decimal as many are there with minimum digits after decimal i.e. in addition or subtraction, the final result should retain as many decimal places as are there in the number with the least decimal places.

Example

• Addition
  When we add two measurements 12.2 and 1302.56, we get 1314.76 as the answer but the minimum decimal digit in the two measurements is 1 (12.2), so we will round off our answer to only 1 decimal place i.e. 1314.8
• Substraction
  When we find the difference of two measurements 13.12 and 9.056, as the minimum decimal digit in the two measurements is 2 (13.12), so we will round off the other measurement to 2 decimal places i.e. 9.06 and then we will perform subtraction and it given 4.06 as the answer.

Significant Figures after Calculation Multiplication

The significant figures of product or quotient is kept equal to minimum number of significant figures of the quantities being operated i.e. in multiplication or division, the final result should retain as many significant figures as are there in the original number with the least significant figures.

Example

• Multiplication
  When we multiply two measurements such as 2.14 and 6, then by mathematics we get the answer as 12.84, but while working with measurements in physics we consider the minimum significant figure, for our example it is 1 (6), so the final product is 1 × 10¹
• Division
  When we divide two measurements such as mass = 4.273 g and volume = 2.51 cm³ to find density, then by mathematics we get the answer as 1.68804780876 g/cm³, but while working with measurements in physics we consider the minimum significant figure, for our example it is 3 (2.51), so the final product is 1.69 g/cm³.

When we reduce a number to the required significant figures we drop some of its digits and round off the number, to perform this process we have a set of rules.

Rules of rounding off a number

• If the first digit to be dropped is more than 5, then preceding digit rise by one.
  Example
  • 9.35672 has to be reduced to only 4 significant figures, then we need to drop the digits 7 and 2, as 7 is greater than 5, the final result becomes 9.357
• If the first digit to be dropped is less than 5, then preceding digit remains as it is.
  Example
  • 9.35321 has to be reduced to only 4 significant figures, then we need to drop the digits 2 and 1, as 2 is less than 5, the final result becomes 9.353
• If the digit to be dropped is exactly than 5, then see the preceding digit.
  • If preceding digit is odd, increase it by one and
  • If preceding digit is even, then leave it as it is.
  Example
• has to be reduced to only 1 significant figures, then we need to drop the digits 5, now we will observe the preceding digit i.e. 3 as it odd, the final result becomes 4.
• has to be reduced to only 1 significant figures, then we need to drop the digits 5, now we will observe the preceding digit i.e. 4 as it even, the final result becomes 4.

Error Cause and Types

Precision

It is the degree of exactness or refinement of a measurement. It defines the limitation of measuring instrument. It is not the result of human error or lack of calibration.

Accuracy

It is the extent to which a reported measurement approaches the true value. It depends upon the number of significant figures in it. The larger the significant digits, the higher the accuracy. Problem with accuracy are due to errors. As we reduce the errors, measurements accuracy increases.

Example − Suppose true value of a certain length is 4.567, let it is measured as 4.4 by an instrument of least count 0.1 cm and 3.32 by an instrument of least count 0.01m.

First reading is more accurate (because it is closer to the true value) but less precise (its resolution is only 0.1 cm). Second reading is less accurate but more precise.

Errors

The uncertainty in a measurement is called ‘error’. It is difference between the measured and the true values of a physical quantity.

In general, the error in measurement can be broadly classified as

• Systematic errors
• Random errors
• Least count errors

Systematic errors

Those errors that tend to be in one direction, either positive or negative. These errors are known errors and these can be corrected.

Some of the sources of systematic errors are

• Instrument errors
  It arises from the errors due to imperfect design or calibration of the measuring instrument.
• Imperfect technique
  Imperfection in experimental technique or procedure causes these type of errors.
• Personnel errors
  These errors are the results of human mistakes made while measuring a physical quantity.

Random errors

The random errors are those errors, which occur irregularly and hence are random with respect to sign and size. These errors arise due to random and unpredictable fluctuations in experimental conduction, personal errors by the observer.

These errors are unknown errors, yet they can be corrected by considering the average of - number of measurements of the same quantity.

Least count errors

The smallest value that can be measured by the measuring instrument is called its least count. All the measured values are good only up to this value.

The least count error is the error associated with the resolution of the instrument.

Accuracy of measurement is related to the systematic errors and its precision is related to the random errors, which include least count error.

Absolute Error

The magnitude of the difference between the true value of the quantity and the individual measurement value is called the absolute error of the measurement.

Absolute error = Measured value - True value

Consider a physical quantity is measured by taking repeated number of observations say a₁, a₂, a₃, a₄ ………a_n

Measured values = a₁, a₂, a₃, a₄,……………,a_n

True value = a₁ + a₂ + a₃ + a₄ + ... + a_nn

Then the errors in the individual measurement values are

Δa₁ = a_mean - a₁,

Δa₂ = a_mean - a₂,

...............

Δa_n = a_mean - a_n

The arithmetic mean of all the absolute errors is the mean absolute error of the value of the physical quantity.

Mean absolute error = (|Δa₁| + |Δa₂| + |Δa₃| + ... + |Δa_n|)n

Δa_mean = 

∑ |Δa_n|n 

Relative Error

The relative error is the ratio of the mean absolute error to the mean value of the quantity measured.

It is also known as fractional error.

Relative error = ± Δa_meana_mean

When the relative error is expressed in percentage, it is called the percentage error.

Percentage error = ± Δa_meana_mean × 100

Sign of error

Error can have either positive or negative sign or it can also have both.

Example − Error = ± 3

Calculating absolute and Relative error

Refractive index of water was measured as 1.29, 1.33, 1.34, 1.35, 1.32, 1.36, 1.30 and 1.33, calculate

• Mean value
• Mean absolute error
• Fractional error
• Percentage error
• Value with error

Solution

Mean value

Mean = ∑an

a_mean = 10.628 = 1.328 = 1.33

Mean absolute value

Errors in the individual measurement values are

Δa₁ = a₁ - a_mean,

Δa₂ = a₂ - a_mean,

.............

Δa_n = a_n - a_mean

So,

Δa₁ = 1.29 - 1.33 = -0.04

Δa₂ = 1.33 - 1.33 = 0.00

Δa₃ = 1.34 - 1.33 = 0.01

Δa₄ = 1.35 - 1.33 = 0.02

Δa₅ = 1.32 - 1.33 = -0.01

Δa₆ = 1.36 - 1.33 = 0.04

Δa₇ = 1.30 - 1.33 = -0.03

Δa₈ = 1.33 - 1.33 = 0.00

Δa_mean = ∑an

Δa_mean = 0.158 = 0.019 ≈ 0.02

Fractional error

Fractional error = Δa_meana_mean

Fractional error = 0.0191.33 = 0.01428 ≈ 0.014

Percentage error

Percentage error = Fractional error × 100

Percentage error = 0.01428 × 100 = 1.428% = 1.4%

Value with error

Value with error = 1.33 ± 0.2

or

Value with error = 1.33 ± 1.4%

Propagation of Error addition

Consider two measurement – A with error ΔA and B with error ΔB that are to be added.

Let C be the resulting measurement with error ΔC

C = A + B

ΔC = ±(ΔA + ΔB)

Note − Error are always added, they are never cancelled.

Example 1

Distance between A to B is measured as 112 ± 2 and the distance between B to C is measured as 150 ± 5

Then distance between A and C is given by −

(112 ± 2) + (150 ± 5) = 262 ± 7

Example 2

Distance between A to C is measured as 272 ± 5 and the distance between A to B is measured as 112 ± 2

Then distance between B and C is given by −

(272 ± 5) - (112 ± 2) = 163 ± 7

Propagation of Error Product

When two measurements with error are multiplied then their product will have fractional error equal to sum of fractional errors of the numbers.

Consider two measurement – a with error Δa and b with error Δb that are to be added.

Let C be the resulting measurement with error ΔC

C = a × b

ΔCC = ±(Δaa + Δbb)

∴ ΔC = ±(Δaa + Δbb) × C

Example 1

Two quantities are measured as 25 ± 2 and 20 ± 1

Then the product of these two measurements is given by −

(25 ± 2) × (20 ± 1)

C = 25 × 20 = 500

ΔCC = ±(225 + 120) = 40 + 25500 = 13100

∴ ΔC = 13100 × C = 13100 × 500 = 65

Hence,

product = 500 ± 65

Or

Product = 500 ± 13%

Example 2

Mass of a liquid is measured as 520 ± 5 and the volume is measured as 130 ± 2

Then density (d) is given by −

520 ± 5130 ± 2

d_m = 520130 = 4

Δd_md = 5520 + 2130 = 13520

Δd_m = 13100 × d = 13520 × 4 = 0.1

Hence,

Density = 4 ± 0.1

Or

Density = 4 ± 2.5%

Propagation of Error Power

When measurements are raised to a power, the fractional error in a physical quantity raised to the power k is the k times the fractional error in the individual quantity.

Consider a measurement – a with error Δa, that is to be raised to a power b.

Let C be the resulting measurement with error ΔC

C = (a ± Δa)^b

ΔCC = b(Δaa)

∴ ΔC = b(Δaa)C

Example 1

Radius is measured as 5 ± 1

Then the area of circle is given by −

π(5 ± 1)²

A = 25π

ΔAA = ±2 ×15 = 25

∴ ΔA = 25 × A = 25 = 25 π = 10 π

Hence,

Area = 25π ± 10 π

Or

Area = 25π ± 40%

Propagation of Error Numerical

A physical quantity P is related to 4 observations a, b, c, d as follows −

P = a³b²√c d

The percentage error in a, b, c, d is 1%, 3%, 4% and 2% respectively. What is the percentage error in the quantity P.

Solution

% error in P

= % error in a³ + % error in b² + % error in √c + % error in d

∴ % error in P = 3 × 1% + 2 × 3% + 12 × 4% + 1 × 2%

∴ % error in P = 13%

Dimensions of a Physical Quantity

Dimensions of physical quantity is the fundamental quantities involved in the given quantity. The number of these dimensions expressed as the number how many times the fundamental quantity is multiplied i.e. all the physical quantities represented by derived units can be expressed in terms of some combination of seven fundamental or base quantities.

Fundamental quantities and their dimensions −

Fundamental Quantity	Dimension	Unit
Mass	M	Kilogram(kg)
Length	L	Meter(m)
Time	T	Second(s)
Temperature	K	Kelvin(K)
Electric Current	A	Ampere(A)

DIMENSIONAL FORMULA

The power (exponent) of base quantity that enters into the expression of a physical quantity, is called the dimensional formula of the quantity.

To make it clear, consider the physical quantity density.

Density = massvolume = ML³ = |M L^-3|

Here,

massvolume = physical formula And

M L^-3 = dimensional formula.

Similarly,

• Dimensional formula for speed is given by −
  Speed = |L T ^-1|
• Dimensional formula for acceleration is given by −
  Acceleration = |L T ^-2|
• Dimensional formula for force is given by −
  Force = |M L T ^-2|

• Physical quantity	Physical formula	Dimensions
  Area	L × W	[M0 L2 T0]
  Volume	L × L × L	[M0 L 3 T 0]
  Density	massvolume	[M 1 L-3 T0]
  Speed	DistanceTime	[M 0 L 1 T -1]
  Acceleration	SpeedTime	[M 0 L1 T -2]
  Momentum	m × V	[M 0 L1 T -1]
  Force	m × a	[M 1 L1 T -2]
  Work	Force × Displacement	[M 1 L2 T -2]
  Energy	12 mV2	[M 1 L2 T -2]
  Pressure	ForceArea	[M 1 L-1 T -2]
  Surface tension	ForceLength	[M 1 L0 T -2]

  Converting Units

• Any physical quantity can be expressed in different measuring systems such as CGS, MKS, etc.

  Q = n₁ U₁ = n₂ U₂

  Where,

  Q = Physical quantity

  n₁ U₁ = Measurement in unit system - U₁

  n₂ U₂ = Measurement in unit system - U₂

  So to convert the measurement of a physical quantity from one system to another, we derive a formula form the above equation −

  n₂ = n₁ × U₁U₂

  n₂ = n_{1 ×}[M₁^a L₁^b T₁^c][M₂^a L₂^b T₂^c]

  n₂ = n₁ × [M₁M₂]^a [L₁L₂]^b [T₁T₂]^c

  Example

  Convert 36 km.Hr⁻¹ into m.s^-1

  n₂ = n₁ × [M₁M₂]^a [L₁L₂]^b [T₁T₂]^c

  n₂ = 36 × [kmm]¹ [Hrs]^-1

  n₂ = 36 × [1000 m1 m]¹ [3600 s1 s]^-1

  n₂ = 36 × [1000 m]¹ [3600 s]^-1

  n₂ = 36 × 1000 m3600 s

  n₂ = 36 × 5 m18 s = 10 m.s^-1

  Hence,

  36 km.Hr^-1 = 10 m.s^-1
• Checking Correctness of Formulas

• Dimensional analysis has the following steps −

  Step 1 − Write the relation with assumed powers and an arbitrary constant.

  Step 2 − Writing dimensions of each quantity.

  Step 3 − Compare similar dimensions.

  Step 4 − Substitute the power of dimensions in the equation formed in step 1.

  Example

  Time of oscillation of pendulum depends upon length of pendulum, mass of bob and acceleration due to gravity.

  To form a new formula we will use dimensional analysis.

  Step 1 − Write the relation with assumed powers and an arbitrary constant.

  T α m^a l^b g^c

  T = k m^a l^b g^c

  Where,

  T = Time of oscillation

  k = Arbitrary constant

  m = Mass of bob

  l = Length of pendulum and

  g = Acceleration due to gravity

  Step 2 − Writing dimensions of each quantity.

  M⁰ L⁰ T¹ = [M]^a [L]^b [L T^-2]^c

  M⁰ L⁰ T¹ = M^a L^b L^c T^-2c

  M⁰ L⁰ T¹ = [M]^a [M]^a [L]^b+c [T]^-2c

  Step 3 − Compare similar dimensions.

  After comparing similar dimensions, we get −

  a = 0,b + c = 0, -2c = 1

  a = 0,b = 12, c = -12

  Step 4 − Substitute the power of dimensions in the equation formed in step 1.

  T = kl ^1/2 g^-1/2

  T = k√lg

• Dimensional Analysis Making new formula

• Dimensional analysis has the following steps −

  Step 1 − Write the relation with assumed powers and an arbitrary constant.

  Step 2 − Writing dimensions of each quantity.

  Step 3 − Compare similar dimensions.

  Step 4 − Substitute the power of dimensions in the equation formed in step 1.

  Example

  Time of oscillation of pendulum depends upon length of pendulum, mass of bob and acceleration due to gravity.

  To form a new formula we will use dimensional analysis.

  Step 1 − Write the relation with assumed powers and an arbitrary constant.

  T α m^a l^b g^c

  T = k m^a l^b g^c

  Where,

  T = Time of oscillation

  k = Arbitrary constant

  m = Mass of bob

  l = Length of pendulum and

  g = Acceleration due to gravity

  Step 2 − Writing dimensions of each quantity.

  M⁰ L⁰ T¹ = [M]^a [L]^b [L T^-2]^c

  M⁰ L⁰ T¹ = M^a L^b L^c T^-2c

  M⁰ L⁰ T¹ = [M]^a [M]^a [L]^b+c [T]^-2c

  Step 3 − Compare similar dimensions.

  After comparing similar dimensions, we get −

  a = 0,b + c = 0, -2c = 1

  a = 0,b = 12, c = -12

  Step 4 − Substitute the power of dimensions in the equation formed in step 1.

  T = kl ^1/2 g^-1/2

  T = k√lg

• Dimensional Analysis Making new formula - 2

• A particle of mass `m` is moving in circular path with constant speed `V` and radius `r`. Find expression of its centripetal force F.

  Solution

  To form a new formula we will use dimensional analysis.

  Step 1 − Write the relation with assumed powers and an arbitrary constant.

  F = km^a V^b r^c

  Where,

  F = Centripetal force

  k = Arbitrary constant

  m = Mass of particle

  V = Speed of particle and

  r = radius of circular path.

  Step 2 − Writing dimensions of each quantity.

  M¹ L¹ T^-2 = [M]^a [L T^-1]^b [L]^c

  M¹ L¹ T^-2 = [M]^a [L]^b+c [T]^-b

  Step 3 − Compare similar dimensions.

  After comparing similar dimensions, we get −

  a = 1, b + c = 1, -b = -2

  a = 1, b = 2, c = -1

  Step 4 − Substitute the power of dimensions in the equation formed in step 1.

  F = km V² r^-1

  F = km V²r

• Dimensional Analysis Making new formula - 3

• Frequency of a vibrating string (v) depends on length of string ( l ), Tension in string ( T ) and Linear mass density ( m ).

  Solution

  To form a new formula we will use dimensional analysis.

  Step 1 − Write the relation with assumed powers and an arbitrary constant.

  v = kl^a T^b m^c

  Where,

  v = Frequency

  k = Arbitrary constant

  l = Length of string

  T = Tension in string and

  m = Linear mass density.

  Step 2 − Writing dimensions of each quantity.

  Frequency = [T^-1]

  Tension (force) = [M L T^-2]

  Linear mass density = [M L^-1]

  Substituting the values −

  M⁰ L⁰ T^-1 = [L]^a [M L T^-2]^b [M L^-1]^c

  M⁰ L⁰ T^-1 = [M]^{b + c} [L]^{a + b -c} [T]^-2b

  Step 3 − Compare similar dimensions.

  After comparing similar dimensions, we get −

  b + c = 0, a + b -c = 0, -2b = -1

  a = -1, b = 12, c = -12

  Step 4 − Substitute the power of dimensions in the equation formed in step 1.

  v = kl^-1 T^1/2 m^-1/2

  v = kl √Tm
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