Units and Measurement

Units and Measurement are essential for expressing physical quantities in a standardized way. The International System of Units (SI) includes base units like meter (m) for length, kilogram (kg) for mass, and second (s) for time. Measurements can involve scalar quantities (e.g., mass) or vector quantities (e.g., velocity). Dimensional analysis helps in converting and verifying units. Accuracy is how close a measurement is to the true value, while precision is the consistency of repeated measurements. Errors can be systematic (due to equipment or methods) or random (due to unpredictable factors).

1.0Fundamental Forces And Its Range

1. Gravitational Force:Gravitational force is the weakest force in nature. It is the force of mutual attraction between two objects due to their masses. This force is universal in nature.

2.Electromagnetic Force:Electromagnetic force is the force between charge particles. When charges are in motion, they generate magnetic fields, which in turn exert a force on other moving charges. Since electric and magnetic effects are generally interrelated, this interaction is referred to as the electromagnetic force.

3. Strong Nuclear Force:The strong nuclear force holds protons and neutrons together in the nucleus. Without this attractive force, the nucleus would be unstable due to the electric repulsion between protons.. The strong nuclear force is the strongest of all fundamental forces. It is charge independent.It is equal for protons and neutrons.Its range is extremely small of the order nuclear dimensions (10-15 10^-15 m).

4. Weak Nuclear Force :The weak nuclear force is involved only in specific nuclear processes, such as the β-decay of a nucleus. During β-decay, the nucleus emits an electron and an uncharged particle known as the anti-neutrino. While the weak nuclear force is not as weak as gravity, it is significantly weaker than the strong nuclear force. Its range is extremely small, on the order of 10-16 10^{-16} meters.

Range of Fundamental Forces

Gravitational Force→Infinite
Electromagnetic Force→ Infinite
Strong Nuclear Force→ Short (Nuclear Size: $10^{-15}$ M)
Weak Nuclear Force→Very Short ( $10^{-16}$ M)

Relative Strength

2.0Physical Quantities

• Physical quantities are those quantities used to describe the laws of physics that can be measured.

Classification On Basis Of Directional Properties

Fundamental or Base Quantities: The quantity which does not depend upon other quantities for their complete definition. 

(1) Mass

(2) Length 

(3) Time 

(4) Temperature

(5) Current 

(6) Luminous Intensity 

(7) Amount of Substance

Derived Quantities:The quantity which can be described in terms of the fundamental quantity.

(1) Area 

(2) Force 

(3) Density

3.0Dimensions and Dimensional Formula

Dimensions:It is a physical quantity are the powers to which the base quantity are raised to express that quantity.

Dimensional Formula:

The expressions which represent how and which of the base quantities are included in that quantity.

NOTE:It is expressed by placing the symbols for the base quantities with the correct powers inside square brackets.

$Mass\to [{\mathbf{M}}^1{\mathbf{L}}^0{\mathbf{T}}^0]$

$Momentum\to [{\mathbf{M}}^1{\mathbf{L}}^1{\mathbf{T}}^{-1}]$

$Force\to [{\mathbf{M}}^1{\mathbf{L}}^1{\mathbf{T}}^{-2}]$

Dimensional Equation:The equation formed by equating a physical quantity to its dimensional formula is known as a dimensional equation.

$Mass\to [{\mathbf{M}}^1{\mathbf{L}}^0{\mathbf{T}}^0]$

$Length\to [{\mathbf{M}}^0{\mathbf{L}}^1{\mathbf{T}}^0]$

$Time\to [{\mathbf{M}}^0{\mathbf{L}}^0{\mathbf{T}}^1]$

$Temperature\to [{\mathbf{M}}^0{\mathbf{L}}^0{\mathbf{T}}^0{\mathbf{K}}^1]$

$Current\to [{\mathbf{M}}^0{\mathbf{L}}^0{\mathbf{T}}^0{\mathbf{A}}^1]$

$LuminousIntensity\to [{\mathbf{M}}^0{\mathbf{L}}^0{\mathbf{T}}^0\mathbf{C}{\mathbf{d}}^1]$

$AmountofSubstance\to [{\mathbf{M}}^0{\mathbf{L}}^0{\mathbf{T}}^0\mathbf{mo}{\mathbf{l}}^1]$

Dimensions of differential coefficients and integrals

$\left[\frac{d^{n}y}{d{x}^n}\right]=\left[\frac{y}{x^n}\right]and\left[\int ydx\right]=[yx]$

4.0Rule of Dimensions

Only SAME physical quantities can be added or subtracted.

A + B = C – D

Dimensionless Quantities

Dimensionless Quantities are:

• Ratio of physical quantities with same dimensions.

• All mathematical constants.

• All standard mathematical functions and their inputs (exponential, logarithmic, trigonometric & inverse trigonometric).

5.0Conversion between System of Units

Magnitude = numeric value (n) × unit (u) = constant

$n_1{u}_1=n_2{u}_2$

$n_2=n_1(\frac{u_1}{u_2})$

So, if a quantity is represented by [MaLbTc] then:

$n_2=n_1(\frac{M_1}{M_2}{)}^a(\frac{L_1}{L_2}{)}^b(\frac{T_1}{T_2}{)}^c$

$n_1=\text{numerical value in I system}$

$n_2=\text{numerical value in II system}$

$M_1=\text{unit of mass in I system}$

$M_2=\text{unit of mass in II system}$

$L_1=\text{unit of Length in I system}$

$L_2=\text{unit of Length in II system}$

$T_1=\text{unit of time in I system}$

$T_2=\text{unit of time in II system}$

6.0Law of Homogeneity and Dimensions of Unknown Quantities

Applications of Dimensional Formula

To verify the dimensional accuracy of a given physical equation:

If in a given relation, terms on both the sides have the same dimensions, then the relation is dimensionally correct. This is known as the Principle of Homogeneity of Dimensions.

To verify the dimensional correctness of a given physical relationship:

If in a given relation, the terms on both the sides have the same dimensions, then the relation is dimensionally correct. This is known as the principle of homogeneity of dimensions.

Relation among Physical Quantities

To derive relationships between different physical quantities: Using the same principle of homogeneity of dimensions new relations among physical quantities can be derived if the dependent quantities are known.

7.0Limitations of Dimensional Analysis

1.In Mechanics the formula for a physical quantity depending on more than three physical quantities cannot be derived. It can only be checked.

2. This method can be used only if the dependency is of multiplication type.

3. The formulae containing exponential, trigonometrical and logarithmic functions cannot be derived using this method.

4. Formulae containing more than one term which are added or subtracted like $s=ut+\frac{1}{2}a{t}^2$ also cannot be derived.

5. The relation derived from this method gives no information about the dimensionless constants.

6. If dimensions are given, physical quantities may not be unique as many physical quantities have the same dimensions.

7. It gives no information whether a physical quantity is a scalar or a vector.

8.0Significant Figures, Rounding off and Order of Magnitude

Significant Figures or Digits

Significant figures (SF) in a measurement are the figures or digits that are known with certainty plus one that is uncertain (i.e. Last digit).

The significant figures in a measured value of a physical quantity indicate the number of digits that are considered reliable. The greater the number of significant figures in a measurement, the higher its accuracy, and vice versa.

Rules to find out the number of significant figures

I Rule : All the non-zero digits are significant e.g. 1984 has 4 SF.

II Rule : All the zeros between two non-zero digits are significant. e.g. 10806 has 5 SF.

III Rule : All the zeros to the left of the first non-zero digit are not significant. e.g.00108 has 3 SF.

IV Rule : If a number is less than 1, the zeros to the right of the decimals point but to the left of the first non-zero digits are not considered significant. For example, the number 0.002308 has 4 significant figures.

V Rule : The trailing zeros (zeros to the right of the last non-zero digit) in a number with a decimal point are significant. e.g. 01.080 has 4 SF.

VI Rule : The trailing zeros in a number without a decimal point may not be significant e.g. 010100 has 3 SF.

VII Rule : When the number is expressed in exponential form, the exponential term does not affect the number of S.F. For example, in

$x=12.3=1.23\times 10^1=0.123\times 10^2=0.0123\times 10^3=123\times 10^{-}1$ Each term has 3 SF only.

Rules for arithmetical operations with significant figures

I Rule :In addition or subtraction, the result should have the same no. of decimal places as the term with the fewest decimal place in the operation. For example, in the calculation 12.587−12.5=0.08712.587 - 12.5 = 0.087, the result is rounded to 0.10.1 because the second term has only one decimal place.

II Rule : In multiplication or division, the number of SF in the product or quotient is the same as the  smallest number of SF in any of the factors. e.g. 2.4 × 3.65 = 8.8

Rounding off

To represent the result of any computation containing more than one uncertain digit, it is rounded off to an adequate number of significant figures.

Rules for rounding off the numbers:

I Rule : If the digit to be rounded off is more than 5, then the preceding digit is increased by one. e.g. 6.87 ≈ 6.9

II Rule : If the digit to be rounded off is less than 5, then the preceding digit is left unchanged. e.g. 3.94 ≈ 3.9

III Rule : If the digit to be rounded off is 5 then the preceding digit is increased by one if it is odd and is left unchanged if it is even. e.g. 14.35 ≈ 14.4 and 14.45 ≈ 14.4

Order of Magnitude

The order of magnitude of a quantity is the power of 10 needed to express that quantity. This power is determined after properly rounding the value of the quantity. When rounding, if the last digit is less than 5, it is ignored, while if it is 5 or greater, the digit is increased by one

• When a number is divided by 10x (where x is the order of magnitude of the number), the result will always fall between 0.5 and 5, i.e., 0.5 ≤N/$10^x$< 5.

9.0Types and Representation of Errors

Errors in Measurement

Every measurement made with a measuring instrument carries some degree of uncertainty, which is referred to as an error. The error in a measurement is the difference between the true value and the measured value of a quantity.

Error = True value – Measured value

Types of Errors

1. Systematic Errors:- Systematic errors are errors with known causes, and they can be either positive or negative. Since their causes are understood, these errors can be minimized. Systematic errors can be further classified into three types:

(i) Instrumental Errors : Error due to imperfect design or calibration of the measuring instrument.

(ii) Environmental Errors : These errors result from variations in outer environmental conditions, such as temperature, pressure, humidity, dust, vibrations.

(iii) Observational Errors :These errors occur due to faulty setup of the apparatus or Carelessness in recording observations.

2. Random Errors: These errors occurred from unknown factors, making them unpredictable and variable in both magnitude and direction. Since their exact causes are not fully understood, they cannot be entirely eliminated.

3. Gross Errors:-Gross errors occur due to human mistakes or carelessness in taking readings, as well as errors in calculating or recording measurement results.

10.0Representation Of Errors

1.Absolute Error ($\Delta a$): The  difference between the true value and an individual measured value of the quantity.

Absolute Error = True Value – Measured Value

$\Delta a=a_T-a$

If the true value of a quantity is not given then the mean of all the measured values is taken.

$a_m=\frac{1}{n}{\sum }_{i=0}^n{a}_i$

2.Mean Absolute Erroram: The arithmetic mean of all the absolute errors(magnitudes) is defined as the final or mean absolute error.

$\text{Mean Absolute Error}=(\Delta a{)}_m=\frac{1}{n}{\sum }_{i=0}^n|a_i|$

3.Relative Or Fractional Error:The ratio of the mean absolute error to either the true value or the average value of the measured quantity.

$\text{Relative Error}=\frac{\text{Mean Absolute Error}}{\text{True Value}}\Rightarrow \text{Relative Error}=\frac{(\Delta a{)}_m}{a_m}$

4.Percentage Error: When the relative error is represented as a percentage, it is referred to as the percentage error.

$\text{Percentage Error}=\text{Relative Error}\times 100\%$

$PercentageError=\frac{\text{Mean Absolute Error}}{\text{True Value}}\times 100\%$

5.Least Count: It is the smallest value of a physical quantity that can be calculated accurately by an instrument.

Least Count Error: This is related to the resolution of the measuring instrument.If the instrument has unknown least count the absolute error is taken to be equal to the least count unless otherwise stated.

Rule 1: - Addition or Subtraction of Quantities:

The maximum absolute error in the sum or difference of two quantities is equal to the sum of the absolute errors in each of the individual quantities.

X=A+B or X=A-B than maximum absolute error in X is $\Delta X=\Delta A+\Delta B$

Maximum percentage error = $\frac{\Delta X}{X}\times 100\%$

Result will be written as $X\mp \Delta X$ ((in terms of absolute error)

$X\mp \left(\frac{\Delta X}{X}\times 100\%\right)$(in terms of PercentageError)

Rule 2:Multiplication or Division of Quantities:

The maximum fractional or relative error in the product or quotient of quantities is the sum of the fractional or relative errors of the individual quantities involved.

If  $X=AB\text{or}X=\frac{A}{B}\text{then}\frac{\Delta X}{X}=\mp \left(\frac{\Delta A}{A}+\frac{\Delta B}{B}\right)$

Rule 3:The maximum fractional error in a quantity raised to power n is n times the fractional error in the quantity itself

1. If $X=A^{n}then\frac{\Delta X}{X}=n\left(\frac{\Delta A}{A}\right)$

2. If $X=A^p{B}^q{C}^{r}then\frac{\Delta X}{X}=\left[p\left(\frac{\Delta A}{A}\right)+q\left(\frac{\Delta B}{B}\right)+r\left(\frac{\Delta C}{C}\right)\right]$

3. If $X=\frac{A^p{B}^q}{C^r}then\frac{\Delta X}{X}=\left[p\left(\frac{\Delta A}{A}\right)+q\left(\frac{\Delta B}{B}\right)+r\left(\frac{\Delta C}{C}\right)\right]$

11.0Vernier Callipers

• A measuring instrument used to measure linear dimensions, the Vernier caliper is also employed to measure the diameters of round objects using its measuring jaws. The Vernier scale, invented by French mathematician Pierre Vernier in 1631, enhances the accuracy of measurements. The primary advantage of the Vernier caliper over the main scale is its ability to provide highly accurate and precise measurements.

Least Count of Vernier Callipers

Let the size of one main scale division (M.S.D.) be M units, and the size of one Vernier scale division (V.S.D.) be V units. Additionally, assume that the length of 'a' main scale divisions is equal to the length of 'b' Vernier scale divisions.

$aM=bV\Rightarrow V=\frac{a}{b}M$

∴ $M-V=M-\frac{a}{b}M=\left(\frac{b-a}{b}\right)M$

$[M\to MSD,V\to VSD]$

$L.C=M-V=\left(\frac{b-a}{b}\right)M$

Least Count(x)=Difference between MSD & VSD

10 VSD=9 MSD

1 VSD=0.9 MSD

x=1 MSD-1 VSD=1 MSD -0.9 MSD=0.1 MSD

Generally,1 MSD=1mm

x=0.1 mm

Measurement with Vernier Calliper

Reading=Main Scale Reading+Vernier Scale Reading

Reading=MSR+V S coincided division with MS ✕  LC

Zero Error

If the zero marking of the main scale and Vernier calliper do not coincide, necessary correction has to be made for this error which is known as zero error of the instrument.                                  

1. Positive Zero Error :If the zero of the Vernier scale is to the right of the zero on the main scale, the zero error is considered positive. To obtain the corrected value, the zero error must always be subtracted from the reading.

Positive Zero Error Correction

READING = (Main Scale Reading) + (Vernier Scale Reading) – ZE

Positive zero error = VS Coincided division with MS × LC

2. Negative Zero Error

If the zero of the Vernier scale is to the left of the zero of the main scale the

zero error is said to be negative.

The zero error is always subtracted from the reading to get the corrected

value.

Negative Zero Error Correction

Negative zero error = –(Total VSD – VS Coincided division with MS) × LC

12.0Screw Gauge

• A screw gauge is the most accurate instrument for measuring spherical or cylindrical objects. However, an intricately calibrated screw gauge can be challenging for untrained hands to use effectively. The goal of this article is to familiarize individuals with the fundamentals of using a screw gauge.

Pitch of Screw Gauge:

The pitch of the instrument is the distance between two consecutive threads of the screw which is equal to the distance moved by the screw due to one complete rotation of the cap.

Least Count of Screw Gauge

The minimum (or least) measurement (or count) of length is equal to one division on the main scale which is equal to pitch divided by the total cap divisions.

•  Pitch of screw gauge = P
•  Number of division on circular scale = N
• $\text{ Least count of screw gauge}=\frac{\text{Pitch}}{\text{Total number of divisions on circular scale}}$

Length as measured by Screw Gauge

The formula for measuring the length is, L = (n × pitch) + (f × least count)

Where, n = main scale reading;f=Circular scale reading 

Zero Error in Screw Gauge

If there is no object between the studs (i.e., the studs are in contact), the screw gauge should ideally show a zero reading. However, due to the presence of extra material on the studs, the gauge may give a small positive reading even when there is no object. This excess reading is referred to as zero error.

Calculation of zero error for screw gauge: -

Positive zero error = CS Coincided division with MS × LC

Negative zero error = –(Total CSD – CS Coincided division with MS) × LC

Correct reading = (Reading) – (zero error)

13.0Spherometer

14.0Parts of Spherometer

1.Main Scale

2.Circular Scale

Pitch of screw gauge = P

Number of division on circular scale = N

Least count of Spherometer = $\frac{P}{N}$

15.0Accuracy and Precision

Accuracy: The extent of closeness between the measured value and the true value.

It refers to the capability of an instrument to measure the true or correct value. It represents how close the measured value is to the standard or true value. To achieve accuracy, it is important to take multiple measurements, as smaller readings help minimize calculation errors.

Precision: How consistent our results are, regardless of proximity to true value.

The precision of a substance refers to the closeness of two or more measurements to each other. For example, if you weigh a substance five times and obtain 4.3 kg each time, your measurements are highly precise, but this does not necessarily mean they are accurate. Precision is independent of accuracy.

Solved Examples

Q-1 Find the dimensions of unknown quantities p, q in the following:

$U=p\cos \left(qt+\frac{\pi }{6}\right)$ where, U: - Potential energy, t :- Time

Solution:

Dimensions of potential energy $[U]=[M^1{L}^2{T}^{-2}]$

Inside cos function →dimensionless

qt→Dimensionless

$qt=[M^0{L}^0{T}^0]$

$q=\frac{[M^0{L}^0{T}^0]}{[T^1]}=[T^{-1}]$

$U=p=[M^1{L}^2{T}^{-2}]$

Q-2.Given that the time of revolution T of a satellite around the Earth depends on the universal gravitational constant G, the mass of the Earth M, and the radius of the circular orbit RR, derive an expression for T using dimensional analysis.

Solution:

$[T]\propto [G{]}^a[M{]}^b[R{]}^c$

$\Rightarrow [M^0][L^0][T^1]=[M{]}^{-a}[L{]}^{3a}[T{]}^{-2a}\times [M{]}^b[L{]}^c=[M{]}^{b-a}[L{]}^{c+3a}[T{]}^{-2a}$

$[T]1=-2a\Rightarrow a=-\frac{1}{2}$

$[M]0=b-a\Rightarrow b=a=-\frac{1}{2}$

$[L]0=c+3a\Rightarrow c=-3a=\frac{3}{2}$

$[T]\propto [G{]}^{-1/2}[M{]}^{-1/2}[R{]}^{3/2}\Rightarrow T\propto \sqrt{\frac{R^3}{GM}}$

$T=2\pi \sqrt{\frac{R^3}{GM}}$

Q-3.Convert 1 newton (SI unit) into dyne (CGS unit)?

Solution:

$n_1{u}_1=n_2{u}_2\Rightarrow n_2=n_1\left(\frac{u_1}{u_2}\right)$

$n_2=1\left[\frac{M_1}{M_2}\right]\left[\frac{L_1}{L_2}\right]{\left[\left(\frac{T_1}{T_2}\right)\right]}^{-2}=1\left[\frac{\text{kg}}{\text{gm}}\right]\left[\frac{\text{m}}{\text{cm}}\right]{\left[\left(\frac{\text{s}}{\text{s}}\right)\right]}^{-2}$

$n_2=1\left[\frac{1000\text{gm}}{\text{gm}}\right]\left[\frac{100\text{cm}}{\text{cm}}\right]{\left[\left(\frac{\text{s}}{\text{s}}\right)\right]}^{-2}=10^5$

Q-4.If the jaws of the vernier caliper are in contact with each other, calculate the zero error of the vernier caliper, given that the Vernier Scale Division (VSD) is 3 and the least count is 0.1 mm.

Solution:

When the jaws of the vernier caliper are in contact with each other, the least count of the vernier caliper is defined as:

LC = MSD – VSD

LC = 0.1 mm

The main scale reading, MSR = 0 mm

The vernier scale reading, VSR = 3

Therefore, Zero error = MSR + VSR × LC = 0 + 3 × 0.1 = 0.3 mm

Q-5.The screw gauge has a pitch of 1 mm, and there are 100 div. on the circular scale. When measuring the diameter of a sphere, the linear scale reads 6 divisions, while 40 divisions of the circular scale align with the reference line. Calculate the diameter of the sphere.

Solution:

Least count of the screw gauge is L.C = pitch / number of circular divisions = 1/100 = 0.01mm

Linear scale reading (LSR) = 6 × 1mm = 6mm

Circular scale reading (CSR) = 40 × 0.01mm

Diameter of the sphere = LSR + (CSR × L.C) = 6 + (40 × 0.01) = 6.4mm