# UNITS AND MEASUREMENTS – PART 2

## Significant figures

In a measured value the reliable digits and the first uncertain digit are known as significant figures.

## Importance of significant figures

Significant figures indicate the precision of the instrument. Number of significant figures does not change if we measure a physical quantity in different units.

## Rules for determining the significant figures.

• All non-zero digits are significant. Ex: 2341 m → has 4 significant figures. 14.3m → has 3 significant figures.
• All the zeros between two non-zero digits are significant. Ex: 308 m → has 3 significant figures. 23.08 m → has 4 significant figures.
• In a number without a decimal point trailing or terminal zeros are NOT significant. Ex: 12300 m → has 3 significant figures. 104000 m → has 3 significant figures.
• In numbers with a decimal point, trailing or terminal zeros are significant. Ex: 4.700 m → has 4 significant figures. 23.04000 m → has 7 significant figures.
• If the number is less than 1, then zeros on the right of the decimal point but to the left of the first non-zero digit are NOT significant. Ex: 0.067 m → has 2 significant figures. 0.0003080 m → has 4 significant figures.

## Scientific notation

In this notation, every number is expressed as 𝑎 × 10𝑏 , where 𝑎 is a number between 1 and 10 is called base number and 𝑏 is any positive or negative exponent of 10. The power of 10 is irrelevant to the determination of significant figures. However, all zeros appearing in the base number in the scientific notation are significant. Ex: 4.700× 102𝑚 has 4 significant figures. 4.700× 10−3𝑚 has 4 significant figures.

## Rules for arithmetic operations with significant figures

(i) When numbers are added or subtracted, the number of decimal places in the final result should be equal to the smallest number of decimal places of any term. Ex: (a) 436.32 + 227.2 = 663.5 (but not 663.52) (b) 0.3074 – 0.304 = 0.003 (but not 0.0034)

(ii) In multiplication or division, the number of significant figures in the final result should be equal to the number of significant figures in the quantity having the smallest number of significant figures.

Ex: (a) 1.21 ×0.12 = 0.14 (but not 0.1452) (b) 5.74 1.2 = 4.8 (but not 4.7833)

## Rounding of the uncertain digits

(i) If the digit to be dropped in a number is less than 5, then the preceding digit remains unchanged. Ex: 1.344 is rounded as 1.34

(ii) If the digit to be dropped in a number is greater than 5, then the preceding digit is raised by 1. Ex: 1.346 is rounded as 1.35

(iii) If the digit to be dropped in a number is 5, then

(a) the preceding digit remains unchanged if it is EVEN. Ex: 1.345 is rounded as 1.34

(b) the preceding digit is raised by 1, if it is ODD. Ex: 1.375 is rounded as 1.38.

## Dimensions of physical quantities

Dimensions of a physical quantity are the power to which the base quantities are raised to represent the physical quantity.

Note: Dimensions of a physical quantity explain its relationship with fundamental quantities. All the derived physical quantities can be expressed in terms of some combination of seven fundamental quantities. Dimensions of a physical quantity are denoted with square brackets.

## Symbols for dimensions of fundamental quantities

Ex: Dimensions of force are MLT-2. Hence force has one dimension in mass, one dimension in length, and -2 dimensions in time.

## Dimensional formula

The expression of a physical quantity in terms of the base quantities is called the dimensional formula.

Ex: Dimensional formula of volume is [𝑀0𝐿3𝑇0], Dimensional formula of Speed is [𝑀0𝐿𝑇−1]

## Dimensional equation

An equation obtained by equating a physical quantity with its dimensional formula is called a dimensional equation. Ex: [𝐹] = [𝑀𝐿𝑇−2], [𝑉] = [𝑀0𝐿3𝑇0].

Different types of variables and constants

## Dimensional variable

The physical quantities which possess dimensions and have variable values are called dimensional variables. Ex: Area, volume, speed, velocity, acceleration, momentum, force etc.

## Dimensionless variables

The physical quantities which have no dimensions but have variable values are called dimensionless variables. Ex: Angle, specific gravity, strain, 𝑠𝑖𝑛 𝜃, 𝑐𝑜𝑠 𝜃, 𝑡𝑎𝑛 𝜃, etc.

## Dimensional constants

The physical quantities which possess dimensions and have constant values are called dimensional constants. Ex: Planck’s constant, Gravitational constant, speed of light in a vacuum, etc.

## Dimensionless constants

The physical quantities which do not have dimensions but have constant values are called dimensionless constants. Ex: 𝜋, 𝑒, pure numbers like 1, 2, 3…. etc.

## Dimensional analysis

The process of examination of dimensions of various physical quantities involved in a relation is called dimensional analysis.

## Uses of Dimensional analysis

(i) The dimensions of all the terms in an equation must be identical. This principle is called the principle of homogeneity and it is a useful method to check whether an equation may be correct or not.

(ii) Dimensional analysis helps to deduce relations between physical quantities.

(iii)Dimensional analysis helps us to convert the unit of a physical quantity from one system to another.

## Check the correctness of the following equation by dimensional analysis

(i) 𝒗 = 𝒗𝟎 + 𝒂𝒕

𝑣 = 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡/𝑡𝑖𝑚𝑒

[𝑣] = [𝐿] [𝑇] = [𝐿𝑇−1]

𝑣0 = 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 [𝑣0] = [𝐿𝑇−1]

𝑎𝑡 = 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 × 𝑡𝑖𝑚𝑒

𝑎𝑡 = (𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦/𝑡𝑖𝑚𝑒) × 𝑡𝑖𝑚𝑒

𝑎𝑡 = 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦

[𝑎𝑡] = [𝐿𝑇−1]

The dimensions of each term on both sides of the equation are the same. Thus, the equation is dimensionally correct.

[𝑚𝑏] = [(𝑚𝑎𝑠𝑠)𝑏] = [𝑀𝑏]

[𝑔𝑐] = [(𝑎𝑐𝑐𝑒𝑙𝑒𝑎𝑟𝑎𝑡𝑖𝑜𝑛 𝑑𝑢𝑒 𝑡𝑜 𝑔𝑟𝑎𝑣𝑖𝑡𝑦)𝑐] = [(𝐿𝑇−2)𝑐] = [𝐿𝑐𝑇−2𝑐]

Then, from the principle of homogeneity, [𝑀0𝐿0𝑇] = [𝐿𝑎][𝑀𝑏][𝐿𝑐𝑇−2𝑐]

[𝑀0𝐿0𝑇] = [𝐿𝑎𝐿𝑐𝑀𝑏𝑇−2𝑐]

[𝑀0𝐿0𝑇] = [𝑀𝑏𝐿𝑎+𝑐𝑇−2𝑐]

Comparing the exponents on both sides, we have

𝑏 = 0

𝑎 + 𝑐 = 0

−2𝑐 = 1

On solving the above equations, −2𝑐 = 1 ⟹ 𝑐 = − 1/2

and, 𝑎 + 𝑐 = 0

𝑎 – (1/2) = 0 ⟹ 𝑎 = 1/2
Now substituting the values of 𝑎, 𝑏, 𝑎𝑛𝑑 𝑐, in the equation 𝑇 = 𝑘𝑙𝑎𝑚𝑏𝑔𝑐

𝑇 = 𝑘𝑙1/2𝑚0𝑔−1/2

𝑻 = 𝒌√(𝒍/𝒈)

2) The centripetal force(F) acting on a particle moving in a circle depends upon mass(m), velocity(v) and radius of the circle(r). Derive an expression for centripetal force using the method of dimensions.

Given, 𝐹 ∝ 𝑚𝑎𝑣𝑏𝑟𝑐

𝐹 = 𝑘𝑚𝑎𝑣𝑏𝑟𝑐 (where 𝑘 is dimensionless constant)

[𝐹] = [𝑓𝑜𝑟𝑐𝑒] = [𝑀𝐿𝑇−2]

[𝑚𝑎] = [(𝑚𝑎𝑠𝑠)𝑎] = [𝑀𝑎]

[𝑣𝑏] = [(𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦)𝑏] = [(𝐿𝑇−1)𝑏] = [𝐿𝑏𝑇𝑏]

[𝑟𝑐] = [(𝑟𝑎𝑑𝑖𝑢𝑠)𝑐] = [𝐿𝑐]

Then, from the principle of homogeneity, [𝑀𝐿𝑇−2] = [𝑀𝑎][𝐿𝑏𝑇𝑏][𝐿𝑐]

[𝑀𝐿𝑇−2] = [𝑀𝑎𝐿𝑏𝐿𝑐𝑇𝑏]

[𝑀𝐿𝑇−2] = [𝑀𝑎𝐿𝑏+𝑐𝑇𝑏]

On comparing,

𝑎 = 1

𝑏 + 𝑐 = 1

−𝑏 = −2

Solving for b and c, we have −𝑏 = −2 ⟹ 𝑏 = 2

𝑏 + 𝑐 = 1

2 + 𝑐 = 1 ⟹ 𝑐 = 1 − 2 = −1

Now substituting the values of 𝑎, 𝑏, 𝑎𝑛𝑑 𝑐, in the equation

F= 𝑘𝑚𝑎𝑣𝑏𝑟𝑐

𝐹 = 𝑘𝑚1𝑣2 𝑟−1

𝑭 = 𝒌(𝒎𝒗𝟐/𝒓)

## Limitations of dimensional analysis

1) Correctness of the constants appearing in an equation cannot be verified.

2) Dimensionally correct equations need not be actually correct.

3) Equations involving trigonometric and exponential functions cannot be verified.

4) An equation can be derived only if it is of product type.

5) While deriving an equation the value of the constant of proportionality cannot be obtained.

6) This method works only if there are as many equations available as there are unknowns.

## Important questions for the exam.

One mark.

1) Define unit.

2) What are derived units?

3) Define a unified atomic mass unit.

4) How many meters make one parsec?

5) Define the term relative error.

6) What is the dimension of a physical quantity?

7) Write the dimensional formula of the work.

8) Write the dimensional formula for linear momentum.

9) Write the dimensional formula for force.

10) State principle of homogeneity of dimensions.

Two marks.

1) What are fundamental units? Give an example of fundamental units.

2) With a diagram explain the parallax method of measuring a large distance like a planet or a star from the earth.

3) Distinguish between accuracy and precision of measurement.

4) Mention the types of errors.

5) Define (a) error and (b) accuracy.

6) Write any two methods to minimize systemic error.

7) What is the systemic error? Mention any one source of systemic errors.

8) The resistance R =V/I, where V=(100±5) volt and I= (10±0.2) A, Find the percentage error in R.

9) Define the term significant figures with examples.

10) Write the number of significant figures in the following. a) 0.007𝑚2 b) 2.64 𝑘𝑔

11) Give the number of significant figures in a) 0.00603𝑚2 b) 0.0203 𝑘𝑔

12) Write the SI unit and dimensional formula for acceleration.

13) Write a dimensional formula for force and work.

14) Name any two physical quantities, which have the same dimensions.

15) Mention the Physical quantity represented by the dimensional formula [𝑀1𝐿1𝑇−1]

16) What are the advantages of dimensional analysis?

Three marks.

1) Name the SI unit of (i) momentum (ii) luminous intensity (iii) solid angle (iv) plane angle (v) Power (vi) Impulse.

2) Write the dimensional formula for pressure, wavelength and force.

3) Using the method of dimensions, deduce the relation connecting the time period, the mass of the bob, the length of the pendulum, and acceleration due to gravity.

4) Check the correctness of the equation 𝑥 = 𝑣0𝑡 + (1/2) 𝑎𝑡2 by the method of dimensions.

5) Check the correctness of the equation 𝑣2 = 𝑣02 + 2𝑎𝑥 by the method of dimensions.

6) Check the correctness of the equation 𝑣 = 𝑣0 + 𝑎𝑡 by the method of dimensions.

7) Check the correctness of the equation 𝐾𝐸 = (1/2)𝑚𝑣2 by the method of dimensions.

8) Check the correctness of the equation 𝑉 = 𝑚𝑔ℎ by the method of dimensions.

9) Write the limitations of dimensional analysis.

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