Physics

Phys - 1

Units And Measurements

Session Opener

We know an elephant is heavier than a feather

Physics wants to know by how many time, by what standards and with what accuracy.

Session Objectives

Session Objective

Standard and Units

Dimensions

Errors

Significant Figures

Accuracy and Precision

Dimensional Analysis

Standards and Units

Laws of physics : expressed in terms of physical quantities

Physical quantities : expressed in terms of fundamental quantities.

Fundamental quantities : defined by measurements and expressed by standards.

Measurements : comparison with a standard.

Standards are defined and universally accepted by competent authority.

Standards and Units

Physical quantity (q) given by a number and a unit.

q = n . u

n : pure number.

u : unit of the standard.

1n

u

Because q is the same whatever be the standard

Basic physical quantity

Name of SI unit

Symbol of SI unit

1. Length Meter m

2. Mass Kilogram kg

3. Time Second s

4. Electric current Ampere A

5. Temperature Kelvin K

6. Luminous intensity Candela cd

7. Amount of substance Mole mol

SI (International system) units

Dimensions of physical quantities

Number of times a fundamental quantity is repeated in physical quantity q

q M L T ..............

Volume is 3 dimensional in length

3V L

Area is 2 dimensional in length

2A L

a

b

c

Dimension

• Quantities with same dimensions only can be added

• Power of dimension on both sides of an equation must match

Questions

Class Exercise - 7

Dimensionally, specific heat is proportional to dimension of mass as

(a) [M0] (b) [M1]

(c) [M–1] (d) [M2]

Solution - 7

Specific heat is (dimensionally)

Heat (energy) per unit mass

per unit temperature (q)

Then,

[Energy][C]

[M ]

–20 2 –2 –1[MLT ]

[M L T ][M ]

Dimensional Analysis

To test whether a relation is wrong.

For interconversion of units

To justify /derive interrelation of quantities.

Dimensional analysis is a powerful method

Questions

Class Exercise - 8

Show dimensionally which of the following physical quantities have an influence on the time period of a simple pendulum?

(i) Mass of the bob

(ii) Length of the string (l)

(iii) Acceleration due to gravity (g) and (iv) Angular displacement ()

Solution - 8

Time period = T

Then [T] m g

Relation with cannot be found dimensionally.

–2 –2M L LT M L T 0 0 1M L T m g

0, –2 1, – 1 1

or – , 2 2

So T Constantg

Class Exercise - 9

What is the value of a force of 10 N in a system with fundamental units of centimetre, gram and hour?

Solution - 9q = n1u1 = n2u2

10 N = n2 new unit

2Newton

n 10 New unit

–2

2 –2kg m s

n 10 g cm hr

23 210 g 10 cm hr10

1 g 1 cm s

= 106 × 60 × 60 = 3.6 × 109

Class Exercise - 10

Check dimensionally if the relation

is correct. 21s ut at

2

Solution - 10

Dimension of left-hand side (s) = [M0L1T0]

On right-hand side: ut =Velocity × Time

= [M0L1T–1][T]

= [M0L1T0] same as LHS

2 2 0 1 –2 21 1

at Acceleration Time M L T T2 2

1 has no dimension

2= [M0L1T0] [Same as LHS] Equation is dimensionally correct.

Errors

An observation is limited by the least count of instrument

Measured value qm = qreal q

Exact value of qreal is not known

Only mean value of q can be found

Errors

Random errors are expected when several observations (qi) are made

n

2i

i 1x

RMS error :n(n 1)

n

ii 1

i i i

1Meanq q

n

Error in q x q q

Errors

In sums and differences, ABSOLUTE ERRORS are added A B = C

C C = A B ( A + B)

In products or quotients,RELATIVE ERRORS are added

A B C

C A BC A B

o

oC

percentage error 100C

AC

BC A B

C A B

Questions

Class Exercise - 3

The percentage errors of X, Y, X are x, y and z respectively. The total percentage error in the product XYZ is

(a) xyz (b) x + y + z

1 1 1

(c)x y z

xy yz xz(d)

x y z

Percentage errors are added in a product.

Solution :- b

Class Exercise - 6

The least count of a stop watch is 0.2 s. The time of 20 oscillations of a pendulum is measured to be 25 s. The percentage error in the measurement of time is

(a) 8% (b) 1.8%(c) 0.8% (d) 0.1%

Solution

Total time measured is important and not time period.

So percentage error 0.2

100% 0.8%25

(i) Accuracy

Sign has to be retained while expressing accuracy.

Accuracy : degree of agreement of a measurement with the true (accepted) value.

Accuracy and Precision

(ii) Precision

Precision is expressed without any sign.

Precision : degree of agreement between two or more measurements done in an identical manner.

Significant figures

Significant figures in 1.007,12.012 and 10.070 are 4, 5 and 5 respectively.

Significant figures are the meaningful digits in a measured or calculated quantity.

i. All non-zero digits are significant.

Rules to determine significant figures

iv Zeroes to the right of the decimal point are significant.

iii. Zeroes between non-zero digits are significant.

ii. Zeroes to the left of the first non-zero

digit are not significant.

Questions

Class Exercise - 2

Which of the following, in the measurement of length, is most accurate?

(a) 2 × 102 m (b) 200.0 m

(c) 20 × 102 m (d) 200 m

200.0 has four significant figures, which is maximum in the group.

Solution :-

Class Exercise - 5

Which of the following measurements is most precise?

(a) 2345 m (b) 234.5 m

(c) 23.45 m (d) 2.345 m

Solution - d

2.345 m measures till the smallest fraction of a meter.

Class Exercise - 4

With regard to the significant figures,(12.5)2 is equal to

(a) 156.250 (b) 156.25

(c) 156.2 (d) 156

(12.5)2 = 156.25. But as only three significant figures are to be considered, 156 is the right answer.

Solution

Class Exercise - 1

Which of the following statements is false among the statements given below?

(a) All non-zero digits are significant.

(b) Zeroes in the middle of a numerical expression are significant, while those immediately following a decimal point are not.

(c) While counting the number of significant figures, the powers of 10 are to be considered.

(d) Greater the number of significant figures in a measurement, smaller is the percentage error.

Solution - 1

In powers of 10 placed as:

212.2 = 2.122 × 102,,102

is not significant.

Ans. c

Thank you