Notes of CH#1

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BABAR ACADEMY

Notes of Ch#1 (Measurements, 1st year)

Babar Iqbal Awan

BSc honors (Physics and Mathematics)

Forman Christian College Lahore

These Notes Include: i. Solution of exercise and numerical problems

ii. Conceptual MCQ’s

iii. Additional Short Questions

Email address: [email protected] Contact No. 0334 – 7895390

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Babar Academy

Chapter No.1 Measurements

1st

year

Multiple Choice Questions

1. Of which of the following have supplementary unit?

A. Temperature B. Angle C. Intensity of light D. None of these

2. Which of the following is a base quantity?

A. Temperature B. Force C. Pressure D. All of these

3. Which of the following is derived unit?

A. Mol B. Candela C. Kelvin D. Coulomb

4. The dimensions of angle are

A. B. C. D. No dimensions

5. The least count of screw gauge is

A. 0.1m B. 0.1mm C. 0.1cm D. 0.01mm

6. Which of the following can be used to measure the internal diameter of a tube

A. Screw gauge B. Vernier Calipers C. Meter rod D. All of these

7. Light year is the unit of

A. Distance B. Time C. Speed D. None of these

8. According to Einstein‟s mass energy equation, 1kg of mass can be converted to ------- joule

of energy.

A. 1 J B. J C. J D. 3 J

9. Convert 0.1 pm to cm

A. cm B. cm C. cm D. cm

10. Significant figures of a measured value indicate the

A. Precision of the instrument B. Accuracy of the instrument

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C. Both A and B D. None of these

11. 43.852 can be rounded off as

A. 43.8 B. 43.7 C. 43.9 D. 44.0

12. Dimensions of viscosity are

A. B. C. D.

13. Physical dimensions does not tell about

A. Proportionality constant B. Base quantities

C. Both A and B D. None of these

14. An instrument is said to be more precise if

A. All observations are close to one another

B. All observations are close to actual value

C. There is a variation in the values of the observations

D. All of these

15. If cm and cm and

A. cm B. cm C. cm D. cm

16. When two quantities are multiplied or divided then the total uncertainty is computed by

A. Subtracting their percentage uncertainties

B. Multiplying their percentage uncertainties

C. Dividing their percentage uncertainties

D. Adding their percentage uncertainties

17. The percentage uncertainty in the radius „r‟ of sphere is 0.5%, the total uncertainty in the

volume of the sphere is

A. 1% B. 1.5 % C. 0.5% D. 0.00025%

18. The time of ten vibrations of a simple pendulum is 15seconds, as measured by the stop watch

of least count 0.1s. The total uncertainty in the time period of simple pendulum will be

A. 0.15s B. 0.01 s C. 0.0015 s D. 0.001 s

19. The time required for light to reach earth from sun is

A. 8 min 20s B. 1 min 20s C. 5 hours 20s D. 8 minutes

20. Which of the following has maximum number of significant figures?

A. 0.025 B. 25.0 C. 2.5 D. None of these

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21. Steradian is a

A. One dimensional angle B. Two dimensional angle C. Three dimensional angle

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A. .3 B. 50.30 C. 50.303 D. None of these

23. Length and breadth of a rectangular object is 0.24 m and 0.134 m. Its area is given as

A. 0.03216 B. 0.032 C. 0.0322 D. 0.03

24. A student takes 3 observations instead of 1 in an experiment. Now the possible error reduces

to

A. times B. times C. times D. times

25. The dimensions of gravitational acceleration are

A. B. C. D.

26. The scientific notation of 0.0450 is

A. B. C. D.

27. A precise instrument is the one which has

A. Maximum precision B. Least precision C. Both A and B

28. Which of the following statement is correct about precision

A. Instrument with small least count is more precise

B. Instrument with large least count is more precise

C. Meter rod is more precise than vernier calipers

D. Both A and C

29. The formula for the volume of the cylinder with diameter „d‟ is given by

A. B. C. D.

30. Zero is significant only if it lies

A. Between the two non zero digits B. On right side of the decimal place

C. On left side before none zero digit D. Both A and B

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Answers

1 B Angle 2 A Temperature

3 D Coulomb 4 D No Dimensions

5 D 0.01 mm 6 B Vernier Callipers

7 A Distance 8 C J

9 B cm 10 C Both A and B

11 A 43.8 12 C

13 A Proportionality constant 14 A All observations are close to one another

15 A 16 D Adding their percentage uncertainties

17 B 1.5 % 18 B 0.0067s

19 A 8 min 20s 20 B 25.0

21 C Three dimensional angle 22 A 50.3

23 B 0.032 24 A times

25 B 26 C

27 B Least Precision 28 A Instrument with small least

count is more precise

29 A 30 D Both A and B

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Short Questions

1. Define Light year and second.

2. How can we minimize systematic error?

3. Differentiate between precision and accuracy.

4. The base and height of the triangle are cm and cm respectively. Calculate

the area and also the overall uncertainty in the final result.

5. Prove that is dimensionally correct. Where E is Young‟s modulus and „ ‟ is density.

6. In what way significant figures are important in calculations?

7. Compute and express your answer in correct number of significant figures

i. ii.

8. Write the physical dimensions of

i. Momentum ii. Power iii. Young‟s Modulus

9. What are physical dimensions? What is their use?

10. Radius of a small sphere is cm. Calculate the volume of sphere and also the total

uncertainty in it.

11. Length of rod is 10.2 cm by using a meter rod. While the least count of meter rod is 0.1cm.

Calculate its precision and % uncertainty.

12. It is impossible to tell the correct number of significant figures in 2000 kg. Why?

13. Time required for 50 vibrations of the simple pendulum is to be 70.4 seconds. What is its

time period? What is total uncertainty in it? If the least count of timing device is 0.1 s.

Note: The answers of these questions will be given later. Because firstly, it is for the students

to work on these questions so that their performance can be evaluated easily.

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Chapter No.1

Questions

1.1 Name several repetitive phenomenas occurring in nature which could serve as

reasonable time standards.

Ans. Phenomenon which repeats itself again and again after regular time intervals can be taken

as a time standard. The reasonable time standards are earth, simple pendulum, revolution

of the moon around the earth, radioactive decay rate of certain substance, etc.

1.2 Give the drawbacks to use the period of a pendulum as a time standard.

Ans. Time period of simple pendulum is given by T = 2 , Time period depends upon

length „ ‟ and acceleration due to gravity „g‟. At higher altitudes value of „g‟ decreases,

while length also changes in different seasons. The air resistance also plays a vital role in

changing the time period of simple pendulum. All these factors change the value of time

period of simple pendulum in different conditions. So it is not a best time standard.

1.3 Why do we found it useful to have two units for the amount of substance, the

kilogram and the mole?

Ans. In general case we find the amount of substance by weighing it, therefore we use

kilogram as the unit of substance. When we are concerned about the atoms and the

molecules, in this case, the number of atoms or particles matters and gave more useful

information, therefore we use „mol‟ as the unit of substance.

1.4 Three students measured the length of a needle with a scale on which minimum

division is 1mm and recorded as (i) 0.2145 m (ii) 0.21 m (iii) 0.214 m. Which record

is correct and why?

Ans. The error in the measurement depends upon the least count of the instrument by which it

is measured. As the least count of the scale is 1mm or 0.001m, therefore 0.214 m is the

correct reading because it is measured up to three decimal places. All the other

measurements are not according to the least count of the scale.

1.5 An old saying is that “A chain is only as strong as its weakest link”. What analogous

statement can you make regarding experimental data used in a computation?

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Ans. The analogous statement of the old saying “A chain is only as strong as its weakest link”

is that any experimental data that we obtained is as accurate as the least accurate

measurement taken during the computation. The number significant figures of the final

answer is mostly equal to the minimum number of significant figures of the observed

readings.

1.6 The period of a simple pendulum is measured by a stop watch. What type of errors

is possible in the time period?

Ans. Possibly two types of errors could occur. First one is „systematic error‟ which is due to

the fault in the measuring instrument (i.e. zero error). Second one is „Personal error‟

which can occur due to the negligence or inexperience of the observer.

1.7 Does a dimensional analysis give any information on constant of proportionality

that may appear in an algebraic expression? Explain.

Ans. Dimensional analysis does not give any information about the constant of proportionality.

The value of constant of proportionality is calculated through experiment or theoretically.

For example F = , the constant of proportionality „G‟ have no unite. Its value is

determined through experiment.

1.8 Write the dimensions of (i) Pressure and (ii) Density.

Ans. (i) Dimensions Pressure is given by =

Dimension of Force = (Dimension of Mass) (Dimension of acceleration)

=

Dimension of Area =

Now dimensions of Pressure

(ii) Dimension of Density is given by

1.9 The wavelength of a wave depends on the speed „v‟ of the wave and its frequency

„f‟. Knowing that

, and

Decide which of the following is correct. f = or f =

Ans. (i) To check f =

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The dimension of L.H.S =

Dimension of R.H.S =

As the Dimensions of L.H.S Dimensions of R.H.S, therefore f = is not

correct formulae.

(ii) To check f =

The dimension of L.H.S =

Dimension of R.H.S =

As the Dimensions of L.H.S Dimensions of R.H.S, therefore f = is

dimensionally correct.

Numerical Problems

1.1 A light year is the distance light travels in one year. How many meters are there in

one light year: (speed of light = ms-1)

Sol. Given data

Time = t = 1year = 365 days =

= 31536000 = 3.154 107 s.

Speed of light = v = ms-1

Distance = S = ?

Solution;

We know that S = v t

S =

S = m

1.2 a. How many seconds are there in 1 year?

b. How many nanoseconds in one year?

c. How many years in one second?

Sol. We know that

1year = 365 days =

= 31536000 = s.

1 year = s

1 s = = years.

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Now 1s = 109 nanoseconds therefore

1 year = nanoseconds

1.3 The length and width of a rectangular plate are measured to be 15.3 cm and 12.80

cm, respectively. Find the area of the plate.

Sol. Given data

Length = l = 15.3 cm

Width = w = 12.80 cm

Area = A =?

Solution;

Area = A =

1.4 Add the following masses given in kg up to appropriate precision. 2.189, 0.089, 11.8

and 5.32.

Sol. Solution;

Sum =

Since least precise measurement is 11.8, so total mass should be equal to one decimal

place, which is 19.4 kg.

1.5 Find the value of „g‟ and its uncertainty using T = from the following

measurements made during an experiment.

Length of simple pendulum =

Time for 20 vibrations = 40.2 s

Least count of meter-scale = 1mm = 0.1 cm

Least count of stop watch = 0.1 s

Sol. Given data

t = 40.2 s

Time for 20 vibrations = 40.2 s

Least count of meter-scale = 1mm = 0.1 cm

g = ?

Solution;

Given that T = where T =

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Putting values of T, , and

ms-2

Absolute uncertainty in length = 0.1 cm

Average uncertainty in length =

Uncertainty in time measurement =

Average uncertainty in time measurement =

There is the square on the time factor so

Total average uncertainty = 0.1% + 2(0.25%) = 0.6%

As g = 9.76 ms-2 , than its 0.6% will be 0.06 ms-2,

So the exact answer will be g = 9.76 0.06 ms-2

1.6 What are the dimensions and units of gravitational constant G in the formula F =

Sol. Solution;

Units. Given that F =

G =

Dimensions.

1.7 Show that the expression is dimensionally correct, where is the

velocity at t = 0, a is acceleration and is the velocity at time t.

Sol. Given equation is

Dimension of L.H.S =

Dimension of R.H.S =

=

=

So the dimensions of both sides are same. So the given equation is dimensionally correct.

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1.8 The speed v of sound waves through a medium may be assumed to depend on (a) the

density of the medium and (b) its modulus of elasticity E which is the ratio of stress to

strain. Deduce by the method of dimensions, the formula for the speed of sound.

Ans. As velocity depends upon the density , and the elasticity E, therefore

------------------- (i)

We will both sides in terms of dimensions

Stress = , strain has no dimensional unite

Equating the powers of M, L and T on both sides

------------------ (ii)

------------------ (iii)

------------------ (iv)

Put in equation (ii) we get

Putting values of a and b in equation (i)

1.9 Show that the famous “Einstein equation” E = mc2 is dimensionally correct.

Sol. We have the equation E = mc2

Dimensions of L.H.S = E = W = Fd

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Dimensions of R.H.S =

Dimensions of L.H.S and R.H.S are same. So E = mc2 is dimensionally correct.

1.10 Suppose we are told that the acceleration of a particle moving in a circle of radius „r‟ with

uniform speed „v‟ is proportional to some power of r, say rn, and some power of v, say

vm, determine the powers of r and v.

Sol. We are given that we have to find the value of m and n.

Writing the above in terms of physical dimensions

Equating powers of L and T on both sides of equation

--------------- (i)

Putting value of „m‟ in equation (i) we get

So the values of „m‟ and „n‟ are 2 and -1.