## It is claimed that two cesium clocks,

[question] Question. It is claimed that two cesium clocks, if allowed to run for 100 years, free from any disturbance, may differ by only about $0.02 \mathrm{~s}$. What does this imply for the accuracy of the standard cesium clock in measuring a time-interval of 1 s? [/question] [solution] solution: Difference in time of caesium clocks = 0.02 s Time required for this difference = 100 years $=100 \times 365 \times 24 \times 60 \times 60=3.15 \times 10^{9} \mathrm{~s}$ In $3.15 \times 10^{9} \math...

## A man walking briskly in rain with speed v must slant

[question] Question. A man walking briskly in rain with speed $v$ must slant his umbrella forward making an angle $\theta$ with the vertical. A student derives the following relation between $\theta$ and $v \cdot \tan \theta=v$ and checks that the relation has a correct limit: as $v \rightarrow 0, \theta \rightarrow 0$, as expected. (We are assuming there is no strong wind and that the rain falls vertically for a stationary man). Do you think this relation can be correct? If not, guess the corre...

## When the planet Jupiter is at a distance of 824.7 million kilometers

[question] Question. When the planet Jupiter is at a distance of $824.7$ million kilometers from the Earth, its angular diameter is measured to be $35.72^{\prime \prime}$ of arc. Calculate the diameter of Jupiter. [/question] [solution] solution: Distance of Jupiter from the Earth, $D=824.7 \times 10^{6} \mathrm{~km}=824.7 \times 10^{9} \mathrm{~m}$ Angular diameter $=35.72^{\prime \prime}=35.72 \times 4.874 \times 10^{-6} \mathrm{rad}$ Diameter of Jupiter = d Using the relation, $\theta=\frac{d...

## A book with many printing errors contains four different

[question] Question. A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain periodic motion: (a) $y=a \sin \left(\frac{2 \pi t}{T}\right)$ (b) $y=a \sin v t$ (c) $y=\left(\frac{a}{T}\right) \sin \frac{t}{a}$ (d) $y=(a \sqrt{2})\left(\sin \frac{2 \pi t}{T}+\cos \frac{2 \pi t}{T}\right)$ (a = maximum displacement of the particle, v = speed of the particle. T = time-period of motion). Rule out the wrong formulas on dimensional gro...

## A physical quantity $P$ is related to four observables $a, b, c$ and $d$ as follows:

[question] Question. A physical quantity $P$ is related to four observables $a, b, c$ and $d$ as follows: $P=\frac{a^{3} b^{2}}{(\sqrt{c} d)}$ The percentage errors of measurement in $a, b, c$ and $d$ are $1 \%, 3 \%, 4 \%$ and $2 \%$, respectively. What is the percentage error in the quantity $P ?$ If the value of $P$ calculated using the above relation turns out to be $3.763$, to what value should you round off the result? [/question] [solution] solution: $P=\frac{a^{3} b^{2}}{(\sqrt{c} d)}$ $...

## The mass of a box measured by a grocer’s balance is 2.300 kg.

[question] Question. The mass of a box measured by a grocer’s balance is 2.300 kg. Two gold pieces of masses 20.15 g and 20.17 g are added to the box. What is (a) the total mass of the box, (b) the difference in the masses of the pieces to correct significant figures? [/question] [solution] solution: Mass of grocer’s box = 2.300 kg Mass of gold piece I = 20.15g = 0.02015 kg Mass of gold piece II = 20.17 g = 0.02017 kg (a) Total mass of the box = 2.3 + 0.02015 + 0.02017 = 2.34032 kg In addition, ...

## The length, breadth and thickness of a rectangular sheet of metal

[question] Question. The length, breadth and thickness of a rectangular sheet of metal are $4.234 \mathrm{~m}, 1.005 \mathrm{~m}$, and $2.01 \mathrm{~cm}$ respectively. Give the area and volume of the sheet to correct significant figures. [/question] [solution] solution: Length of sheet, $I=4.234 \mathrm{~m}$ Breadth of sheet, $b=1.005 \mathrm{~m}$ Thickness of sheet, $h=2.01 \mathrm{~cm}=0.0201 \mathrm{~m}$ The given table lists the respective significant figures: Hence, area and volume both mu...

## State the number of significant figures in the following:

[question] Question. State the number of significant figures in the following: (a) $0.007 \mathrm{~m}^{2}$ (b) $2.64 \times 10^{24} \mathrm{~kg}$ (c) $0.2370 \mathrm{~g} \mathrm{~cm}^{-3}$ (d) $6.320 \mathrm{~J}$ (e) $6.032 \mathrm{~N} \mathrm{~m}^{-2}$ (f) $0.0006032 \mathrm{~m}^{2}$ [/question] [solution] solution: (a) Answer: 1 The given quantity is $0.007 \mathrm{~m}^{2}$. If the number is less than one, then all zeros on the right of the decimal point (but left to the first non-zero) are in...

## Answer the following:

[question] Question. Answer the following: (a) You are given a thread and a metre scale. How will you estimate the diameter of the thread? (b) A screw gauge has a pitch of 1.0 mm and 200 divisions on the circular scale. Do you think it is possible to increase the accuracy of the screw gauge arbitrarily by increasing the number of divisions on the circular scale? (c) The mean diameter of a thin brass rod is to be measured by vernier callipers. Why is a set of 100 measurements of the diameter expe...

## Which of the following is the most precise device for measuring length:

[question] Question. Which of the following is the most precise device for measuring length: (a) a vernier callipers with 20 divisions on the sliding scale (b) a screw gauge of pitch 1 mm and 100 divisions on the circular scale (c) an optical instrument that can measure length to within a wavelength of light ? [/question] [solution] solution: A device with minimum count is the most suitable to measure length. (a) Least count of vernier callipers $=1$ standard division $(S D)-1$ vernier division ...

## A new unit of length is chosen such that the speed of light in vacuum is unity.

[question] Question. A new unit of length is chosen such that the speed of light in vacuum is unity. What is the distance between the Sun and the Earth in terms of the new unit if light takes 8 min and 20 s to cover this distance? [/question] [solution] solution: Distance between the Sun and the Earth: = Speed of light × Time taken by light to cover the distance Given that in the new unit, speed of light = 1 unit Time taken, $t=8 \mathrm{~min} 20 \mathrm{~s}=500 \mathrm{~s}$ ∴Distance between th...

## Explain this statement clearly:

[question] Question. Explain this statement clearly: “To call a dimensional quantity ‘large’ or ‘small’ is meaningless without specifying a standard for comparison”. In view of this, reframe the following statements wherever necessary: (a) atoms are very small objects (b) a jet plane moves with great speed (c) the mass of Jupiter is very large (d) the air inside this room contains a large number of molecules (e) a proton is much more massive than an electron (f) the speed of sound is much smalle...

## Fill in the blanks by suitable conversion of units:

[question] Question. Fill in the blanks by suitable conversion of units: (a) $1 \mathrm{~kg} \mathrm{~m}^{2} \mathrm{~s}^{-2}=\ldots . \mathrm{g} \mathrm{cm}^{2} \mathrm{~s}^{-2}$ (b) $1 \mathrm{~m}=\ldots . .1 \mathrm{y}$ (c) $3.0 \mathrm{~ms}^{-2}=\ldots . \mathrm{km} \mathrm{h}^{-2}$ (d) $G=6.67 \times 10^{-11} \mathrm{~N} \mathrm{~m}^{2}(\mathrm{~kg})^{-2}=\ldots(\mathrm{cm})^{3} \mathrm{~s}^{-2} \mathrm{~g}^{-1}$ [solution] solution: (a) $1 \mathrm{~kg}=10^{3} \mathrm{~g}$ $1 \mathrm{~m}^{2...

## Fill in the blanks

[question] Question. Fill in the blanks (a) The volume of a cube of side $1 \mathrm{~cm}$ is equal to..... $\mathrm{m}^{3}$ (b) The surface area of a solid cylinder of radius $2.0 \mathrm{~cm}$ and height $10.0 \mathrm{~cm}$ is equal to $(\mathrm{mm})^{2}$ (c) A vehicle moving with a speed of $18 \mathrm{~km} \mathrm{~h}^{-1}$ covers... $\mathrm{m}$ in $1 \mathrm{~s}$ (d) The relative density of lead is $11.3 .$ Its density is .... $\mathrm{g} \mathrm{cm}^{-3}$ or .... $\mathrm{kg} \mathrm{m}^{-...