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Questions:

Multiple Choice Question

50 x 1 = 50

If the equation of the wave is represented by \(\mathbf{y}=10^{-2} \sin \left(500 t-\frac{x}{20}\right) \mathbf{m}\)then the velocity of the wave is_______m/s.

(a)

10,000

(b)

500

(c)

(d)

1000

Two vibrating tuning forks produce waves given by Y₁= 2 sin 400 \(\pi t\) and y₂= 2 sin 406\(\pi t\) The number of beats produced per second is ____________

(a)

(b)

(c)

1.5

(d)

The ratio between the rms speed and most probable speed of gas molecules at a given temperature is

(a)

\(\begin{equation} 2 \sqrt{2}: \sqrt{1} \end{equation}\)

(b)

\(\begin{equation} \sqrt{3}: \sqrt{2} \end{equation}\)

(c)

\(\begin{equation} \sqrt{2}: \sqrt{3} \end{equation}\)

(d)

\(\begin{equation} \sqrt{1}: 2 \sqrt{2} \end{equation}\)

The ratio of energy of radiation emitted by a black body at 27^oC to 927^oC is

(a)

1:4

(b)

1:16

(c)

1:64

(d)

1:256

The potential energy possessed by a, drop bubble, having surface tension equal to 0.04 Nm^-1 of diameter 1 cm is

(a)

8\(\pi\)\(\times\)10^-6J

(b)

2\(\pi\)\(\times\)10^-6J

(c)

4\(\pi\)\(\times\)10^-6J

(d)

16\(\pi\)\(\times\)10^-6J

The diameters of two planets are in the ratio of 4 : 1. Their mean densities are in the ratio 1 : 2. The ratio of accelerations due, to gravity on the planet is

(a)

2:1

(b)

1:2

(c)

4:1

(d)

1:4

An air column in a pipe, which is closed at one end, will be in resonance with a vibrating tuning fork of frequency 256 Hz, if the length of the column in centimeter is ____________ (velocity of sound in air = 340 m/s)

(a)

21.25

(b)

125

(c)

62.50

(d)

33.2

Beats occur because of ____________.

(a)

interference

(b)

reflection

(c)

refraction

(d)

Doppler effect

A transverse wave propagating on a stretched string of linear density 3\(\times\)10^-4 kg m^-1 is represented by the equation. y = 0.2 sin (1.5x + 60t)
Where x is in metres and t is in seconds. The tension in the string (in newtons) is _______________.

(a)

0.48

(b)

0.24

(c)

0.20

(d)

1.80

10.

A closed bottle containing water at room temperature is taken to the moon and then the lid is opened. The water will______.

(a)

freeze

(b)

boil

(c)

decompose into hydrogen and oxygen

(d)

not change at all.

11.

The mass of 1 litre of helium under a pressure of 2 atm and at a temperature of 27°C is _______________.

(a)

0.16 g

(b)

0.32 g

(c)

0.48 g

(d)

0.64 g

12.

Orbital velocity of an artificial satellite does not depend upon ________.

(a)

mass of Earth

(b)

mass of satellite

(c)

radius of Earth

(d)

acceleration doe to gravity

13.

A satellite which is geostationary in a particular orbit is taken to another orbit. Its distance from the center of earth in new orbit is two times of the earlier orbit. The time period in second orbit is ________ hours.

(a)

4.8

(b)

48\(\sqrt{2}\)

(c)

(d)

24\(\sqrt{2}\)

14.

Which of the following statement/statements are correct?
(i) Pressure of the gas molecules increases with increases of number density.
(ii) Pressure decreases with increases of mean square speed of the gas molecules.
(iii) Larger mass of the molecule will have larger momentum for fixed speed.

(a)

i) ii)

(b)

ii) iii)

(c)

i) iii)

(d)

all are correct

15.

Universal gas constant is _________________.

(a)

C_p/C_v

(b)

C_p - C_v

(c)

C_p + C_v

(d)

C_v/C_p

16.

If more air is pushed in a soap bubble, the pressure in it will _______________.

(a)

decreases

(b)

increases

(c)

remains same

(d)

becomes zero

17.

A person standing between two parallel hills fires a gun and hears the first echo after t₁ sec and the second echo after t₂ sec. The distance between the two hills is

(a)

\(\frac { v({ t }_{ 1 }-{ t }_{ 2 }) }{ 2 } \)

(b)

\(\frac { v\left( { t }_{ 1 }{ t }_{ 2 } \right) }{ 2({ t }_{ 1 }+{ t }_{ 2 }) } \)

(c)

v (t₁+ t₂)

(d)

\(\frac { v({ t }_{ 1 }+{ t }_{ 2 }) }{ 2 } \)

18.

A transverse wave moves from a medium A to a medium B. In medium A, the velocity of the transverse wave is 500 ms^-1 and the wavelength is 5 m. The frequency and the wavelength of the wave in medium B when its velocity is 600 ms^-1, respectively are

(a)

120 Hz and 5 m

(b)

100 Hz and 5 m

(c)

120 Hz and 6 m

(d)

100 Hz and 6 m

19.

A simple pendulum has a time period T₁. When its point of suspension is moved vertically upwards according as y = k t², where y is vertical distance covered and k = 1 ms⁻², its time period becomes T₂. Then, \(\frac { { T }_{ 1 }^{ 2 } }{ { T }_{ 2 }^{ 2 } } \) is (g = 10 m s⁻²).

(a)

\(\frac{5}{6}\)

(b)

\(\frac{11}{10}\)

(c)

\(\frac{6}{5}\)

(d)

\(\frac{5}{4}\)

20.

The length of a second’s pendulum on the surface of the Earth is 0.9 m. The length of the same pendulum on surface of planet X such that the acceleration of the planet X is n times greater than the Earth is

(a)

0.9n

(b)

\(\frac{0.9}{n}\)m

(c)

0.9n²m

(d)

\(\frac{0.9}{n^2}\)

21.

An ideal gas is maintained at constant pressure. If the temperature of an ideal gas increases from 100K to 1000K then the rms speed of the gas molecules

(a)

increases by 5 times

(b)

increases by 10 times

(c)

remains same

(d)

increases by 7 times

22.

A hot cup of coffee is kept on the table. After some time it attains a thermal equilibrium with the surroundings. By considering the air molecules in the room as a thermodynamic system, which of the following is true

(a)

ΔU > 0, Q = 0

(b)

ΔU > 0, W < 0

(c)

ΔU > 0, Q > 0

(d)

ΔU = 0, Q > 0

23.

In hot summer after a bath, the body’s

(a)

internal energy decreases

(b)

internal energy increases

(c)

heat decreases

(d)

no change in internal energy and heat

24.

If the acceleration due to gravity becomes 4 times its original value, then escape speed

(a)

remains same

(b)

2 times of original value

(c)

becomes halved

(d)

4 times of original value

25.

The gravitational potential energy of the Moon with respect to Earth is

(a)

always positive

(b)

always negative

(c)

can be positive or negative

(d)

always zero

26.

The linear momentum and position vector of the planet is perpendicular to each other at

(a)

perihelion and aphelion

(b)

at all points

(c)

only at perihelion

(d)

no point

27.

A fly wheel rotating about fixed axis has a kinetic energy of 360 joule when its angular speed is 30 radian/sec, The moment of inertia of the wheel about the ax is of rotation is _____________.

(a)

0.6 kgm²

(b)

0.15 kgm²

(c)

0.8 kgm²

(d)

0.75 kgm²

28.

A sphere rolls on a horizontal plane without slipping. The percentage of kinetic energy which is rotational is ______________.

(a)

57%

(b)

28.05%

(c)

14.28%

(d)

58.5%

29.

Moment of inertia of a solid cylinder having radius R₀ is ______________.

(a)

\(MR_{0}^{2}\)

(b)

\(MR_{0}^{{2\over3}}\)

(c)

\(MR_{0}^{{2\over4}}\)

(d)

\(MR_{0}^{{2\over 2}}\)

30.

The centre of mass of a system of particles does not depend on ______________.

(a)

position of the particles

(b)

relative distance between the particles

(c)

masses of the particles

(d)

forces acting on the particles

31.

Centre of mass of the earth and the moon system lies ____________.

(a)

Closer to the earth

(b)

Closer to the moon

(c)

at the mid-point of line joining the earth and the moon

(d)

cannot be predicted

32.

The fraction of \(1\over 273.16\) temperature of the triple point of water represents __________.

(a)

celsius

(b)

fahrenheit

(c)

centigrade

(d)

kelvin

33.

If a person standing on a rotating disc stretches out his hands the angular speed will ____________.

(a)

Increase

(b)

Decrease

(c)

Remain same

(d)

None

34.

Which of the following has largest M.I _________________.

(a)

Ring about its axis perpendicular to its plane

(b)

Disc about its axis perpendicular to its plane

(c)

Solid sphere

(d)

Bar magnet

35.

If force acts on a body, whose line of action does not pass through its CG, then the body will experience ______________

(a)

angular acceleration

(b)

lineal acceleration

(c)

both (a) and (b)

(d)

none

36.

From a disc of radius R a mass M, a circular hole of diameter R, whose rim passes through the center is cut. What is the moment of inertia of the remaining part of the disc about a perpendicular axis passing through it

(a)

15MR²/32

(b)

13MR²/32

(c)

11MR²/32

(d)

9MR²/32

37.

A particle which is constrained to move along x-axis, is subjected to a force in the same direction which varies with the distance x of the particle from the origin as F(x) = kx + ax³. Here, k and a are positive constants. For x ≥ 0, the functional form of the potential, energy U(x) of the particles

(a)

(b)

(c)

(d)

38.

If the potential energy of the particle is \(\alpha -\frac { \beta }{ 2 } { x }^{ 2 }\), then force experienced by the particle is

(a)

F = \(\frac { \beta }{ 2 } { x }^{ 2 }\)

(b)

F = βx

(c)

F = -βx

(d)

F = -\(\frac { \beta }{ 2 } { x }^{ 2 }\)

39.

A couple produces,

(a)

pure rotation

(b)

pure translation

(c)

rotation and translation

(d)

no motion

40.

The fractional error \(\left( {{\triangle x}\over{x}} \right)\) _____________.

(a)

\(\pm\left( {{\triangle x}\over{x}} \right)\)

(b)

\(\pm n\left( {{\triangle a}\over{a}}\right)\)

(c)

\(\pm n \log_e \left( {{\triangle a}\over{a}}\right)\)

(d)

\(\pm n \log_{10}{{\triangle a}\over{a}}\)

41.

An object of mass m begins to move on the plane inclined at an angle θ. The coefficient of static friction of inclined surface is μ_s. The maximum static friction experienced by the mass is

(a)

(b)

μ_s mg

(c)

μ_s mg sin ፀ

(d)

μ_s mg cos ፀ

42.

A book is at rest on the table which exerts a normal force on the book. If this force is considered as reaction force, what is the action force according to Newton's third law?

(a)

Gravitational force exerted by Earth on the book

(b)

Gravitational force exerted by the book on Earth

(c)

Normal force exerted by the book on the table

(d)

None of the above

43.

How many AU makes one metre?

(a)

3.26 \(\times\) 10¹¹AU

(b)

1.496 \(\times\) 10¹¹AU

(c)

3.08 \(\times\) 10¹⁶AU

(d)

6.684 \(\times\) 10^-12AU

44.

Match the following fundamental forces with respect to relative strengths.

(1)	Gravitational force	(a)	1
(2)	Electromagnetic force	(b)	10^-39
(3)	Weak nuclear force	(c)	10^-2
(4)	Strong nuclear force	(d)	10^-13

(a)

(1)	(2)	(3)	(4)
a	d	b	c

(b)

(1)	(2)	(3)	(4)
b	c	d	a

(c)

(1)	(2)	(3)	(4)
c	d	a	b

(d)

(1)	(2)	(3)	(4)
c	a	b	d

45.

Astronomical Scale is dealt with the _________ Physics

(a)

Mesoscopic

(b)

Microscopic

(c)

Macroscopic

(d)

None

46.

An object is dropped in an unknown planet from height 50 m, it reaches the ground in 2 s. The acceleration due to gravity in this unknown planet is

(a)

g = 20 ms^-2

(b)

g = 25 ms^-2

(c)

g = 15 ms^-2

(d)

g = 30 ms^-2

47.

A ball is projected vertically upwards with a velocity v. It comes back to ground in time t. Which v-t graph shows the motion correctly?

(a)

(b)

(c)

(d)

48.

Which one of the following Cartesian coordinate systems is not followed in physics?

(a)

(b)

(c)

(d)

49.

The dimensional formula for gravitational constant G is

(a)

[ML³T^-2]

(b)

[M^-1L³T^-2]

(c)

[M^-1L^-3T^-2]

(d)

[ML^-3T²]

50.

One of the combinations from the fundamental physical constants is \({{hc}\over{G}},\) The unit of this expression is

(a)

Kg²

(b)

m³

(c)

S^-1

(d)

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Answers Key:

Multiple Choice Question

50 x 1 = 50

\(y =a \sin (\omega t-k x) \)
\(y =10^{-2} \sin \left(500 t-\frac{x}{20}\right) \)
\(\therefore \omega=500 \quad k=\frac{1}{20} \)
\(\therefore \text { Velocity } V =\frac{\omega}{k} \)
\(=\frac{500}{1 / 20} =1000 \mathrm{~m} / \mathrm{s} \)

\(\mathrm{y}_{1}=2 \sin 400 \pi t \)
\(\mathrm{y}_{2}=2 \sin 406 \pi t \)
\(\omega_{1}=2 \pi f_{1}=400 \pi \quad \therefore f_{1}=200 \mathrm{~Hz} \)
\(\omega_{2}=2 \pi f_{2}=406 \pi \quad \therefore f_{2}=203 \mathrm{~Hz} \)
\(\text { Number of Beats }=f_{2}-f_{1} \)
\(=203-200=3 \text { beats } / \mathrm{second} \)

(b)
\(\begin{equation} \sqrt{3}: \sqrt{2} \end{equation}\)

\(\mathrm{E} \propto \mathrm{T}^{4} \)
\(\frac{E_{1}}{E_{2}}=\frac{T_{1}}{T_{2}}=\frac{300}{1200}=\frac{1}{4} \)
\(\therefore \mathrm{E}_{1}: \mathrm{E}_{2}=1: 4 \)

\(\text { P.E }=U =2 \times 4 \pi R^{2} \rho \)
\(=8 \pi \times\left(0.5 \times 10^{-2}\right)^{2} \times 0.04 J\)
\(=8 \pi \times 10^{-6} \mathrm{~J} \)

\(g=\frac{2}{3} \pi G D \rho \)
\(\frac{g_{1}}{g_{2}}=\frac{D_{1} \rho_{1}}{D_{2} \rho_{2}}=\frac{4}{1} \times \frac{1}{2}=\frac{2}{1} \)

(d)
33.2

(a)
interference

(a)
0.48

10.

(b)
boil

11.

(b)
0.32 g

12.

(a)
mass of Earth

13.

(b)
48\(\sqrt{2}\)

14.

(a)
i) ii)

15.

(b)
C_p - C_v

16.

(a)
decreases

17.

\(2 d_{1}=v t_{1}-\text { first echo }\)
\(2 d_{2}=v t_{2}-\text { second echo }\)
\(\therefore \mathrm{d}=\mathrm{d}_{1}+\mathrm{d}_{2} \quad \therefore \frac{v\left(t_{1}+t_{2}\right)}{2}\)

18.

\(v_{\mathrm{A}}=500 \mathrm{~ms}^{-1} \quad \lambda_{A}=5 \mathrm{~m}\)
Frequency in medium B
\(\mathrm{f}_{\mathrm{B}}=\frac{v_{A}}{\lambda_{A}}=\frac{500}{5}=100 \mathrm{~Hz}\)
Wavelength in medium B
\(\lambda_{B}=\frac{v_{s}}{f_{s}}=\frac{600}{100}=6 \mathrm{~Hz}\)

19.

\(\mathrm{T}= \mathrm{T}_{1}=2 \pi \sqrt{\frac{l}{g}} \)
\(T_{1}^{2}=4 \pi^{2} \frac{l}{g} \)
\(\mathrm{y}=\mathrm{kt}^{2} \quad \mathrm{k}=1 \mathrm{~m} \mathrm{~s}^{-2} \)
\(\therefore \mathrm{y}=\mathrm{t}^{2} \quad \mathrm{y} \propto \mathrm{t}^{2} \)
\(\frac{y_{1}}{y_{2}}=\frac{t_{1}^{2}}{t_{2}^{2}} \)
\(\frac{T_{1}^{2}}{T_{2}^{2}}=\frac{y_{1}}{y_{2}}=\frac{6}{5} \)

20.

\(l =0.9 \)
\(\mathrm{~T} =2 \pi \frac{l}{g} \)
\(T^{\prime} =2 \pi \frac{l_{X}}{g_{X}} \)
\(\mathrm{~g} \mathrm{x} =\mathrm{ng} \)
\(\therefore \mathrm{x} =0.9 \mathrm{n} \)

21.

\(v_{m s}=1.73 \sqrt{\frac{k T}{m}}\)
\(\text { AT Increased by } 10 \text { times }\)
\(v_{\mathrm{ms}} \propto \Delta T\)
\(\text { RMS speed increases by } 10 \text { times. }\)

22.

During the thermal equilibrium with surroundings internal energy is increased and heat energy will be increased.
\(\Delta U>0 Q>0\)

23.

(a)
internal energy decreases

24.

\(\text { Escape speed } v_{e}=\sqrt{2 g R}\)
\(\text { If } g^{\prime}=4 \mathrm{~g}\)
\(\text { Then } v_{e}^{\prime}=\sqrt{2(4 g) R}\)
\(=\sqrt{2 g R} \times 2 \)
\(=2 v_{e} \)

25.

(b)
always negative

26.

(a)
perihelion and aphelion

27.

28.

(b)
28.05%

29.

(d)
\(MR_{0}^{{2\over 2}}\)

30.

(d)
forces acting on the particles

31.

32.

(d)
kelvin

33.

(b)
Decrease

34.

(a)
Ring about its axis perpendicular to its plane

35.

36.

Moment of inertia of a disc
\(\mathrm{I}_{1}=\frac{M R^{2}}{2}\)
\(\text { Mass of small disc }=\frac{M}{\pi R^{2}} \times \pi \times\left(\frac{R}{2}\right)^{2}\)
\(=\frac{M}{\pi R^{2}} \times \frac{\pi R^{2}}{4}=\frac{M}{4}\)
By the theorem of parallel axis, the moment of inertia of the small disc. About an axis passing through 0 is
\(I_{2} =\frac{1}{2} \times \frac{M}{4}\left(\frac{R}{2}\right)^{2}+\frac{M}{4}\left(\frac{R}{2}\right)^{2} \)
\(=\frac{M}{8} \times \frac{R^{2}}{4}+\frac{M}{4} \times \frac{R^{2}}{4} \)
\(=\frac{M R^{2}}{32}+\frac{M R^{2}}{16}=\frac{M R^{2}+2 M R^{2}}{32} \)
\(I_{2} =\frac{3 M R^{2}}{32} \)
Moment of inertia of the remaining part is I= I₁ - I₂
\(=\frac{M R^{2}}{2}-\frac{3 M R^{2}}{32} \)
\(=\frac{16 M R^{2}-3 M R^{2}}{32}=\frac{13 M R^{2}}{32}\)
\(I =\frac{13 M R^{2}}{32} \)

37.

\(F=-\frac{d u}{d x} \quad F(x) =k x+a x^{3} \)
\(d u =-F d x \)
\(u(x) =-\int_{0}^{x}\left(-k x+a x^{3}\right) d x \)
\(=\int_{0}^{x} k x d x-a \int_{0}^{x} x^{3} d x \)
\(=\frac{k x^{2}}{2}-\frac{a x^{4}}{2} \)
\(U(x) =\frac{x^{2}}{2}\left(k-\frac{a x^{2}}{2}\right) \)
\(u(x)=0 \text { at } x=0 \text { and }\)
\(U(x) =0 ; k-\frac{a x^{2}}{2}=0 \)
\(=\frac{a}{2} x^{2}=-k \)
\(x^{2} =\frac{2 k}{a} \)
\(\therefore x =\sqrt{\frac{2 k}{a}} \)
\(\text { Clearly } u(x)=0 \text { at } x=0 \text { and }\)
\(x=\sqrt{\frac{2 k}{a}}\)
\(\text { For } x>\sqrt{\frac{2 k}{a}} U(x) \text { will be negative. } \)
\(\text { At } x=0 ; F=\frac{-d u}{d x}=0\)
(i.e.,) Slope of V - x graph is zero at x = 0
Hence the most appropriate answer is d.

38.

\(\text {Potential energy } P . E=\alpha-\frac{\beta}{2} x^{2}\)
P.E = Work = Fx
\(P=\alpha-\frac{\beta}{2} x^{2}\)
\(\text {Force }=\frac{d p}{d x}=\frac{d}{d x}\left(\alpha-\frac{\beta}{2} x^{2}\right)\)
\(=0-\frac{\beta}{2} \times 2 x =-\beta x \)

39.

(a)
pure rotation

40.

(b)
\(\pm n\left( {{\triangle a}\over{a}}\right)\)

41.

(d)
μ_s mg cos ፀ

42.

43.

(d)
6.684 \(\times\) 10^-12AU

44.

(b)

(1) (2) (3) (4)

b c d a

45.

46.

(b)
g = 25 ms^-2

47.

Initially velocity has maximum value and at maximum height velocity becomes zero. After that the velocity becomes negative

48.

Answers (a), (b) and (c) are all anticlockwiseand answer (d) alone is in the clockwise direction

49.

\(\text { Gravitational constant } G=\frac{F r^{2}}{m_{l} m_{2}}\)
\(\text { Dimensional formula of } \mathrm{G}=\frac{\left[\mathrm{MLT}^{-2}\right]\left[\mathrm{L}^{2}\right]}{[\mathrm{M}][\mathrm{M}]}\)
\(=\frac{\mathrm{ML}^{3} \mathrm{~T}^{-2}}{\mathrm{M}^{2}}=\mathrm{M}^{-1} \mathrm{~L}^{3} \mathrm{~T}^{-2}\)

50.

Unit of a (Planck's constant) - Js
Unit of c (Velocity of light) - ms^-1
Unit of G (Gravitational Constant) - \(\frac{\mathrm{Nm}^{2}}{\mathrm{Kg}^{2}}\)
\(\therefore \text { Unit of } \frac{h c}{G} \text { is }=\frac{J s \times m s^{-1}}{N m^{2} / k g^{2}} \)
\(=\frac{N m s \times m s^{-1} \times k g^{2}}{N m^{2}}[J=N m] =\mathrm{kg}^{2}\)