Plus One Public Exam March 2019 Important One Mark Questions

Download Tamil Nadu 11th Standard Physics question papers, model tests, one-mark questions, important questions, and public exam papers in PDF format. Free study materials and answer keys for TN State Board students.

Questions:

Multiple Choice Question

100 x 1 = 100

Moment of inertia of a thin uniform rod about an axis passing through the center of mass and perpendicular to the length is _____________.

(a)

\(\frac{1}{3}Ml^{2}\)

(b)

\(\frac{1}{12}Ml^{2}\)

(c)

\(\frac{1}{2}m(l^{2}+b^{2})\)

(d)

ml²

The torque in rotational motion is analogous to __________ in translational motion

(a)

linear momentum

(b)

mass

(c)

couple

(d)

force

A solid cylinder of mass 3 kg and radius 10 cm is rotating about its axis with a frequency of 20/\(\pi \). The rotational kinetic energy of the cylinder _____________.

(a)

10\(\pi \)J

(b)

12 J

(c)

\(\frac { 6\times { 10 }^{ 2 } }{ \pi } \)J

(d)

3 J

A gun fires two bullets with same velocity at 60° and 30° with horizontal. The bullets strike at the same horizontal distance. The ratio of maximum height for the two bullets is in the ratio of _____________.

(a)

1: 2

(b)

3: 1

(c)

2: 1

(d)

1: 3

In pure rolling, rotational velocity of points at its edges is equal to ____________.

(a)

Rω

(b)

velocity of centre of mass

(c)

translational velocity

(d)

all the above

While rolling, the path of center of mass of an object is _______________.

(a)

straight line

(b)

parabola

(c)

hyperbola

(d)

circle

Rotational Kinetic energy of a body is ____________.

(a)

\(\frac { 1 }{ 2 } \)mr²

(b)

\(\frac { 1 }{ 2 } \)Iω²

(c)

\(\frac { 1 }{ 2 } \)Iv²

(d)

\(\frac { 1 }{ 2 } \)mω²

In non-uniform circular motion, the resultant acceleration makes an angle with the radius vector is ___________.

(a)

\(\tan^{-1}(\frac{ra_t}{V^2})\)

(b)

\(\tan_{-1}(\frac{a_t}{(\frac{r}{v^2})})\)

(c)

\(\tan^{-1}(\frac{rv^2}{at})\)

(d)

\(\tan^{-1}(\frac{r+at^2}{v^2})\)

The slope of velocity-time graph gives _____________.

(a)

velocity

(b)

acceleration

(c)

force

(d)

displacement

10.

The first derivative of position vector with respect to time is _____________.

(a)

velocity

(b)

acceleration

(c)

force

(d)

displacement

11.

The dimension of point mass is _____________.

(a)

(b)

(c)

(d)

12.

Centrifugal force is a _____________.

(a)

pseudo force

(b)

real force

(c)

forced acting towards centre

(d)

none of the above

13.

Which of the following pairs of materials has minimum amount of coefficient of static friction is ___________.

(a)

Glass and glass

(b)

wood and wood

(c)

ice and ice

(d)

steel and steel

14.

If the lines of forces act in the same plane, they can be ___________.

(a)

concurrent forces

(b)

coplanar forces

(c)

either concurrent force or coplanar forces

(d)

Lami's force

15.

The kinetic energy is not conserved in ______________.

(a)

Elastic collision

(b)

Inelastic collision

(c)

both (a) and (b)

(d)

none

16.

A bullet is fired normally on an immovable wooden plank of thickness 2 m. It loses 20% of its kinetic energy in penetrating a thickness 0.2 m of the plank. The distance penetrated by the bullet inside the wooden plank is ____________.

(a)

0.2 m

(b)

0.8 m

(c)

1 m

(d)

1.5 m

17.

The unit of power is _______________.

(a)

(b)

(c)

Js^-1

(d)

both (b) and (c)

18.

Number of significant digits in 2030______________.

(a)

(b)

(c)

(d)

19.

Number of significant digits in 32005_______________.

(a)

(b)

(c)

(d)

20.

The triple point temperature of water is_______

(a)

-273.16 K

(b)

0 K

(c)

273.16 K

(d)

100 K

21.

If a car and a scooter have the same momentum, then which one is having greater speed?

(a)

scooter

(b)

car

(c)

both have same velocity

(d)

data insufficient

22.

If the brake is applied in the moving bus suddenly, passengers move forward is an example for _______________.

(a)

Inertia of motion

(b)

Inertia of direction

(c)

Inertia of rest

(d)

back pull

23.

When a bus starts to move from rest, the passengers experience a sudden backward push is an example for ____________.

(a)

Inertia of motion

(b)

Inertia of direction

(c)

Inertia of rest

(d)

back pull

24.

Inertia means ______________.

(a)

inability

(b)

resistance to change its state

(c)

movement

(d)

inertial frame

25.

A force acts on a 3 g particle, in such a way that the position of the particle as a function of time is given by x = 3t - 4t² + t³, where x is in metres and t is in seconds. The work done during the first 4 second is _____________.

(a)

490 mJ

(b)

450 mJ

(c)

576 mJ

(d)

530 mJ

26.

Which of the following potential energy curve describes the elastic collision of two billiard balls?(R=radius of each ball)

(a)

(b)

(c)

(d)

27.

A car is moving along a straight road with a uniform acceleration. It passes through two points P and Q separated by a distance with velocity 30 km/h and 40 km/h respectively. The velocity of the car midway between P and Q is ______________.

(a)

33.3km/h

(b)

20\(\sqrt{2}\) km/h

(c)

25\(\sqrt{2}\)km/h

(d)

35km/h

28.

Dimensional formula of self inductance is _______________.

(a)

[MLT^-2A^-2]

(b)

[ML²T^-1A^-2]

(c)

[ML²T^-2A^-2]

(d)

[ML^-2T^-2A^-1]

29.

The dimensional formula of permeability of free space m₀ is ______________.

(a)

[MLT^-2A^-2]

(b)

[M⁰L¹T]

(c)

[M⁰L²T^-1A²]

(d)

none of these

30.

The position x of a particle with respect to time t along x-axis is given by x = 9t² + t³ where x is in metres and t in seconds. What will be the position of this particle when it achieves maximum speed along the +x direction?

(a)

54m

(b)

81m

(c)

24m

(d)

32m

31.

A block of mass m is placed on a smooth wedge of inclination θ. The whole system is accelerated horizontally so that the block does not slip on the wedge: The force exerted by the wedge on the block will be ____________ (g is acceleration due to gravity)

(a)

mg cosθ

(b)

mg sinθ

(c)

(d)

mg/cosθ

32.

Identify the pair of physical quantities having the same dimensions.

(a)

Light year and period of a pendulum

(b)

Angular momentum and torque

(c)

Energy and Modulus of elasticity

(d)

Torque and work

33.

Which is a vector quantity?

(a)

Angular momentum

(b)

Work

(c)

Potential energy

(d)

Electric energy

34.

A solid sphere is rotating in free space. If the radius of the sphere is increased keeping mass same, which one of the following will not be affected?

(a)

Moment of inertia

(b)

Angular momentum

(c)

Angular velocity

(d)

Rotational kinetic energy

35.

A point in the system at which whole mass of the body is supposed to be concentrated is called _____________.

(a)

centre of gravity

(b)

centre of mass

(c)

centre of energy

(d)

centre of buoyancy

36.

Principle of conservation of linear momentum ______________.

(a)

is applicable when time of impact between 2 colliding particles is extremely small

(b)

when time of impact is moderately small

(c)

when time of impact is large

(d)

is independent of time of impact

37.

Which of the following curves does not represent motion in one dimension?

(a)

(b)

(c)

(d)

38.

If the force and the displacement are at an angle of 180°, then work done is ___________.

(a)

F.S

(b)

-F.S

(c)

(d)

F-S

39.

Length can't be measured by _____________.

(a)

fermi

(b)

micron

(c)

debye

(d)

light year

40.

The prefix of 10^-18 is ______________.

(a)

femto

(b)

nano

(c)

giga

(d)

atto

41.

An engine pumps water continuously through a hose. Water leaves the hose with a velocity v and m is the mass per unit length of the water of the jet. What is the rate at which kinetic energy is imparted to water?

(a)

\({{1}\over{2}}{m}{v}^{2}\)

(b)

mv³

(c)

\({{1}\over{2}}{m}{v}^{3}\)

(d)

\({{1}\over{2}}{m}{v}^{2}\)

42.

M.I. of a ring of mass M and radius R about an axis passing through the centre & perpendicular to the plane is I. What is M.I. about its diameter?

(a)

(b)

I/2

(c)

\(I/\sqrt{2}\)

(d)

I + MR²

43.

Two rings of radius R & nR made from the same wire have the ratio of moments of inertia about an axis passing through their centre equal to 1 : 8. The value of n is

(a)

(b)

2\(\sqrt{2}\)

(c)

(d)

1/2

44.

A spacecraft of mass M moving with velocity v in free space explodes and breaks into two pieces. After the explosion a mass m of the spacecraft is left stationary. The velocity of other part is _______________.

(a)

\(\frac { mv }{ M-n } \)

(b)

\(\frac { M+n }{ Mv } \)

(c)

\(\frac { Mv }{ M-m } \)

(d)

\(\frac { Mv }{ m } \)

45.

When a body falls freely towards the earth, then its T.E. ______________.

(a)

increases

(b)

decreases

(c)

remains constant

(d)

first increases and then decreases

46.

A body is falling from a height h. After it has fallen a height \(\frac{h}{2}\) it will possess _______________

(a)

only Potential Energy

(b)

only Kinetic Energy

(c)

half potential and half kinetic energy

(d)

more kinetic and less potential energy

47.

A force\(\overrightarrow { F } =3\hat { i } +c\hat { j } +2\hat { k } \) acting on a particle causes a displacement \(\overrightarrow { S } =(2\hat { i } -3\hat { j } +4\hat { k } )\) in its own direction. If the work done is 8J, then the value of c is ______________.

(a)

(b)

(c)

(d)

48.

A ball moves on a frictionless inclined table without slipping. The work done by the table surface on the ball is ____________.

(a)

positive

(b)

negative

(c)

zero

(d)

none

49.

Action and reaction ___________

(a)

act on two different objects

(b)

have opposite direction

(c)

have equal magnitude

(d)

all of these

50.

A particle moves along x-axis from x = 0 to x = 7m under the influence of a force given by f(x) = 12-2x+3x² then the work done is__________________

(a)

205 J

(b)

390 J

(c)

378 J

(d)

291 J

51.

A round object of mass M and radius R rolls down without slipping along an inclined plane. The frictional force,

(a)

dissipates kinetic energy as heat

(b)

decreases the rotational motion

(c)

decreases the rotational and transnational motion

(d)

converts transnational energy into rotational energy

52.

Two discs of same moment of inertia rotating about their regular axes passing through center and perpendicular to the plane of the disc with angular velocities ω₁ and ω₁. They are brought in to contact face to face coinciding with the axis of rotation. The expression for loss of energy during this process is

(a)

\(\frac{1}{4}\)\(I(\omega _{1}-\omega _{2})^2\)

(b)

\(I(\omega _{ 1 }-\omega _{ 2 })^{ 2 }\)

(c)

\(\frac{1}{8}\)\(I(\omega _{1}-\omega _{2})^2\)

(d)

\(\frac{1}{2}I\)\((\omega _{1}-\omega _{2})^2\)

53.

The speed of the center of a wheel rolling on a horizontal surface is v_o. A point on the rim in level with the center will be moving at a speed of,

(a)

zero

(b)

v_o

(c)

\(\sqrt{2}\)v_o

(d)

2v_o

54.

The speed of a solid sphere after rolling down from rest without sliding on an inclined plane of vertical height h is,

(a)

\( \sqrt \frac{4}{3}gh\)

(b)

\( \sqrt \frac{10}{7}gh\)

(c)

\(\sqrt{2gh}\)

(d)

\( \sqrt \frac{1}{2}gh\)

55.

The ratio of the acceleration for a solid sphere (mass m and radius R) rolling down an incline of angle \(\theta\) without slipping and slipping down the incline without rolling is,

(a)

5: 7

(b)

2: 3

(c)

2: 5

(d)

7: 5

56.

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

57.

A disc of the moment of inertia I_a is rotating in a horizontal plane about its symmetry axis with a constant angular speed \(\omega\). Another disc initially at rest of moment of inertia I_b is dropped coaxially on to the rotating disc. Then, both the discs rotate with the same constant angular speed. The loss of kinetic energy due to friction in this process is,

(a)

\(\frac { 1 }{ 2 } \frac { { I }_{ b }^{ 2 } }{ 2({ I }_{ a }+{ I }_{ b }) } { \omega }^{ 2 }\)

(b)

\(\frac { { I }_{ b }^{ 2 } }{ ({ I }_{ a }+{ I }_{ b }) } { \omega }^{ 2 }\)

(c)

\(\frac { { ({ I }_{ b }-{ I }_{ a }) }^{ 2 } }{ ({ I }_{ a }+{ I }_{ b }) } { \omega }^{ 2 }\)

(d)

\(\frac { 1 }{ 2 } \frac { { { I }_{ b }{ I }_{ b } } }{ ({ I }_{ a }+{ I }_{ b }) } { \omega }^{ 2 }\)

58.

A spring of force constant k is cut into two pieces such that one piece is double the length of the other. Then, the long piece will have a force constant of

(a)

\(\frac{2}{3}\)k

(b)

\(\frac{3}{2}\)k

(c)

(d)

59.

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)

60.

A particle is placed at the origin and a force F = kx is acting on it (where k is a positive constant). If U (0) = 0, the graph of U(x) versus x will be (where U, is the potential , energy function)

(a)

(b)

(c)

(d)

61.

What is the minimum velocity with which a body of mass m must enter a vertical loop of radius R so that it can complete the loop?

(a)

\(\sqrt{2gR}\)

(b)

\(\sqrt{3gR}\)

(c)

\(\sqrt{5gR}\)

(d)

\(\sqrt{gR}\)

62.

A particle undergoes uniform circular motion. The angular momentum of the particle remain conserved about,

(a)

the center point of the circle

(b)

the point on the circumference of the circle

(c)

any point inside the circle

(d)

any point outside the circle

63.

A rigid body rotates with an angular momentum L. If its kinetic energy is halved, the angular momentum becomes,

(a)

(b)

L/2

(c)

(d)

L/\(\sqrt{2}\)

64.

A closed cylindrical container is partially filled with water. As the container rotates in a horizontal plane about a perpendicular bisector, its moment of inertia

(a)

increases

(b)

decreases

(c)

remains constant

(d)

depends on direction of rotation

65.

A rope is wound around a hollow cylinder of mass 3 kg and radius 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force 30 N?

(a)

0.25 rad s^-2

(b)

25 rad s^-2

(c)

5 ms^-2

(d)

25 ms^-2

66.

A particle is moving with a constant velocity along a line parallel to positive X-axis. The magnitude of its angular momentum with respect to the origin is

(a)

zero

(b)

increasing with x

(c)

decreasing with x

(d)

remaining constant

67.

A body of mass 1 kg is thrown upwards with a velocity 20 ms^-1. It momentarily comes to rest after attaining a height of 18 m. How much energy is lost due to air friction?(Take g = 10 ms^-2)

(a)

20 J

(b)

30 J

(c)

40 J

(d)

10 J

68.

A ball of mass 1 kg and another of mass 2 kg are dropped from a tall building whose height is 80 m. After, a fall of 40 m each towards Earth, their respective kinetic energies will be in the ratio of

(a)

\(\sqrt2:1\)

(b)

\(1:\sqrt2\)

(c)

2:1

(d)

1:2

69.

A uniform force of (2\(\hat { i }\)+\(\hat { j }\)) N acts on a particle of mass 1 kg. The particle displaces from position (3\(\hat { j }\)+\(\hat { k }\)) m to (5\(\hat { i }\)+3\(\hat { j }\)) m. The work done by the force on the particle is

(a)

9 J

(b)

6 J

(c)

10 J

(d)

12 J

70.

A couple produces,

(a)

pure rotation

(b)

pure translation

(c)

rotation and translation

(d)

no motion

71.

The center of mass of a system of particles does not depend upon,

(a)

position of particles

(b)

relative distance between particles

(c)

masses of particles

(d)

force acting on particle

72.

The displacement of a particle moving along x-axis with respect to times is given by x = at + bt²-ct³. The dimensions of b are ______________.

(a)

L^oT^-3

(b)

L^oT^-3

(c)

LT^-2

(d)

LT^-3

73.

The number of significant figures in 2.64\(\times\)10⁴ kg is ______________.

(a)

(b)

(c)

(d)

74.

If a person moving from pole to equator, the centrifugal force acting on him

(a)

increases

(b)

decreases

(c)

remains the same

(d)

increases and then decreases

75.

The centrifugal force appears to exist

(a)

only in inertial frames

(b)

only in rotating frames

(c)

in any accelerated frame

(d)

both in inertial and non-inertial frames

76.

When an object is at rest on the inclined rough surface ____________.

(a)

static and kinetic frictions acting on the object is zero

(b)

static friction is zero but kinetic friction is not zero

(c)

static friction is not zero and kinetic friction is zero

(d)

static and kinetic frictions are not zero

77.

When the object is moving at constant velocity on the rough surface

(a)

net force on the object is zero

(b)

no force acts on the object

(c)

only external force acts on the object

(d)

only kinetic friction acts on the object

78.

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

79.

Force acting on the particle moving with constant speed is

(a)

always zero

(b)

need not be zero

(c)

always non zero

(d)

cannot be concluded

80.

A particle of mass m sliding on the smooth double inclined plane (shown in figure) will experience

(a)

greater acceleration along the path AB

(b)

greater acceleration along the path AC

(c)

same acceleration in both the paths

(d)

no acceleration in both the paths

81.

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

82.

A vehicle is moving along the positive x direction, if sudden brake is applied, then

(a)

frictional force acting on the vehicle is along negative x direction

(b)

frictional force acting on the vehicle is along positive x direction

(c)

no frictional force acts on the vehicle

(d)

frictional force acts in downward direction

83.

When a car takes a sudden left turn in the curved road, passengers are pushed towards the right due to

(a)

inertia of direction

(b)

inertia of motion

(c)

inertia of rest

(d)

absence of inertia

84.

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

85.

Two objects are projected at angles 30° and 60° respectively with respect to the horizontal direction. The range of two objects are denoted as R_30° and R_30°. Choose the correct relation from the following

(a)

R_30° = R₆_0°

(b)

R_30° = 4R₆_0°

(c)

R_30° =\(\frac{R_{60°}}{2}\)

(d)

R_30° = 2R₆_0°

86.

If an object is thrown vertically up with the initial speed u from the ground, then the time taken by the object to return back to ground is

(a)

\(\frac{u^2}{2g}\)

(b)

\(\frac{u^2}{g}\)

(c)

\(\frac{u}{2g}\)

(d)

\(\frac{2u}{g}\)

87.

If a particle executes uniform circular motion in the xy plane in clockwise direction, then the angular velocity is in

(a)

+y direction

(b)

+z direction

(c)

-z direction

(d)

-x direction

88.

A ball is dropped from some height towards the ground. Which one of the following represents the correct motion of the ball?

(a)

(b)

(c)

(d)

89.

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)

90.

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

(a)

(b)

(c)

(d)

91.

The dimension of \({\left( {\mu}_{0}{\epsilon}_{0} \right)}^{{{1}\over{2}}}\) is

(a)

length

(b)

time

(c)

velocity

(d)

force

92.

If the force is proportional to square of velocity, then the dimension of proportionality constant is

(a)

[MLT⁰]

(b)

[MLT^-1]

(c)

[MLT^-2T]

(d)

[MLT^-1T⁰]

93.

The dimensional formula for gravitational constant G is

(a)

[ML³T^-2]

(b)

[M^-1L³T^-2]

(c)

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

(d)

[ML^-3T²]

94.

The velocity of a particle v at an instant t is given by v = at + br². The dimensions of b is

(a)

[L]

(b)

[LT^-1]

(c)

[LT^-2]

(d)

[LT^-3]

95.

The dimensional formula of Planck's constant h is

(a)

[ML²T^-1]

(b)

[ML²T³]

(c)

[MLT^-1]

(d)

[ML³T^-3]

96.

Which of the following pairs of physical quantities have same dimension?

(a)

force and power

(b)

torque and energy

(c)

torque and power

(d)

force and torque

97.

If \(\pi=3.14,\) then the value of \({\pi}^{2}\) is

(a)

9.8596

(b)

9.860

(c)

9.86

(d)

9.9

98.

Which of the following has the highest number of significant figures?

(a)

0.007 m²

(b)

2.64\(\times\)10²⁴kg

(c)

0.0006032 m²

(d)

6.3200 J

99.

If the length and time period of an oscillating pendulum have errors of 1% and 3% respectively then the error in measurement of acceleration due to gravity is

(a)

(b)

(c)

(d)

100.

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

100 x 1 = 100

(b)
\(\frac{1}{12}Ml^{2}\)

(d)
force

(b)
12 J

(b)
3: 1

(a)
Rω

(a)
straight line

(b)
\(\frac { 1 }{ 2 } \)Iω²

(a)
\(\tan^{-1}(\frac{ra_t}{V^2})\)

(b)
acceleration

10.

(a)
velocity

11.

(a)
0

12.

(a)
pseudo force

13.

14.

(a)
concurrent forces

15.

(b)
Inelastic collision

16.

17.

(d)
both (b) and (c)

18.

(d)
4

19.

20.

(d)
100 K

21.

(a)
scooter

22.

(a)
Inertia of motion

23.

24.

(b)
resistance to change its state

25.

26.

27.

28.

29.

(a)
[MLT^-2A^-2]

30.

(a)
54m

31.

(d)
mg/cosθ

32.

(d)
Torque and work

33.

(a)
Angular momentum

34.

(b)
Angular momentum

35.

(b)
centre of mass

36.

(a)
is applicable when time of impact between 2 colliding particles is extremely small

37.

38.

(b)
-F.S

39.

40.

(d)
atto

41.

Mass per unit length = m
Velocity = v
(mass of water pumped in one second)
M = mv
\(\text { Kinetic energy }=\frac{1}{2} M v^{2}\)
\(=\frac{1}{2} m v^{3}\)

42.

(a)
I

43.

The moment of inertia (I) of circular ring whose axis of rotation is passing through its center, \(\mathrm{I}_{1}=\mathrm{m}_{2} \mathrm{R}^{2} . \text { Also } \mathrm{I}_{2}=\mathrm{m}_{2}(\mathrm{n} \mathrm{R})^{2}\)
\(\frac{I}{I_{2}}=\frac{M R^{2}}{n M(n R)^{2}}=\frac{1}{n^{3}}=\frac{1}{8} \)
\(\therefore n^{3}=8 \)
\(\therefore n=\sqrt[3]{8}=2 \)

44.

45.

46.

47.

48.

49.

(d)
all of these

50.

(b)
390 J

51.

(d)
converts transnational energy into rotational energy

52.

Moment of inertia of a disc passing through
\(\text { centre } I=\frac{1}{2} M R^{2}\)
\(\text { Energy of the first disc }=\frac{1}{4} I \omega_{1}^{2}\)
\(\text { Energy of the second disc }=\frac{1}{4} I \omega_{2}^{2}\)
\(\text { Loss of energy }=\frac{1}{4} I\left(\omega_{1}-\omega_{2}\right)^{2}\)

53.

\(v_{0}=r \omega ; \quad \therefore v_{0} \alpha r\)
For a wheel (uniform ring) the distance of a point on the rim in level with the center
\(\text { [i.e., radius] is } \sqrt{2} r\)
\(\therefore \text { The speed of the center is } \sqrt{2} v_{0}\)

54.

Potential energy = Translational kinetic energy + Rotational kinetic energy
\(m g h=\frac{1}{2} m v^{2}+\frac{1}{2} I \omega^{2} \)
\(=\frac{1}{2} m v^{2}+\frac{1}{2} \times \frac{2}{5} M R^{2} \times \frac{v^{2}}{R^{2}}\left[\omega=\frac{v}{R}\right] \)
\(=\frac{1}{2} m v^{2}+\frac{1}{5} m v^{2} \)
\(=\frac{5 m v^{2}+2 m v^{2}}{10}=\frac{7 m v^{2}}{10} \)
\(m g h=\frac{7 m v^{2}}{10} \)
\(g h=\frac{7 v^{2}}{10} \)
\(\therefore v^{2}=\frac{10 g h}{7} \)
\(\therefore v=\frac{\sqrt{10 g h}}{7} \)

55.

Acceleration of the solid sphere while rolling down without slipping
\(a_{1}=\frac{g \sin \theta}{1+\frac{k^{2}}{r^{2}}}\)
Acceleration developed while slipping down \(a_{2}=g \sin \theta\)
\(\text { Required ratio } \frac{a_{1}}{a_{2}}=\frac{g \sin \theta}{1+\frac{k^{2}}{r^{2}}} / g \sin \theta\)
\(\frac{a_{1}}{a_{2}}=\frac{1}{1+\frac{k^{2}}{r^{2}}}\)
\(\text { For a solid sphere } \frac{k^{2}}{r^{2}}=\frac{2}{5}\)
\(\therefore \text { Ratio of accelerations } \frac{a_{1}}{a_{2}}=\frac{1}{1+\frac{2}{5}}\)
\(=\frac{1}{5+\frac{2}{5}}=\frac{1}{\frac{7}{5}}=\frac{5}{7}\)
\(\therefore a_{1}: a_{2}=5: 7 \)

56.

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
\(=\frac{M R^{2}}{2}-\frac{3 M R^{2}}{32} \)
\(=\frac{16 M R^{2}-3 M R^{2}}{32}=\frac{13 M R^{2}}{32}\)
\(I =\frac{13 M R^{2}}{32} \)

57.

The moments of inertia of two discs are I_a and I_b respectively The angular velocity of the disc A is \(\omega\).
The sum of kinetic energies of two discs before coming in contact is \(k_{1}=\frac{1}{2} I_{a} \omega_{1}^{2}+\frac{1}{2} I_{b} \omega_{2}^{2}\)
\(\text { But angular velocity of the disc be is } \omega_{2}=0 \ \text {(rest)}\)
\(\therefore k_{1}=\frac{1}{2} I_{a} \omega_{1}^{2}\)
The final kinetic energy of the two discs system \(k_{2}=\frac{1}{2} \frac{I_{a}^{2} \omega_{1}^{2}}{I_{a}+I_{b}}\)
The loss of kinetic energy is
\(k_{1}-k_{2} =\frac{1}{2} I_{a} \omega^{2}-\frac{1}{2}\left[\frac{I a^{2} \omega_{1}^{2}}{I_{a}+I_{2 b}}\right] \)
\(=\frac{1}{2} \frac{\left[I_{1}\left(I_{a}+I_{b}\right) \omega^{2}-I_{a}^{2} \omega^{2}\right]}{I_{a}+I_{b}} \)
\(k_{1}-k_{2} =\frac{1}{2} \frac{I_{a} b}{\left(I_{a}+I_{b}\right)} \omega^{2} \)

58.

For any spring kl = constant
Length of the longer piece
\(=\frac{2 l}{3} \)
\(\therefore k^{1} \times \frac{2 l}{3} =k l \)
\(\therefore k^{1}=\frac{k l \times 3}{2 l}=\frac{3}{2} k \)
\(\therefore k^{1}=\frac{3}{2} k \)

59.

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

60.

For a conservative force
\(F=-\frac{d v}{d t} \)
\(\int_{0}^{u(x)} d v=-\int_{0}^{x} F d x=-\int_{0}^{x} k x d x \)
\(\text { As } v(0)=0\)
\(U(x)=-\frac{k x^{2}}{2}\)
Thus, the graph of U(x) versus (x) will be a parabola, symmetric about U - ax is bying below x - ax is with its vertex at the origin. Hence the correct answer is C

61.

Radius =R
\(v_{1}^{2}-v_{2}^{2}=4 g R\)
\(\text { Tension } T_{2}=\frac{m v_{2}^{2}}{R_{2}}-m g\)
\(\text {To find minimum speed, let } T_{2}=0\)
\(0 =\frac{m v_{2}^{2}}{R}-m g \)
\(\frac{m v^{2}}{R} =m g \)
\(v_{2}^{2}=R g \ v_{2} =\sqrt{g R} \)
\(\text { sub (2) in the eqn (1) we get }\)
\(v_{1}^{2}-(\sqrt{g R})^{2} =4 g R \)
\(v_{1}^{2}-g R =4 g R \)
\(v_{1}^{2} =4 g R+g R \)
\(=5 g R \)
\(v_{1} =\sqrt{5 g R} \)

62.

(a)
the center point of the circle

63.

\(K \cdot E=\frac{1}{2} I \omega^{2} ; \quad L=I \omega ; \quad K \cdot E=\frac{2^{2}}{2^{2}} \)
\(\therefore K \cdot E \alpha L^{2} \quad E_{1} \alpha L_{1}^{2} \quad E_{2} \alpha L_{2}^{2}\)
\(\frac{E_{1}}{E_{2}}=\left(\frac{L_{1}}{L_{2}}\right)^{2} \)
\(\text { Here } E_{1}=E \quad E_{2}=\frac{E}{2} \)
\(L_{1}=L \quad \quad L_{2}=? \)
\(\frac{E}{\frac{E}{2}}=\left(\frac{L}{L_{2}}\right)^{2} \)
\(\frac{2 E}{E}=\left(\frac{L}{L_{2}}\right)^{2}\left(\frac{L_{1}}{L_{2}}\right)^{2}=2 \)
\(\therefore \frac{L}{L_{2}}=\sqrt{2} \)
\(L_{2}=\frac{L}{\sqrt{2}} \)

64.

(a)
increases

65.

\(m=3 \mathrm{~kg} \quad r=40 \times 10^{-2} \mathrm{~m}=0.4 \mathrm{~m}\)
\(\text { Force }=30 N\)
Moment of inertia of a hollow-cylinder I= MR²
\(=3 \times\left(40 \times 10^{-2}\right)^{2} \)
\(=3 \times 0.4 \times 0.4=0.48 \mathrm{kgm}^{2} \)
\( F R =I \alpha \)
\(30 \times 40 \times 10^{-2}=0.48 d \)
\(d=\frac{12}{0.48}=\frac{1200}{48}=25 \mathrm{rad} \mathrm{s}^{-2} \)
\(\alpha=25 \mathrm{rad} \mathrm{s}^{-2} \)

66.

(d)
remaining constant

67.

Mass = 1kg
Velocity v = 20m/s
Height h = 18 m
\(g=10 \mathrm{~m} / \mathrm{s}^{2}\)
\(\text { Potential energy } P . E=m g h\)
\(=1 \times 10 \times 18 =180 \mathrm{~J} \)
\(\text { Kinetic energy } K . E=\frac{1}{2} m v^{2}\)
\(K . E=\frac{1}{2} \times 1(20)^{2}\)
\(=\frac{1}{2} \times 20 \times 20=200 J\)
\(\text { Loss of energy due to air friction }= K.E - P.E\)
\(=200-180=20 \mathrm{~J}\)

68.

\(m_{1} =1, \quad m_{2}=2 \)
\(K . E . =m g(h-x) \)
\(\text { For both balls }(h-x)\)
\(=40 \text { i.e.) Same }\)
\(g=\text { constant }\)
\(\therefore K \cdot E_{1}=m_{1} g(h-x)=m_{1} g \times 40 \)
\(K \cdot E_{2}=m_{2} g(h-x)=m_{2} g \times 40 \)
\(\therefore \frac{K \cdot E_{1}}{K \cdot E_{2}}=\frac{m_{1} g \times 40}{m_{2} g \times 40}=\frac{m_{1}}{m_{2}} \)
\( \therefore K \cdot E_{1}: K \cdot E_{2}=1: 2 \)

69.

\(\text { Force } \overrightarrow{\mathbf{F}}=(2 i+\vec{j}) N\)
\(\text { Displacement } d=(5 \vec{i}+3 \vec{j})-(3 \vec{j}+\vec{k})\)
\(=(5 i-k) m\)
\(\text { Work done } W=F . d\)
\(=(2 \vec{i}+\vec{j})(5 i-k)\)
\(=10-0-0=10 J \)

70.

(a)
pure rotation

71.

(d)
force acting on particle

72.

73.

(d)
3

74.

(a)
increases

75.

(b)
only in rotating frames

76.

77.

(a)
net force on the object is zero

78.

(d)
μ_s mg cos ፀ

79.

(b)
need not be zero

80.

(b)
greater acceleration along the path AC

81.

82.

(a)
frictional force acting on the vehicle is along negative x direction

83.

(a)
inertia of direction

84.

(b)
g = 25 ms^-2

85.

Range is same for the angle if projection \(\theta \text { and } 90-\theta\)

86.

\(\text {Time of flight }=\frac{2 u}{g}\)

87.

Use thumb rule

88.

Distance travelled \(s=\frac{1}{2} g t^{2} ; s \alpha t^{2}\). The ratio of distances travelled by a freely falling body will be with ratio 1:4:9:...

89.

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

90.

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

91.

\(\text { Velocity of light } c=\frac{1}{\sqrt{\mu_{0} \varepsilon_{0}}}\)
\(c=\left(\mu_{0} \varepsilon_{0}\right)^{-\frac{1}{2}}\)
\(\text { Hence dimension }\left(\mu_{0} \varepsilon_{0}\right)^{-\frac{1}{2}} \text { is that of velocity. }\)

92.

F = kv²
Dimensional of k
\(=\frac{\text { Dimension of } \mathrm{F}}{\text { Dimension of }(v)^{2}}\)
\(=\frac{\mathrm{MLT}^{-2}}{\left(\mathrm{LT}^{-1}\right)^{2}}=\frac{\mathrm{MLT}^{-2}}{\mathrm{~L}^{2} \mathrm{~T}^{-2}} \)
\(=\left[\mathrm{ML}^{-1-2} \mathrm{~T}^{-2+2}\right] \)
Dimension of proportionality constant \(=\left[\mathrm{ML}^{-1} \mathrm{~T}^{0}\right]\)

93.

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

94.

\(v=a t+b t^{2}\)
\(\text { Dimensional equation is } \mathrm{LT}^{-1}\)
\(=a T=b T^{2}\)
\(\therefore \text { The dimension of } b=\frac{\mathrm{LT}^{-1}}{\mathrm{~T}^{2}}=\mathrm{LT}^{-3}\)

95.

Dimensional formula of Planck's
\(\text { constant }=\frac{\text { Energy }}{\text { Frequency }}=\frac{\mathrm{ML}^{2} \mathrm{~T}^{-2}}{\mathrm{~T}^{-1}}\)
\(=M L^{2} T^{-2+1}=M L^{2} T^{-1}\)

96.

\(\text { Torque }=\text { Force } \times \mathrm{r}\)
\(\text { Dimension of torque }\)
\(=\left[\mathrm{MLT}^{-2}\right][\mathrm{L}]=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\)
\(\text { Energy }=\text { Force } \times \text { displacement }\)
\(\text { Dimension of Energy }\)
\(=\left[\mathrm{MLT}^{-2}\right][\mathrm{L}]=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\)

97.

\(\pi=3.14 \)
\(\pi^{2} =3.14 \times 3.14 \)
\(=9.8596=9.86 \)

98.

The number of significant figures of 6.3200 J is 5

99.

\(T =2 \pi \sqrt{\frac{l}{g}} \)
\(g =4 \pi^{2} l / T^{2} \)
\(\frac{d g}{g} =\frac{d l}{l}-\frac{2 d T}{T} \)
\(\frac{d g}{g} \% =\left(\frac{d l}{l}\right) \%-2\left(\frac{d T}{T}\right) \% \)
\(=1 \%-2 \times(-3 \%) \)
\(=1+6=7 \% \)

100.

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