164. A body of mass $2\,\text{kg}$ slides on an inclined surface that makes an angle of $30°$ with the horizontal, moving downward at constant speed. If in this motion the body is displaced $2\,\text{m}$, how much work (in Joules) does the friction force do? $\left(g = 10\,\dfrac{\text{m}}{\text{s}^2}\right)$
$-25\sqrt{3}\ (1$ $-10\sqrt{3}\ (2$ $-10\ (3$ $-20\ (4$

164. According to the figure below, a object of mass $250\,\text{g}$ is placed on top of a spring whose spring constant is $2.5\,\dfrac{\text{N}}{\text{cm}}$, is released, and after hitting the spring, the spring compresses $12\,\text{cm}$. The work done by gravity on the object from the moment of release to the moment the spring reaches maximum compression is how many joules? (Air resistance is negligible and $g = 10\,\dfrac{\text{m}}{\text{s}^2}$.)
[Figure: A block resting on a spring attached to the ground]

• [(1)] $0.3$
• [(2)] $1.2$
• [(3)] $1.8$
• [(4)] $3.6$

43-- A football with mass $45\,\text{g}$ is shot from a penalty spot at a speed of $20\,\dfrac{\text{m}}{\text{s}}$ toward the goal and hits the goalkeeper at a speed of $16\,\dfrac{\text{m}}{\text{s}}$. What is the total work done on the ball?
(1) $-10$ (2) $-16.2$ (3) $-32.4$ (4) $-64.8$

Disc A has moment of inertia $I$ and angular velocity $\omega$. Disc B has moment of inertia $2I$ and is initially at rest. When disc B is brought in contact with disc A, they acquire a common angular velocity. The loss of kinetic energy during the above process is
(A) $\frac{I\omega^2}{2}$
(B) $\frac{I\omega^2}{3}$
(C) $\frac{I\omega^2}{4}$
(D) $\frac{I\omega^2}{6}$

A spring of force constant $800 \mathrm{~N/m}$ has an extension of 5 cm. The work done in extending it from 5 cm to 15 cm is
(1) 16 J
(2) 8 J
(3) 32 J
(4) 24 J

A spring of spring constant $5 \times 10^{3} \mathrm{~N/m}$ is stretched initially by 5 cm from the unstretched position. Then the work required to stretch it further by another 5 cm is
(1) $12.50 \mathrm{~N-m}$
(2) $18.75 \mathrm{~N-m}$
(3) $25.00 \mathrm{~N-m}$
(4) $6.25 \mathrm{~N-m}$

A uniform chain of length 2 m is kept on a table such that a length of 60 cm hangs freely from the edge of the table. The total mass of the chain is 4 kg . What is the work done in pulling the entire chain on the table?
(1) 7.2 J
(2) 3.6 J
(3) 120 J
(4) 1200 J

A force $\vec { F } = ( 5 \hat { i } + 3 \hat { j } + 2 \hat { k } ) N$ is applied over a particle which displaces it from its origin to the point $\vec { r } = ( 2 \hat { i } - \hat { j } ) m$. The work done on the particle in joules is
(1) - 7
(2) + 7
(3) + 10
(4) + 13

If $g$ is the acceleration due to gravity on the earth's surface, the gain in the potential energy of object of mass $m$ raised from the surface of the earth to a height equal to the radius $R$ of the earth is
(1) 2 mgR
(2) $\frac { 1 } { 2 } \mathrm { mgR }$
(3) $\frac { 1 } { 4 } \mathrm { mgR }$
(4) mgR

A spherical ball of mass 20 kg is stationary at the top of a hill of height 100 m. It rolls down a smooth surface to the ground, then climbs up another hill of height 30 m and finally rolls down to a horizontal base at a height of 20 m above the ground. The velocity attained by the ball is
(1) $40 \mathrm{~m}/\mathrm{s}$
(2) $20 \mathrm{~m}/\mathrm{s}$
(3) $10 \mathrm{~m}/\mathrm{s}$
(4) $10\sqrt{30} \mathrm{~m}/\mathrm{s}$

A player caught a cricket ball of mass 150 g moving at a rate of $20$ m/s. If the catching process is completed in 0.1 s, the force of the blow exerted by the ball on the hand of the player is equal to
(1) 300 N
(2) 150 N
(3) 3 N
(4) 30 N

A particle of mass 100 g is thrown vertically upwards with a speed of $5$ m/s. The work done by the force of gravity during the time the particle goes up is
(1) 0.5 J
(2) $-0.5$ J
(3) $-1.25$ J
(4) 1.25 J

A ball of mass 0.2 kg is thrown vertically upwards by applying a force by hand. If the hand moves 0.2 m while applying the force and the ball goes upto 2 m height further, find the magnitude of the force. Consider $g = 10$ m/s$^2$
(1) 22 N
(2) 4 N
(3) 16 N
(4) 20 N

The potential energy of a 1 kg particle free to move along the $x$-axis is given by $$V(x) = \left(\frac{x^4}{4} - \frac{x^2}{2}\right) \text{J}$$ The total mechanical energy of the particle is 2 J. Then, the maximum speed (in m/s) is
(1) 2
(2) $3/\sqrt{2}$
(3) $\sqrt{2}$
(4) $1/\sqrt{2}$

A spring is compressed between two blocks of masses $m_{1}$ and $m_{2}$ placed on a horizontal frictionless surface as shown in the figure. When the blocks are released, they have initial velocity of $v_{1}$ and $v_{2}$ as shown. The blocks travel distances $x_{1}$ and $x_{2}$ respectively before coming to rest. The ratio $\left(\frac{x_{1}}{x_{2}}\right)$ is
(1) $\frac{m_{2}}{m_{1}}$
(2) $\frac{m_{1}}{m_{2}}$
(3) $\sqrt{\frac{m_{2}}{m_{1}}}$
(4) $\sqrt{\frac{m_{1}}{m_{2}}}$

A spring of unstretched length $l$ has a mass $m$ with one end fixed to a rigid support. Assuming spring to be made of a uniform wire, the kinetic energy possessed by it if its free end is pulled with uniform velocity $v$ is:
(1) $\frac{1}{2}mv^{2}$
(2) $mv^{2}$
(3) $\frac{1}{3}mv^{2}$
(4) $\frac{1}{6}mv^{2}$

A particle is moving in a circle of radius $r$ under the action of a force $F = \alpha r ^ { 2 }$ which is directed towards centre of the circle. Total mechanical energy (kinetic energy + potential energy) of the particle is (take potential energy $= 0$ for $r = 0$ ):
(1) $\frac { 5 } { 6 } \alpha r ^ { 3 }$
(2) $\alpha r ^ { 3 }$
(3) $\frac { 1 } { 2 } \alpha r ^ { 3 }$
(4) $\frac { 4 } { 3 } \alpha r ^ { 3 }$

A body of mass $m$ starts moving from rest along $x$-axis so that its velocity varies as $v = a \sqrt { s }$ where $a$ is a constant and $s$ is the distance covered by the body. The total work done by all the forces acting on the body in the first $t$ second after the start of the motion is
(1) $8 m a ^ { 4 } t ^ { 2 }$
(2) $\frac { 1 } { 4 } m a ^ { 4 } t ^ { 2 }$
(3) $4 m a ^ { 4 } t ^ { 2 }$
(4) $\frac { 1 } { 8 } m a ^ { 4 } t ^ { 2 }$

A thin rod MN , free to rotate in the vertical plane about the fixed end N , is held horizontal. When the end M is released the speed of this end, when the rod makes an angle $\alpha$ with the horizontal, will be proportional to: [Figure]
(1) $\sqrt { \cos \alpha }$
(2) $\cos \alpha$
(3) $\sin \alpha$
(4) $\sqrt { \sin \alpha }$

A block of mass $m$, lying on a smooth horizontal surface, is attached to a spring (of negligible mass) of spring constant $k$. The other end of the spring is fixed, as shown in the figure. The block is initially at rest in its equilibrium position. If now the block is pulled with a constant force $F$, the maximum speed of the block is:
(1) $\frac { F } { \sqrt { m k } }$
(2) $\frac { 2 F } { \sqrt { m k } }$
(3) $\frac { \pi F } { \sqrt { m k } }$
(4) $\frac { F } { \pi \sqrt { m k } }$

A particle which is experiencing a force, given by $\vec { F } = 3 \hat { \mathrm { i } } - 12 \hat { \mathrm { j } }$, undergoes a displacement of $\vec { d } = 4 \hat { \mathrm { i } }$. If the particle had a kinetic energy of 3 J at the beginning of the displacement, what is its kinetic energy at the end of the displacement?
(1) 9 J.
(2) 15 J.
(3) 12 J.
(4) 10 J.

Consider a force $\vec { F } = - x \hat { i } + y \hat { j }$. The work done by this force in moving a particle from point $A ( 1,0 )$ to $B ( 0,1 )$ along the line segment is : (all quantities are in SI units)
(1) 2
(2) $\frac { 1 } { 2 }$
(3) 1
(4) $\frac { 3 } { 2 }$

A person pushes a box on a rough horizontal platform surface. He applies a force of $200$ N over a distance of 15 m. Thereafter, he gets progressively tired and his applied force reduces linearly with distance to 100 N. The total distance through which the box has been moved is 30 m. What is the work done by the person during the total movement of the box?
(1) 3280 J
(2) 2780 J
(3) 5690 J
(4) 5250 J

A particle ($\mathrm { m } = 1 \mathrm {~kg}$) slides down a frictionless track (AOC) starting from rest at a point $A$ (height 2 m). After reaching $C$, the particle continues to move freely in air as a projectile. When it reaching its highest point P (height 1 m), the kinetic energy of the particle (in J) is: (Figure drawn is schematic and not to scale; take $g = 10 \mathrm {~ms} ^ { - 2 }$) $\_\_\_\_$.

A boy is rolling a 0.5 kg ball on the frictionless floor with the speed of $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The ball gets deflected by an obstacle on the way. After deflection it moves with $5\%$ of its initial kinetic energy. What is the speed of the ball now?
(1) $19.0 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(2) $4.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(3) $14.41 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(4) $1.00 \mathrm {~m} \mathrm {~s} ^ { - 1 }$