Given below is the plot of a potential energy function $\mathrm { U } ( \mathrm { x } )$ for a system, in which a particle is in one dimensional motion, while a conservative force $\mathrm { F } ( \mathrm { x } )$ acts on it. Suppose that $\mathrm { E } _ { \text {mech} } = 8 \mathrm {~J}$, the incorrect statement for this system is:
[where K.E. = kinetic energy]
(1) at $\mathrm { x } > \mathrm { x } _ { 4 }$, K. E. is constant throughout the region.
(2) at $\mathrm { x } < \mathrm { x } _ { 1 }$, K. E. is smallest and the particle is moving at the slowest speed.
(3) at $\mathbf { x } = \mathbf { x } _ { 2 }$, K. E. is greatest and the particle is moving at the fastest speed.
(4) at $x = x _ { 3 }$, K.E. $= 4 \mathrm {~J}$

A body of mass $m$ dropped from a height $h$ reaches the ground with a speed of $0.8 \sqrt { g h }$. The value of work done by the air-friction is:
(1) $- 0.68 m g h$
(2) $m g h$
(3) 0.64 mgh
(4) 1.64 mgh

Two persons $A$ and $B$ perform same amount of work in moving a body through a certain distance $d$ with application of forces acting at angles $45^{\circ}$ and $60^{\circ}$ with the direction of displacement respectively. The ratio of force applied by person $A$ to the force applied by person $B$ is $\frac{1}{\sqrt{x}}$. The value of $x$ is $\_\_\_\_$.

As shown in the figure, a particle of mass 10 kg is placed at a point $A$. When the particle is slightly displaced to its right, it starts moving and reaches the point $B$. The speed of the particle at $B$ is $x \mathrm {~m} \mathrm {~s} ^ { - 1 }$. (Take $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$) The value of $x$ to the nearest integer is [Figure]

A pendulum bob has a speed of $3\mathrm{~m~s}^{-1}$ at its lowest position. The pendulum is 50 cm long. The speed of bob, when the length makes an angle of $60^{\circ}$ to the vertical will be ($g = 10\mathrm{~m~s}^{-2}$) \_\_\_\_ $\mathrm{m~s}^{-1}$.

A ball is projected with kinetic energy $E$, at an angle of $60 ^ { \circ }$ to the horizontal. The kinetic energy of this ball at the highest point of its flight will become :
(1) Zero
(2) $\frac { E } { 2 }$
(3) $\frac { E } { 4 }$
(4) $E$

A particle of mass 500 g is moving in a straight line with velocity $v = \mathrm { b } x ^ { \frac { 5 } { 2 } }$. The work done by the net force during its displacement from $x = 0$ to $x = 4 \mathrm {~m}$ is (Take $\mathrm { b } = 0.25 \mathrm {~m} ^ { \frac { - 3 } { 2 } } \mathrm {~s} ^ { - 1 }$).
(1) 2 J
(2) 4 J
(3) 8 J
(4) 16 J

A pendulum is suspended by a string of length 250 cm. The mass of the bob of the pendulum is 200 g. The bob is pulled aside until the string is at $60^\circ$ with vertical as shown in the figure. After releasing the bob, the maximum velocity attained by the bob will be $\_\_\_\_$ $\mathrm{m\,s^{-1}}$. (if $g = 10\mathrm{~m\,s^{-2}}$)

A block of mass '$m$' (as shown in figure) moving with kinetic energy $E$ compresses a spring through a distance 25 cm when, its speed is halved. The value of spring constant of used spring will be $nE$ N$^{-1}$ for $n =$ \_\_\_\_. [Figure]

A block is fastened to a horizontal spring. The block is pulled to a distance $x = 10 \mathrm {~cm}$ from its equilibrium position (at $x = 0$ ) on a frictionless surface from rest. The energy of the block at $x = 5 \mathrm {~cm}$ is 0.25 J . The spring constant of the spring is $\_\_\_\_$ $\mathrm { N } \mathrm { m } ^ { - 1 }$.

A force $F = \left( 5 + 3 y ^ { 2 } \right)$ acts on a particle in the $y$-direction, where $F$ is newton and $y$ is in meter. The work done by the force during a displacement from $y = 2 \mathrm {~m}$ to $y = 5 \mathrm {~m}$ is $\_\_\_\_$ J.

The surface tension of soap solution is $3.5 \times 10 ^ { - 2 } \mathrm {~N} \mathrm {~m} ^ { - 1 }$. The amount of work done required to increase the radius of soap bubble from 10 cm to 20 cm is $\_\_\_\_$ $\times 10 ^ { - 4 } \mathrm {~J}$. $\left($ take $\left. \pi = \frac { 22 } { 7 } \right)$

A car accelerates from rest to $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The energy spent in this process is $E \mathrm {~J}$. The energy required to accelerate the car from $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to $2u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is $nE \mathrm {~J}$. The value of $n$ is $\_\_\_\_$.

A block of mass 10 kg is moving along $x$-axis under the action of force $F = 5 x \mathrm {~N}$. The work done by the force in moving the block from $x = 2 \mathrm {~m}$ to 4 m will be $\_\_\_\_$ J.

A force $\vec{F} = (2 + 3x)\hat{i}$ acts on a particle in the $x$ direction where $F$ is in Newton and $x$ is in meter. The work done by this force during a displacement from $x = 0$ to $x = 4$ m is $\_\_\_\_$ J.

The bob of a pendulum was released from a horizontal position. The length of the pendulum is 10 m. If it dissipates $10 \%$ of its initial energy against air resistance, the speed with which the bob arrives at the lowest point is: [Use, $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$]
(1) $6 \sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(2) $5 \sqrt { 6 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(3) $5 \sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(4) $2 \sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$

A particle is placed at the point A of a frictionless track ABC as shown in figure. It is gently pushed towards right. The speed of the particle when it reaches the point $B$ is: (Take $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ ).
(1) $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(2) $\sqrt { 10 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(3) $2 \sqrt { 10 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(4) $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

A particle of mass $m$ moves on a straight line with its velocity increasing with distance according to the equation $v = \alpha \sqrt { x }$, where $\alpha$ is a constant. The total work done by all the forces applied on the particle during its displacement from $x = 0$ to $x = \mathrm { d }$, will be :
(1) $\frac { m } { 2 \alpha ^ { 2 } d }$
(2) $\frac { \mathrm { md } } { 2 \alpha ^ { 2 } }$
(3) $2 m \alpha ^ { 2 } d$
(4) $\frac { m \alpha ^ { 2 } d } { 2 }$

A body of mass 50 kg is lifted to a height of 20 m from the ground in the two different ways as shown in the figures. The ratio of work done against the gravity in both the respective cases, will be: Case 1: Vertically upward Case 2: Along the ramp
(1) $1:2$
(2) $\sqrt{3}:2$
(3) $2:1$
(4) $1:1$

If a rubber ball falls from a height $h$ and rebounds upto the height of $h / 2$. The percentage loss of total energy of the initial system as well as velocity of ball before it strikes the ground, respectively, are :
(1) $50 \% , \sqrt { 2 \mathrm { gh } }$
(2) $50 \% , \sqrt { \mathrm { gh } }$
(3) $40 \% , \sqrt { 2 \mathrm { gh } }$
(4) $50 \% , \sqrt { \frac { \mathrm { gh } } { 2 } }$

A solid sphere and a hollow cylinder roll up without slipping on same inclined plane with same initial speed $v$. The sphere and the cylinder reaches upto maximum heights $h _ { 1 }$ and $h _ { 2 }$, respectively, above the initial level. The ratio $h _ { 1 } : h _ { 2 }$ is $\frac { n } { 10 }$. The value of $n$ is $\_\_\_\_$.

Q4. A body of $m \mathrm {~kg}$ slides from rest along the curve of vertical circle from point $A$ to $B$ in friction less path. The [Figure] velocity of the body at $B$ is: (given, $R = 14 \mathrm {~m} , g = 10 \mathrm {~m} / \mathrm { s } ^ { 2 }$ and $\sqrt { 2 } = 1.4$ )
(1) $16.7 \mathrm {~m} / \mathrm { s }$
(2) $19.8 \mathrm {~m} / \mathrm { s }$
(3) $10.6 \mathrm {~m} / \mathrm { s }$
(4) $21.9 \mathrm {~m} / \mathrm { s }$

Q4. A particle of mass $m$ moves on a straight line with its velocity increasing with distance according to the equation $v = \alpha \sqrt { x }$, where $\alpha$ is a constant. The total work done by all the forces applied on the particle during its displacement from $x = 0$ to $x = \mathrm { d }$, will be :
(1) $\frac { m } { 2 \alpha ^ { 2 } d }$
(2) $\frac { \mathrm { md } } { 2 \alpha ^ { 2 } }$
(3) $2 m \alpha ^ { 2 } d$
(4) $\frac { m \alpha ^ { 2 } d } { 2 }$

Q4. A satellite of $10 ^ { 3 } \mathrm {~kg}$ mass is revolving in circular orbit of radius $2 R$. If $\frac { 10 ^ { 4 } R } { 6 } J$ energy is supplied to the satellite, it would revolve in a new circular orbit of radius (use $g = 10 \mathrm {~m} / \mathrm { s } ^ { 2 } , R =$ radius of earth)
(1) $2.5 R$
(2) $3 R$
(3) $4 R$
(4) $6 R$

Q5. A body of mass 50 kg is lifted to a height of 20 m from the ground in the two different ways as shown in the figures. The ratio of work done against the gravity in both the respective cases, will be :
[Figure]
(1) $1 : 2$
(3) $2 : 1$
[Figure]
Case $- 2 \rightarrow$ Along the ramp
(2) $\sqrt { 3 } : 2$
(4) $1 : 1$