Let $x \in C ^ { 2 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { n } \right) \right. \right.$ be a solution of the differential equation (1). For each real $t \geq 0$ we set, $T \left( x ^ { \prime } \right) ( t ) = \frac { 1 } { 2 } \left\langle A x ^ { \prime } ( t ) ; x ^ { \prime } ( t ) \right\rangle$ and $U ( x ) ( t ) = \frac { 1 } { 2 } \langle K x ( t ) ; x ( t ) \rangle$. Show then that the quantity $T \left( x ^ { \prime } \right) ( t ) + U ( x ) ( t )$ does not depend on $t \in [ 0 , + \infty [$.

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 block rests on a rough inclined plane making an angle of $30 ^ { \circ }$ with the horizontal. The coefficient of static friction between the block and the plane is 0.8 . If the frictional force on the block is 10 N , the mass of the block (in kg ) is (take $\mathrm { g } = 10 \mathrm {~m} / \mathrm { s } ^ { 2 }$ )
(1) 2.0
(2) 4.0
(3) 1.6
(4) 2.5

A particle moves in a straight line with retardation proportional to its displacement. Its loss of kinetic energy for any displacement x is proportional to
(1) $x ^ { 3 }$
(2) $e ^ { x }$
(3) $x$
(4) $\log _ { e } x$

An annular ring with inner and outer radii $R_1$ and $R_2$ is rolling without slipping with a uniform angular speed. The ratio of the forces experienced by the two particles situated on the inner and outer parts of the ring, $F_1/F_2$ is
(1) $\frac{R_2}{R_1}$
(2) $\left(\frac{R_1}{R_2}\right)^2$
(3) 1
(4) $\frac{R_1}{R_2}$

A bullet fired into a fixed target loses half of its velocity after penetrating 3 cm. How much further it will penetrate before coming to rest assuming that it faces constant resistance to motion?
(1) 3.0 cm
(2) 2.0 cm
(3) 1.5 cm
(4) 1.0 cm

A solid sphere is rolling on a surface as shown in figure, with a translational velocity $v \mathrm{~ms}^{-1}$. If it is to climb the inclined surface continuing to roll without slipping, then minimum velocity for this to happen is
(1) $\sqrt{2gh}$
(2) $\sqrt{\frac{7}{5}gh}$
(3) $\sqrt{\frac{7}{2}gh}$
(4) $\sqrt{\frac{10}{7}gh}$

A small ball of mass $m$ starts at a point $A$ with speed $v_o$ and moves along a frictionless track $AB$ as shown. The track BC has coefficient of friction $\mu$. The ball comes to stop at C after travelling a distance $L$ which is:
(1) $\frac{2\mathrm{~h}}{\mu} + \frac{\mathrm{v}_\mathrm{o}^2}{2\mu\mathrm{~g}}$
(2) $\frac{\mathrm{h}}{\mu} + \frac{\mathrm{v}_0^2}{2\mu\mathrm{~g}}$
(3) $\frac{\mathrm{h}}{2\mu} + \frac{\mathrm{v}_\mathrm{o}^2}{\mu\mathrm{g}}$
(4) $\frac{\mathrm{h}}{2\mu} + \frac{\mathrm{v}_\mathrm{o}^2}{2\mu\mathrm{g}}$

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 bullet of mass 20 g has an initial speed of $1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, just before it starts penetrating a mud wall of thickness 20 cm . If the wall offers a mean resistance of $2.5 \times 10 ^ { - 2 } \mathrm {~N}$, the speed of the bullet after emerging from the other side of the wall is close to:
(1) $0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(2) $0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(3) $0.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(4) $0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

A body of mass 1 kg falls freely from a height of 100 m, on a platform of mass 3 kg which is mounted on a spring having spring constant $\mathrm { k } = 1.25 \times 10 ^ { 6 } \mathrm {~N} / \mathrm { m }$. The body sticks to the platform and the spring's maximum compression is found to be $x$. Given that $g = 10 \mathrm {~ms} ^ { - 2 }$, the value of x will be close to:
(1) 40 cm
(2) 4 cm
(3) 80 cm
(4) 8 cm

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.

A cricket ball of mass 0.15 kg is thrown vertically up by a bowling machine so that it rises to a maximum height of 20 m after leaving the machine. If the part pushing the ball applies a constant force $F$ on the ball and moves horizontally a distance of 0.2 m while launching the ball, the value of $F$ (in N) is $\left( g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 } \right)$

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) $1.00 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

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

A cord is wound round the circumference of wheel of radius $r$, The axis of the wheel is horizontal and the moment of inertia about it is $I$. A weight $m g$ is attached to the cord at the end. The weight falls from rest. After falling through a distance h , the square of angular velocity of wheel will be
(1) $\frac { 2 m g h } { I + m r ^ { 2 } }$
(2) $\frac { 2 m g h } { I + 2 m r ^ { 2 } }$
(3) $2 g h$
(4) $\frac { 2 g h } { I + m r ^ { 2 } }$

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 $\_\_\_\_$.

A particle experiences a variable force $\overrightarrow { \mathrm { F } } = \left( 4 x \hat { i } + 3 y ^ { 2 } \hat { j } \right)$ in a horizontal $x - y$ plane. Assume distance in meters and force is newton. If the particle moves from point $( 1,2 )$ to point $( 2,3 )$ in the $x - y$ plane, then Kinetic Energy changes by :
(1) 25 J
(2) 50 J
(3) 12.5 J
(4) 0 J

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

In the given figure, the block of mass $m$ is dropped from the point $|A|$. The expression for kinetic energy of block when it reaches point $|B|$ is
(1) $mgy_0$
(2) $\frac{1}{2}mgy_0^2$
(3) $\frac{1}{2}mgy^2$
(4) $mg(y - y_0)$

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.