STATEMENT-1
A block of mass $m$ starts moving on a rough horizontal surface with a velocity $v$. It stops due to friction between the block and the surface after moving through a certain distance. The surface is now tilted to an angle of $30^\circ$ with the horizontal and the same block is made to go up on the surface with the same initial velocity $v$. The decrease in the mechanical energy in the second situation is smaller than that in the first situation. because STATEMENT-2 The coefficient of friction between the block and the surface decreases with the increase in the angle of inclination.
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
(B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
(C) Statement-1 is True, Statement-2 is False
(D) Statement-1 is False, Statement-2 is True

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 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 small ball of mass $m$ is thrown upward with velocity $u$ from the ground. The ball experiences a resistive force $m k v ^ { 2 }$ where $v$ is it speed. The maximum height attained by the ball is:
(1) $\frac { 1 } { 2 k } \tan ^ { - 1 } \frac { k u ^ { 2 } } { g }$
(2) $\frac { 1 } { k } \ln \left( 1 + \frac { k u ^ { 2 } } { 2 g } \right)$
(3) $\frac { 1 } { k } \tan ^ { - 1 } \frac { k u ^ { 2 } } { 2 g }$
(4) $\frac { 1 } { 2 k } \ln \left( 1 + \frac { k u ^ { 2 } } { g } \right)$

A satellite of mass $M$ is launched vertically upwards with an initial speed $u$ from the surface of the earth. After it reaches height $R$ ($R =$ radius of the earth), it ejects a rocket of mass $\frac { M } { 10 }$ so that subsequently the satellite moves in a circular orbit. The kinetic energy of the rocket is ($G$ is the gravitational constant; $M _ { e }$ is the mass of the earth):
(1) $\frac { M } { 20 } \left( u ^ { 2 } + \frac { 113 } { 200 } \frac { G M _ { e } } { R } \right)$
(2) $5 M \left( u ^ { 2 } - \frac { 119 } { 200 } \frac { G M _ { e } } { R } \right)$
(3) $\frac { 3 M } { 8 } \left( u + \sqrt { \frac { 5 G M _ { e } } { 6 R } } \right) ^ { 2 }$
(4) $\frac { M } { 20 } \left( u - \sqrt { \frac { 2 G M _ { e } } { 3 R } } \right) ^ { 2 }$

A body of mass 2 kg is driven by an engine delivering a constant power of $1\,\mathrm{J\,s^{-1}}$. The body starts from rest and moves in a straight line. After 9 s, the body has moved a distance (in m) ...

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) $\frac { 2 g h } { I + m r ^ { 2 } }$

An engine is attached to a wagon through a shock absorber of length 1.5 m. The system with a total mass of $40,000 \mathrm {~kg}$ is moving with a speed of $72 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ when the brakes are applied to bring it to rest. In the process of the system being brought to rest, the spring of the shock absorber gets compressed by 1.0 m. If $90 \%$ of energy of the wagon is lost due to friction, the spring constant is $\_\_\_\_$ $\times 10 ^ { 5 } \mathrm {~N} \mathrm {~m} ^ { - 1 }$.

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 space ship of mass $2 \times 10 ^ { 4 } \mathrm {~kg}$ is launched into a circular orbit close to the earth surface. The additional velocity to be imparted to the space ship in the orbit to overcome the gravitational pull will be (if $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ and radius of earth $= 6400 \mathrm {~km}$):
(1) $11.2 ( \sqrt { 2 } - 1 ) \mathrm { km } \mathrm { s } ^ { - 1 }$
(2) $8 ( \sqrt { 2 } - 1 ) \mathrm { km } \mathrm { s } ^ { - 1 }$
(3) $7.9 ( \sqrt { 2 } - 1 ) \mathrm { km } \mathrm { s } ^ { - 1 }$
(4) $7.4 ( \sqrt { 2 } - 1 ) \mathrm { km } \mathrm { s } ^ { - 1 }$

A particle of mass 10 g moves in a straight line with retardation $2x$, where $x$ is the displacement in SI units. Its loss of kinetic energy for above displacement is $\frac{10^{-n}}{x}\mathrm{~J}$. The value of $n$ will be $\_\_\_\_$.

An astronaut takes a ball of mass $m$ from earth to space. He throws the ball into a circular orbit about earth at an altitude of 318.5 km. From earth's surface to the orbit, the change in total mechanical energy of the ball is $x \frac { \mathrm { GM } _ { \mathrm { e } } \mathrm { m } } { 21 \mathrm { R } _ { \mathrm { e } } }$. The value of $x$ is (take $\mathrm { R } _ { \mathrm { e } } = 6370 \mathrm {~km}$) :
(1) 10
(2) 12
(3) 9
(4) 11

A wire of length $L$ and radius $r$ is clamped at one end. If its other end is pulled by a force $F$, its length increases by $l$. If the radius of the wire and the applied force both are reduced to half of their original values keeping original length constant, the increase in length will become:
(1) 3 times
(2) $\frac { 3 } { 2 }$ times
(3) 4 times
(4) 2 times

A small liquid drop of radius $R$ is divided into 27 identical liquid drops. If the surface tension is $T$, then the work done in the process will be:
(1) $8 \pi R ^ { 2 } T$
(2) $3 \pi R ^ { 2 } T$
(3) $\frac { 1 } { 8 } \pi R ^ { 2 } T$
(4) $4 \pi R ^ { 2 } T$

Two metallic wires $P$ and $Q$ have same volume and are made up of same material. If their area of cross sections are in the ratio $4 : 1$ and force $F _ { 1 }$ is applied to $P$, an extension of $\Delta l$ is produced. The force which is required to produce same extension in Q is $F _ { 2 }$. The value of $\frac { F _ { 1 } } { F _ { 2 } }$ is $\_\_\_\_$ .

Q6. An astronaut takes a ball of mass $m$ from earth to space. He throws the ball into a circular orbit about earth at an altitude of 318.5 km . From earth's surface to the orbit, the change in total mechanical energy of the ball is $x \frac { \mathrm { GM } _ { \mathrm { e } } \mathrm { m } } { 21 \mathrm { R } _ { \mathrm { e } } }$. The value of $x$ is (take $\mathrm { R } _ { \mathrm { e } } = 6370 \mathrm {~km}$ ) :
(1) 10
(2) 12
(3) 9
(4) 11