Mass per unit area of a circular disc of radius a depends on the distance $r$ from its centre as $\sigma ( r ) = A + B r$. The moment of inertia of the disc about the axis, perpendicular to the plane and passing through its centre is:
(1) $2 \pi a ^ { 4 } \left( \frac { A } { 4 } + \frac { a B } { 5 } \right)$
(2) $2 \pi a ^ { 4 } \left( \frac { a A } { 4 } + \frac { B } { 5 } \right)$
(3) $\pi a ^ { 4 } \left( \frac { A } { 4 } + \frac { a B } { 5 } \right)$
(4) $2 \pi \mathrm { a } ^ { 4 } \left( \frac { \mathrm {~A} } { 4 } + \frac { \mathrm { B } } { 5 } \right)$

As shown in figure. When a spherical cavity (centred at $O$) of radius 1 is cut out of a uniform sphere of radius $R$ (centred at $C$), the centre of mass of remaining (shaded part of sphere) is at $G$, i.e., on the surface of the cavity. $R$ can be determined by the equation:
(1) $\left(R^{2} + R + 1\right)(2 - R) = 1$
(2) $\left(R^{2} - R - 1\right)(2 - R) = 1$
(3) $\left(R^{2} - R + 1\right)(2 - R) = 1$
(4) $\left(R^{2} + R - 1\right)(2 - R) = 1$

Three solid spheres each of mass $m$ and diameter $d$ are stuck together such that the lines connecting the centres form an equilateral triangle of side of length $d$. The ratio $\frac { I _ { 0 } } { I _ { A } }$ of moment of inertia $I _ { 0 }$ of the system about an axis passing the centroid and about center of any of the spheres $I _ { A }$ and perpendicular to the plane of the triangle is:
(1) $\frac { 13 } { 23 }$
(2) $\frac { 15 } { 13 }$
(3) $\frac { 23 } { 13 }$
(4) $\frac { 13 } { 15 }$

Blocks of masses $\mathrm { m } , 2 \mathrm {~m} , 4 \mathrm {~m}$ and 8 m are arranged in a line of a frictionless floor. Another block of mass m , moving with speed $v$ along the same line (see figure) collides with mass m in perfectly inelastic manner. All the subsequent collisions are also perfectly inelastic. By the time the last block of mass 8 m starts moving the total energy loss is $\mathrm { p } \%$ of the original energy. Value of 'p' is close to:
(1) 77
(2) 94
(3) 37
(4) 87

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

The acceleration due to gravity on the earth's surface at the poles is $g$ and angular velocity of the earth about the axis passing through the pole is $\omega$. An object is weighed at the equator and at a height $h$ above the poles by using a spring balance. If the weights are found to be same, then $h$ is: ($h \ll R$, where $R$ is the radius of the earth)
(1) $\frac{R^2\omega^2}{2g}$
(2) $\frac{R^2\omega^2}{g}$
(3) $\frac{R^2\omega^2}{4g}$
(4) $\frac{R^2\omega^2}{8g}$

Consider a uniform rod of mass $M = 4 m$ and length $l$ pivoted about its centre. A mass $m$ moving with velocity $v$ making angle $\theta = \frac { \pi } { 4 }$ to the rod's long axis collides with one end of the rod and sticks to it. The angular speed of the rod-mass system just after the collision is:
(1) $\frac { 3 } { 7 \sqrt { 2 } } \frac { v } { l }$
(2) $\frac { 3 } { 7 } \frac { v } { l }$
(3) $\frac { 3 \sqrt { 2 } } { 7 } \frac { v } { l }$
(4) $\frac { 4 } { 7 } \frac { v } { l }$

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 box weighs 196 N on a spring balance at the north pole. Its weight recorded on the same balance if it is shifted to the equator is close to (Take $\mathrm { g } = 10 \mathrm {~ms} ^ { - 2 }$ at the north pole and the radius of the earth $= 6400 \mathrm {~km}$):
(1) 195.66 N
(2) 194.32 N
(3) 194.66 N
(4) 195.32 N

A particle of mass $m$ is dropped from a height $h$ above the ground. At the same time another particle of the same mass is thrown vertically upwards from the ground with a speed of $\sqrt{2gh}$. If they collide head-on completely inelastically, the time taken for the combined mass to reach the ground, in units of $\sqrt{\frac{h}{g}}$ is:
(1) $\sqrt{\frac{1}{2}}$
(2) $\sqrt{\frac{3}{4}}$
(3) $\frac{1}{2}$
(4) $\sqrt{\frac{3}{2}}$

A body A of mass $m$ is moving in a circular orbit of radius $R$ about a planet. Another body B of mass $\frac { m } { 2 }$ collides with A with a velocity which is half $\left( \frac { \vec { v } } { 2 } \right)$ the instantaneous velocity $\vec { v }$ of A. The collision is completely inelastic. Then, the combined body:
(1) continues to move in a circular orbit
(2) Escapes from the Planet's Gravitational field
(3) Falls vertically downwards towards the planet
(4) starts moving in an elliptical orbit around the planet

On the $x$-axis and at a distance $x$ from the origin, the gravitational field due to a mass distribution is given by $\frac { A x } { \left( x ^ { 2 } + a ^ { 2 } \right) ^ { 3 / 2 } }$ in the $x$-direction. The magnitude of the gravitational potential on the $x$-axis at a distance $x$, taking its value to be zero at infinity is:
(1) $\frac { A } { \left( x ^ { 2 } + a ^ { 2 } \right) ^ { 1 / 2 } }$
(2) $\frac { A } { \left( x ^ { 2 } + a ^ { 2 } \right) ^ { 3 / 2 } }$
(3) $A \left( x ^ { 2 } + a ^ { 2 } \right) ^ { 1 / 2 }$
(4) $A \left( x ^ { 2 } + a ^ { 2 } \right) ^ { 3 / 2 }$

Consider two uniform discs of the same thickness and different radii $R _ { 1 } = R$ and $R _ { 2 } = \alpha R$ made of the same material. If the ratio of their moments of inertia $I _ { 1 }$ and $I _ { 2 }$, respectively, about their axes is $I _ { 1 } : I _ { 2 } = 1 : 16$ then the value of $\alpha$ is:
(1) $2 \sqrt { 2 }$
(2) $\sqrt { 2 }$
(3) 2
(4) 4

In an experiment to verify Stokes law, a small spherical ball of radius $r$ and density $\rho$ falls under gravity through a distance $h$ in air before entering a tank of water. If the terminal velocity of the ball inside water is same as its velocity just before entering the water surface, then the value of $h$ is proportional to: (ignore viscosity of air)
(1) $r^4$
(2) $r$
(3) $r^3$
(4) $r^2$

Consider two solid spheres of radii $R _ { 1 } = 1 \mathrm {~m} , R _ { 2 } = 2 \mathrm {~m}$ and masses $M _ { 1 }$ and $M _ { 2 }$, respectively. The gravitational field due to sphere (1) and (2) are shown. The value of $\frac { M _ { 1 } } { M _ { 2 } }$ is:
(1) $\frac { 2 } { 3 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 1 } { 3 }$

An ideal fluid flows (laminar flow) through a pipe of non-uniform diameter. The maximum and minimum diameters of the pipes are 6.4 cm and 4.8 cm, respectively. The ratio of the minimum and the maximum velocities of fluid in this pipe is:
(1) $\frac { 9 } { 16 }$
(2) $\frac { \sqrt { 3 } } { 2 }$
(3) $\frac { 3 } { 4 }$
(4) $\frac { 81 } { 256 }$

A uniform sphere of mass 500 g rolls without slipping on a plane horizontal surface with its centre moving at a speed of $5.00\,\mathrm{cm\,s^{-1}}$. Its kinetic energy is:
(1) $8.75 \times 10^{-4}\,\mathrm{J}$
(2) $8.75 \times 10^{-3}\,\mathrm{J}$
(3) $6.25 \times 10^{-4}\,\mathrm{J}$
(4) $1.13 \times 10^{-3}\,\mathrm{J}$

For a uniform rectangular sheet shown in the figure, the ratio of moments of inertia about the axes perpendicular to the sheet and passing through O (the centre of mass) and $\mathrm { O } ^ { \prime }$ (corner point) is:
(1) $2/3$
(2) $1/4$
(3) $1/8$
(4) $1/2$

Two liquids of densities $\rho_{1}$ and $\rho_{2}$ ($\rho_{2} = 2\rho_{1}$) are filled up behind a square wall of side 10 m as shown in figure. Each liquid has a height of 5 m. The ratio of the forces due to these liquids exerted on upper part MN to that at the lower part NO is (Assume that the liquids are not mixing):
(1) $\frac{1}{3}$
(2) $\frac{2}{3}$
(3) $\frac{1}{2}$
(4) $\frac{1}{4}$

A body is moving in a low circular orbit about a planet of mass $M$ and radius $R$. The radius of the orbit can be taken to be $R$ itself. Then the ratio of the speed of this body in the orbit to the escape velocity from the planet is:
(1) $\frac { 1 } { \sqrt { 2 } }$
(2) 2
(3) 1
(4) $\sqrt { 2 }$

Consider a solid sphere of radius $R$ and mass density $\rho ( r ) = \rho _ { 0 } \left( 1 - \frac { r ^ { 2 } } { R ^ { 2 } } \right) , 0 < r \leq R$. The minimum density of a liquid in which it will float is:
(1) $\frac { \rho _ { 0 } } { 3 }$
(2) $\frac { \rho _ { 0 } } { 5 }$
(3) $\frac { 2 \rho _ { 0 } } { 5 }$
(4) $\frac { 2 \rho _ { 0 } } { 3 }$

Two ideal Carnot engines operate in cascade (all heat given up by one engine is used by the other engine to produce work) between temperatures, $\mathrm { T } _ { 1 }$ and $\mathrm { T } _ { 2 }$. The temperature of the hot reservoir of the first engine is $\mathrm { T } _ { 1 }$ and the temperature of the cold reservoir of the second engine is $T _ { 2 }$. $T$ is temperature of the sink of first engine which is also the source for the second engine. How is T related to $\mathrm { T } _ { 1 }$ and $\mathrm { T } _ { 2 }$, if both the engines perform equal amount of work?
(1) $\mathrm { T } = \frac { 2 \mathrm {~T} _ { 1 } \mathrm {~T} _ { 2 } } { \mathrm {~T} _ { 1 } + \mathrm { T } _ { 2 } }$
(2) $\mathrm { T } = \frac { \mathrm { T } _ { 1 } + \mathrm { T } _ { 2 } } { 2 }$
(3) $T = \sqrt { T _ { 1 } T _ { 2 } }$
(4) $\mathrm { T } = 0$

A ring is hung on a nail. It can oscillate, without slipping or sliding (i) in its plane with a time period $T_1$ and (ii) back and forth in a direction perpendicular to its plane, with a period $T_2$. The ratio $\frac{T_1}{T_2}$ will be:
(1) $\frac{2}{\sqrt{3}}$
(2) $\frac{2}{3}$
(3) $\frac{3}{\sqrt{2}}$
(4) $\frac{\sqrt{2}}{3}$

A stationary observer receives sound from two identical tuning forks, one of which approaches and the other one recedes with the same speed (much less than the speed of sound). The observer hears 2 beats/sec. The oscillation frequency of each tuning fork is $v _ { 0 } = 1400 \mathrm {~Hz}$ and the velocity of sound in air is $350 \mathrm {~m} / \mathrm { s }$. The speed of each tuning fork is close to:
(1) $\frac { 1 } { 2 } \mathrm {~m} / \mathrm { s }$
(2) $1 \mathrm {~m} / \mathrm { s }$
(3) $\frac { 1 } { 4 } \mathrm {~m} / \mathrm { s }$
(4) $\frac { 1 } { 8 } \mathrm {~m} / \mathrm { s }$

An emf of 20 V is applied at time $t = 0$ to a circuit containing in series 10 mH inductor and $5 \Omega$ resistor. The ratio of the currents at time $t = \infty$ and at $t = 40 \mathrm {~s}$ is close to: (Take $e ^ { 2 } = 7.389$)
(1) 1.06
(2) 1.15
(3) 1.46
(4) 0.84