A metal coin of mass 5 g and radius 1 cm is fixed to a thin stick $AB$ of negligible mass as shown in the figure. The system is initially at rest. The constant torque, that will make the system rotate about AB at 25 rotations per second in 5 s , is close to:
(1) $1.6 \times 10 ^ { - 5 } \mathrm {~N} \mathrm {~m}$
(2) $2.0 \times 10 ^ { - 5 } \mathrm {~N} \mathrm {~m}$
(3) $7.9 \times 10 ^ { - 6 } \mathrm {~N} \mathrm {~m}$
(4) $4.0 \times 10 ^ { - 6 } \mathrm {~N} \mathrm {~m}$

A rigid massless rod of length $3l$ has two masses attached at each end as shown in the figure. The rod is pivoted at point $P$ on the horizontal axis. When released from the initial horizontal position, its instantaneous angular acceleration will be
(1) $\frac { g } { 2l }$
(2) $\frac { 7g } { 3l }$
(3) $\frac { g } { 3l }$
(4) $\frac { g } { 13l }$

A satellite is revolving in a circular orbit at a height $h$ from the earth surface, such that $h \ll R$ where $R$ is the radius of the earth. Assuming that the effect of earth's atmosphere can be neglected the minimum increase in the speed required so that the satellite could escape from the gravitational field of earth is
(1) $\sqrt { 2gR }$
(2) $\sqrt { gR }$
(3) $\sqrt { \frac { gR } { 2 } }$
(4) $\sqrt { gR } ( \sqrt { 2 } - 1 )$

A straight rod of length $L$ extends from $x = a$ to $x = L + a$. The gravitational force it exerts on a point mass '$m$' at $x = 0$, if the mass per unit length of the rod is $A + B x ^ { 2 }$, is given by:
(1) $G m \left[ A \left( \frac { 1 } { a + L } - \frac { 1 } { a } \right) + B L \right]$
(2) $G m \left[ A \left( \frac { 1 } { a + L } - \frac { 1 } { a } \right) - B L \right]$
(3) $G m \left[ A \left( \frac { 1 } { a } - \frac { 1 } { a + L } \right) - B L \right]$
(4) $G m \left[ A \left( \frac { 1 } { a } - \frac { 1 } { a + L } \right) + B L \right]$

A circular disc of radius $b$ has a hole of radius $a$ at its centre. If the mass per unit area of the disc varies as $\frac{\sigma_0}{r}$ then, the radius of gyration of the disc about its axis passing through the center is
(1) $\frac{a+b}{3}$
(2) $\sqrt{\frac{a^2+b^2+ab}{3}}$
(3) $\frac{a+b}{2}$
(4) $\sqrt{\frac{a^2+b^2+ab}{2}}$

The ratio of the weights of a body on Earth's surface to that on the surface of a planet is $9 : 4$. The mass of the planet is $\frac { 1 } { 9 }$th of that of the Earth. If $R$ is the radius of the Earth, what is the radius of the planet? (Take the planets to have the same mass density)
(1) $\frac { R } { 4 }$
(2) $\frac { R } { 2 }$
(3) $\frac { R } { 3 }$
(4) $\frac { R } { 9 }$

The mass and the diameter of a planet are three times the respective values for the Earth. The period of oscillation of a simple pendulum on the Earth is 2 s. The period of oscillation of the same pendulum on the planet would be:
(1) $\frac { \sqrt { 3 } } { 2 } \mathrm {~s}$
(2) $\frac { 2 } { \sqrt { 3 } } \mathrm {~s}$
(3) $\frac { 3 } { 2 } s$
(4) $2 \sqrt { 3 } \mathrm {~s}$

The ratio of surface tensions of mercury and water is given to be 7.5, while the ratio of their densities is 13.6. Their contact angles, with glass, are close to $135 ^ { \circ }$ and $0 ^ { \circ }$, respectively. If it is observed that mercury gets depressed by an amount $h$ in a capillary tube of radius $r _ { 1 }$, while water rises by the same amount $h$ in a capillary tube of radius $r _ { 2 }$, then the ratio $\frac { r _ { 1 } } { r _ { 2 } }$ is close to
(1) $\frac { 3 } { 5 }$
(2) $\frac { 2 } { 3 }$
(3) $\frac { 4 } { 5 }$
(4) $\frac { 2 } { 5 }$

The time dependence of the position of a particle of mass $m = 2$ is given by $\vec { r } t = 2 t \hat { i } - 3 t ^ { 2 } \hat { j }$. Its angular momentum, with respect to the origin, at time $\mathrm { t } = 2$ is:
(1) 36 k
(2) $48 \hat { i } + \hat { j }$
(3) $- 48 \hat{k}$
(4) $- 34 \mathrm { k } - \hat { \mathrm { i } }$

Two stars of masses $3 \times 10 ^ { 31 } \mathrm {~kg}$ each, and at distance $2 \times 10 ^ { 11 } \mathrm {~m}$ rotate in a plane about their common centre of mass O. A meteorite passes through O moving perpendicular to the stars' rotation plane. In order to escape from the gravitational field of this double star, the minimum speed that meteorite should have at O is (Take Gravitational constant $G = 6.67 \times 10 ^ { - 11 } \mathrm {~N} \mathrm {~m} ^ { 2 } \mathrm {~kg} ^ { - 2 }$)
(1) $2.4 \times 10 ^ { 4 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(2) $3.8 \times 10 ^ { 4 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(3) $2.8 \times 10 ^ { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(4) $1.4 \times 10 ^ { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$

A liquid of density $\rho$ is coming out of a hose pipe of radius a with horizontal speed $v$ and hits a mesh. $50\%$ of the liquid passes through the mesh unaffected. $25\%$ looses all of its momentum and $25\%$ comes back with the same speed. The resultant pressure on the mesh will be:
(1) $\frac { 1 } { 4 } \rho v ^ { 2 }$
(2) $\frac { 3 } { 4 } \rho v ^ { 2 }$
(3) $\frac { 1 } { 2 } \rho v ^ { 2 }$
(4) $\rho v ^ { 2 }$

A solid sphere, of radius R acquires a terminal velocity $v _ { 1 }$ when falling (due to gravity) through a viscous fluid having a coefficient of viscosity $\eta$. The sphere is broken into 27 identical solid spheres. If each of these spheres acquires a terminal velocity, $v _ { 2 }$, when falling through the same fluid, the ratio $\left( \frac { v _ { 1 } } { v _ { 2 } } \right)$ equals:
(1) $\frac { 1 } { 9 }$
(2) 27
(3) $\frac { 1 } { 27 }$
(4) 9

If the angular momentum of a planet of mass $m$, moving around the Sun in a circular orbit is $L$, about the center of the Sun, its areal velocity is:
(1) $\frac { L } { m }$
(2) $\frac { 4 L } { m }$
(3) $\frac { m } { 2 m }$
(4) $\frac { m } { m }$

n moles of an ideal gas with constant volume heat capacity $\mathrm { C } _ { \mathrm { v } }$ undergo an isobaric expansion by certain volume. The ratio of the work done in the process, to the heat supplied is:
(1) $\frac { 4 n R } { \mathrm { C } _ { \mathrm { v } } + n R }$
(2) $\frac { 4 n R } { C _ { v } - n R }$
(3) $\frac { n R } { C _ { v } + n R }$
(4) $\frac { n R } { \mathrm { C } _ { \mathrm { v } } - n R }$

A spaceship orbits around a planet at a height of 20 km from its surface. Assuming that only gravitational field of the planet acts on the spaceship, what will be the number of complete revolutions made by the spaceship in 24 hours around the planet? [Given: Mass of planet $= 8 \times 10 ^ { 22 } \mathrm {~kg}$, Radius of planet $= 2 \times 10 ^ { 6 } \mathrm {~m}$, Gravitational constant $\mathrm { G } = 6.67 \times 10 ^ { - 11 } \mathrm { Nm } ^ { 2 } / \mathrm { kg } ^ { 2 }$ ]
(1) 17
(2) 9
(3) 13
(4) 11

A heavy ball of mass $M$ is suspended from the ceiling of a car by a light string of mass $m$ ($m \ll M$). When the car is at rest, the speed of transverse waves in the string is $60 \mathrm {~ms} ^ { - 1 }$. When the car has acceleration $a$, the wavespeed increases to $60.5 \mathrm {~ms} ^ { - 1 }$. The value of $a$, in terms of gravitational acceleration $g$, is closest to
(1) $\frac { g } { 10 }$
(2) $\frac { g } { 20 }$
(3) $\frac { g } { 5 }$
(4) $\frac { g } { 30 }$

A cylinder with fixed capacity of 67.2 litre contains helium gas at STP. The amount of heat needed to raise the temperature of the gas by $20 ^ { \circ } \mathrm { C }$ is: [Given that $\mathrm { R } = 8.31 \mathrm {~J} \mathrm {~mol} ^ { - 1 } \mathrm {~K} ^ { - 1 }$]
(1) 748 J
(2) 700 J
(3) 350 J
(4) 374 J

The elastic limit of brass is 379 MPa . The minimum diameter of a brass rod if it is to support a 400 N load without exceeding its elastic limit will be
(1) 1.00 mm
(2) 1.36 mm
(3) 1.16 mm
(4) 0.90 mm

A $25 \times 10 ^ { - 3 } \mathrm {~m} ^ { 3 }$ volume cylinder is filled with 1 mol of $\mathrm { O } _ { 2 }$ gas at room temperature (300 K). The molecular diameter of $\mathrm { O } _ { 2 }$, and its root mean square speed, are found to be 0.3 nm and $200 \mathrm {~m} / \mathrm { s }$, respectively. What is the average collision rate (per second) for an $\mathrm { O } _ { 2 }$ molecule?
(1) $\sim 10 ^ { 11 }$
(2) $\sim 10 ^ { 12 }$
(3) $\sim 10 ^ { 10 }$
(4) $\sim 10 ^ { 13 }$

In an experiment, brass and steel wires of length 1 m each with areas of cross section $1 \mathrm {~mm} ^ { 2 }$ are used. The wires are connected in series and one end of the combined wire is connected to a rigid support and other end is subjected to elongation. The stress required to produce a net elongation of 0.2 mm is, [Given, the Young's Modulus for steel and brass are, respectively, $120 \times 10 ^ { 9 } \mathrm {~N} / \mathrm { m } ^ { 2 }$ and $60 \times 10 ^ { 9 } \mathrm {~N} / \mathrm { m } ^ { 2 }$ ]
(1) $8.0 \times 10 ^ { 6 } \mathrm {~N} / \mathrm { m } ^ { 2 }$
(2) $1.2 \times 10 ^ { 6 } \mathrm {~N} / \mathrm { m } ^ { 2 }$
(3) $0.2 \times 10 ^ { 6 } \mathrm {~N} / \mathrm { m } ^ { 2 }$
(4) $1.8 \times 10 ^ { 6 } \mathrm {~N} / \mathrm { m } ^ { 2 }$

The displacement of a damped harmonic oscillator is given by $x ( \mathrm { t } ) = \mathrm { e } ^ { - 0.1 \mathrm { t } } \cos ( 10 \pi \mathrm { t } + \varphi )$. Here t is in seconds. The time taken for its amplitude of vibration to drop to half of its initial value is close to:
(1) 27 s
(2) 4 s
(3) 13 s
(4) 7 s

A cubical block of side 0.5 m floats on water with $30\%$ of its volume under water. What is the maximum weight that can be put on the block without fully submerging it under water? [Take, density of water $= 10 ^ { 3 } \mathrm {~kg} / \mathrm { m } ^ { 3 }$ ]
(1) 87.5 kg
(2) 65.4 kg
(3) 30.1 kg
(4) 46.3 kg

A stationary source emits sound waves of frequency 500 Hz. Two observers moving along a line passing through the source detect sound to be of frequencies 480 Hz and 530 Hz. Their respective speeds are, in $\mathrm { m } \mathrm {~s} ^ { - 1 }$, (Given speed of sound $= 300 \mathrm {~m} / \mathrm { s }$)
(1) 16, 14
(2) 12, 16
(3) 12, 18
(4) 8, 18

Water from a tap emerges vertically downwards with an initial speed of $1.0 \mathrm {~ms} ^ { - 1 }$. The cross-sectional area of the tap is $10 ^ { - 4 } \mathrm {~m} ^ { 2 }$. Assume that the pressure is constant throughout the stream of water and that the flow is streamlined. The cross-sectional area of the stream, 0.15 m below the tap would be: (Take $\mathrm { g } = 10 \mathrm {~ms} ^ { - 2 }$ )
(1) $1 \times 10 ^ { - 5 } \mathrm {~m} ^ { 2 }$
(2) $5 \times 10 ^ { - 4 } \mathrm {~m} ^ { 2 }$
(3) $2 \times 10 ^ { - 5 } \mathrm {~m} ^ { 2 }$
(4) $5 \times 10 ^ { - 5 } \mathrm {~m} ^ { 2 }$

Equation of travelling wave on a stretched string of linear density $5 \mathrm {~g} / \mathrm { m }$ is $\mathrm { y } = 0.03 \sin ( 450 \mathrm { t } - 9 \mathrm { x } )$ where distance and time are measured in SI units. The tension in the string is:
(1) 10 N
(2) 7.5 N
(3) 12.5 N
(4) 5 N