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) $\frac { 2 } { 5 }$

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 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

The number density of molecules of a gas depends on their distance r from the origin as, $n ( r ) = n _ { 0 } e ^ { - \alpha r ^ { 4 } }$. Then the number of molecules is proportional to:
(1) $n _ { 0 } \alpha ^ { - 3 }$
(2) $\sqrt { n _ { 0 } } \alpha ^ { \frac { 1 } { 2 } }$
(3) $n _ { 0 } \alpha ^ { \frac { - 3 } { 4 } }$
(4) $n _ { 0 } \alpha ^ { \frac { 1 } { 4 } }$

A mixture of 2 moles of helium gas (atomic mass $= 4 \mathrm { u }$), and 1 mole of argon gas (atomic mass $= 40 \mathrm { u }$) is kept at 300 K in a container. The ratio of their rms speeds $\frac { V _ { \text {rms} } (\text{helium}) } { V _ { \text {rms} } (\text{argon}) }$, is close to:
(1) 0.45
(2) 2.24
(3) 0.32
(4) 3.16

A uniformly charged ring of radius $3a$ and total charge $q$ is placed in $x - y$ plane centred at origin. A point charge $q$ is moving towards the ring along the $z$-axis and has speed $v$ at $z = 4a$. The minimum value of $v$ such that it crosses the origin is:
(1) $\sqrt { \frac { 2 } { m } } \left( \frac { 1 } { 15 } \frac { q ^ { 2 } } { 4 \pi \epsilon _ { 0 } a } \right) ^ { 1 / 2 }$
(2) $\sqrt { \frac { 2 } { m } } \left( \frac { 4 } { 15 } \frac { q ^ { 2 } } { 4 \pi \epsilon _ { 0 } a } \right) ^ { 1 / 2 }$
(3) $\sqrt { \frac { 2 } { m } } \left( \frac { 1 } { 5 } \frac { q ^ { 2 } } { 4 \pi \epsilon _ { 0 } a } \right) ^ { 1 / 2 }$
(4) $\sqrt { \frac { 2 } { m } } \left( \frac { 2 } { 15 } \frac { q ^ { 2 } } { 4 \pi \epsilon _ { 0 } a } \right) ^ { 1 / 2 }$

A submarine experiences a pressure of $5.05 \times 10 ^ { 6 } \mathrm {~Pa}$ at a depth of $\mathrm { d } _ { 1 }$ in a sea. When it goes further to a depth of $\mathrm { d } _ { 2 }$, it experiences a pressure of $8.08 \times 10 ^ { 6 } \mathrm {~Pa}$. Then $\mathrm { d } _ { 2 } - \mathrm { d } _ { 1 }$ is approximately (density of water $= 10 ^ { 3 } \mathrm {~kg} / \mathrm { m } ^ { 3 }$ and acceleration due to gravity $= 10 \mathrm {~ms} ^ { - 2 }$ ):
(1) 600 m
(2) 500 m
(3) 300 m
(4) 400 m

Figure shows charge ($q$) versus voltage ($V$) graph for series and parallel combination of two given capacitors. The capacitances are:
(1) $60 \mu \mathrm {~F}$ and $40 \mu \mathrm {~F}$
(2) $50 \mu \mathrm {~F}$ and $30 \mu \mathrm {~F}$
(3) $20 \mu \mathrm {~F}$ and $30 \mu \mathrm {~F}$
(4) $40 \mu \mathrm {~F}$ and $10 \mu \mathrm {~F}$

A current of 5 A passes through a copper conductor (resistivity $= 1.7 \times 10 ^ { - 8 } \Omega \mathrm {~m}$) of radius of cross-section 5 mm. Find the mobility of the charges if their drift velocity is $1.1 \times 10 ^ { - 3 } \mathrm {~ms} ^ { - 1 }$.
(1) $1.5 \mathrm {~m} ^ { 2 } \mathrm {~V} ^ { - 1 } \mathrm {~s} ^ { - 1 }$
(2) $1.8 \mathrm {~m} ^ { 2 } \mathrm {~V} ^ { - 1 } \mathrm {~s} ^ { - 1 }$
(3) $1.0 \mathrm {~m} ^ { 2 } \mathrm {~V} ^ { - 1 } \mathrm {~s} ^ { - 1 }$
(4) $1.3 \mathrm {~m} ^ { 2 } \mathrm {~V} ^ { - 1 } \mathrm {~s} ^ { - 1 }$

In the given circuit, an ideal voltmeter connected across the $10 \Omega$ resistance reads 2 V. The internal resistance $r$, of each cell is: The circuit has two cells each of EMF 1.5 V and internal resistance $r\,\Omega$.
(1) $0 \Omega$
(2) $1.5 \Omega$
(3) $0.5 \Omega$
(4) $1 \Omega$

A source of sound S is moving with a velocity of $50 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ towards a stationary observer. The observer measures the frequency of the source as 1000 Hz . What will be the apparent frequency of the source when it is moving away from the observer after crossing him? (Take velocity of sound in air is $350 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ )
(1) 750 Hz
(2) 857 Hz
(3) 1143 Hz
(4) 807 Hz

Four equal point charges $Q$ each are placed in the $xy$ plane at $( 0,2 ) , ( 4,2 ) , ( 4 , - 2 )$ and $( 0 , - 2 )$. The work required to put a fifth charge $Q$ at the origin of the coordinate system will be:
(1) $\frac { Q ^ { 2 } } { 4 \pi \epsilon _ { 0 } }$
(2) $\frac { Q ^ { 2 } } { 2 \sqrt { 2 } \pi \epsilon _ { 0 } }$
(3) $\frac { Q ^ { 2 } } { 4 \pi \epsilon _ { 0 } } \left( 1 + \frac { 1 } { \sqrt { 5 } } \right)$
(4) $\frac { Q ^ { 2 } } { 4 \pi \epsilon _ { 0 } } \left( 1 + \frac { 1 } { \sqrt { 3 } } \right)$