Ratio of radius of gyration of a hollow sphere to that of a solid cylinder of equal mass, for moment of Inertia about their diameter axis AB as shown in figure is $\sqrt{8/x}$. The value of $x$ is:
(1) 51
(2) 34
(3) 17
(4) 67

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

If $R$ is the radius of the earth and the acceleration due to gravity on the surface of earth is $g = \pi ^ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }$, then the length of the second's pendulum at a height $h = 2R$ from the surface of earth will be:
(1) $\frac { 2 } { 9 } \mathrm {~m}$
(2) $\frac { 1 } { 9 } \mathrm {~m}$
(3) $\frac { 4 } { 9 } \mathrm {~m}$
(4) $\frac { 8 } { 9 } \mathrm {~m}$

A metal wire of uniform mass density having length $L$ and mass $M$ is bent to form a semicircular arc and a particle of mass $m$ is placed at the centre of the arc. The gravitational force on the particle by the wire is :
(1) $\frac { \mathrm { GmM } \pi ^ { 2 } } { \mathrm {~L} ^ { 2 } }$
(2) $\frac { \mathrm { GMm } \pi } { 2 \mathrm {~L} ^ { 2 } }$
(3) 0
(4) $\frac { 2 \mathrm { GmM } \pi } { \mathrm { L } ^ { 2 } }$

A simple pendulum doing small oscillations at a place R height above earth surface has time period of $T_1 = 4\mathrm{~s}$. $T_2$ would be its time period if it is brought to a point which is at a height 2R from earth surface. Choose the correct relation [$R =$ radius of earth]:
(1) $2\mathrm{~T}_1 = \mathrm{T}_2$
(2) $2\mathrm{~T}_1 = 3\mathrm{T}_2$
(3) $\mathrm{T}_1 = \mathrm{T}_2$
(4) $3\mathrm{T}_1 = 2\mathrm{T}_2$

A sphere of relative density $\sigma$ and diameter $D$ has concentric cavity of diameter $d$. The ratio of $\frac { D } { d }$, if it just floats on water in a tank is :
(1) $\left( \frac { \sigma - 2 } { \sigma + 2 } \right) ^ { 1 / 3 }$
(2) $\left( \frac { \sigma } { \sigma - 1 } \right) ^ { 1 / 3 }$
(3) $\left( \frac { \sigma - 1 } { \sigma } \right) ^ { 1 / 3 }$
(4) $\left( \frac { \sigma + 1 } { \sigma - 1 } \right) ^ { 1 / 3 }$

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) $4 \pi R ^ { 2 } T$

A sample of 1 mole gas at temperature $T$ is adiabatically expanded to double its volume. If adiabatic constant for the gas is $\gamma = \frac { 3 } { 2 }$, then the work done by the gas in the process is:
(1) $\frac { R } { T } [ 2 - \sqrt { 2 } ]$
(2) $\frac { T } { R } [ 2 + \sqrt { 2 } ]$
(3) RT $[ 2 - \sqrt { 2 } ]$
(4) $\mathrm { RT } [ 2 + \sqrt { 2 } ]$

The pressure and volume of an ideal gas are related as $P V ^ { \frac { 3 } { 2 } } = K$ (Constant). The work done when the gas is taken from state $A\left(P _ { 1 } , V _ { 1 } , T _ { 1 }\right)$ to state $B\left(P _ { 2 } , V _ { 2 } , T _ { 2 }\right)$ is:
(1) $2 \left( P _ { 1 } V _ { 1 } - P _ { 2 } V _ { 2 } \right)$
(2) $2 \left( P _ { 2 } V _ { 2 } - P _ { 1 } V _ { 1 } \right)$
(3) $2 \sqrt { P _ { 1 } } V _ { 1 } - \sqrt { P _ { 2 } } V _ { 2 }$
(4) $2 P _ { 2 } \sqrt { V _ { 2 } } - P _ { 1 } \sqrt { V _ { 1 } }$

Match List I with List II:
List I
(A) Kinetic energy of planet
(B) Gravitation Potential energy of sun-planet system
(C) Total mechanical energy of planet
(D) Escape energy at the surface of planet for unit mass object
List II (I) $-\mathrm{GMm}/\mathrm{a}$ (II) $\mathrm{GMm}/2\mathrm{a}$ (III) $\frac{\mathrm{Gm}}{\mathrm{r}}$ (IV) $-\mathrm{GMm}/2\mathrm{a}$ (Where $\mathbf{a} =$ radius of planet orbit, $\mathbf{r} =$ radius of planet, $\mathrm{M} =$ mass of Sun, $\mathrm{m} =$ mass of planet) Choose the correct answer from the options given below:
(1) (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
(2) (A)-(II), (B)-(I), (C)-(IV), (D)-(III)
(3) (A)-(I), (B)-(II), (C)-(III), (D)-(IV)
(4) (A)-(I), (B)-(IV), (C)-(II), (D)-(III)

The volume of an ideal gas ($\gamma = 1.5$) is changed adiabatically from 5 litres to 4 litres. The ratio of initial pressure to final pressure is:
(1) $\frac { 16 } { 25 }$
(2) $\frac { 4 } { 5 }$
(3) $\frac { 8 } { 5 \sqrt { 5 } }$
(4) $\frac { 2 } { \sqrt { 5 } }$

Two moles of a monoatomic gas is mixed with six moles of a diatomic gas. The molar specific heat of the mixture at constant volume is:
(1) $\frac { 9 } { 4 } R$
(2) $\frac { 7 } { 4 } R$
(3) $\frac { 3 } { 2 } R$
(4) $\frac { 5 } { 2 } R$

Two identical capacitors have same capacitance $C$. One of them is charged to the potential $V$ and other to the potential $2V$. The negative ends of both are connected together. When the positive ends are also joined together, the decrease in energy of the combined system is:
(1) $\frac { 1 } { 4 } C V ^ { 2 }$
(2) $2 C V ^ { 2 }$
(3) $\frac { 1 } { 2 } C V ^ { 2 }$
(4) $\frac { 3 } { 4 } C V ^ { 2 }$

A galvanometer has a resistance of $50 \Omega$ and it allows maximum current of 5 mA. It can be converted into voltmeter to measure upto 100 V by connecting in series a resistor of resistance.
(1) $5975 \Omega$
(2) $20050 \Omega$
(3) $19950 \Omega$
(4) $19500 \Omega$

In series LCR circuit, the capacitance is changed from $C$ to $4C$. To keep the resonance frequency unchanged, the new inductance should be:
(1) reduced by $\frac { 1 } { 4 } L$
(2) increased by $2L$
(3) reduced by $\frac { 3 } { 4 } L$
(4) increased to $4L$

A monochromatic light of wavelength $6000 \mathrm {~\AA}$ is incident on the single slit of width 0.01 mm. If the diffraction pattern is formed at the focus of the convex lens of focal length 20 cm, the linear width of the central maximum is:
(1) 60 mm
(2) 24 mm
(3) 120 mm
(4) 12 mm

The de Broglie wavelengths of a proton and an $\alpha$ particle are $\lambda$ and $2\lambda$ respectively. The ratio of the velocities of proton and $\alpha$ particle will be:
(1) $1 : 8$
(2) $1 : 2$
(3) $4 : 1$
(4) $8 : 1$

The minimum energy required by a hydrogen atom in ground state to emit radiation in Balmer series is nearly:
(1) 1.5 eV
(2) 13.6 eV
(3) 1.9 eV
(4) 12.1 eV

An electron rotates in a circle around a nucleus having positive charge Ze. Correct relation between total energy (E) of electron to its potential energy (U) is:
(1) $\mathrm{E} = \mathrm{U}$
(2) $2\mathrm{E} = \mathrm{U}$
(3) $2\mathrm{E} = 3\mathrm{U}$
(4) $\mathrm{E} = 2\mathrm{U}$

10 divisions on the main scale of a Vernier calliper coincide with 11 divisions on the Vernier scale. If each division on the main scale is of 5 units, the least count of the instrument is:
(1) $\frac { 1 } { 2 }$
(2) $\frac { 10 } { 11 }$
(3) $\frac { 50 } { 11 }$
(4) $\frac { 5 } { 11 }$

One main scale division of a vernier caliper is equal to $m$ units. If $\mathrm { n } ^ { \text {th } }$ division of main scale coincides with $( n + 1 ) ^ { \text {th } }$ division of vernier scale, the least count of the vernier caliper is :
(1) $\frac { n } { ( n + 1 ) }$
(2) $\frac { 1 } { ( n + 1 ) }$
(3) $\frac { m } { ( n + 1 ) }$
(4) $\frac { m } { n ( n + 1 ) }$

A particle is moving in a circle of radius 50 cm in such a way that at any instant the normal and tangential components of its acceleration are equal. If its speed at $\mathrm { t } = 0$ is $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the time taken to complete the first revolution will be $\frac { 1 } { \alpha } \left[ 1 - \mathrm { e } ^ { - 2 \pi } \right] \mathrm { s }$, where $\alpha =$ $\_\_\_\_$ .

Two identical spheres each of mass 2 kg and radius 50 cm are fixed at the ends of a light rod so that the separation between the centers is 150 cm. Then, moment of inertia of the system about an axis perpendicular to the rod and passing through its middle point is $\dfrac{x}{20}$ kg m$^2$, where the value of $x$ is

A body of mass 5 kg moving with a uniform speed $3 \sqrt { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in $X - Y$ plane along the line $y = x + 4$. The angular momentum of the particle about the origin will be $\_\_\_\_$ $\mathrm { kg } \mathrm { m } ^ { 2 } \mathrm {~s} ^ { - 1 }$.