A rod of mass $M$ and length $L$ is lying on a horizontal frictionless surface. A particle of mass $m$ travelling along the surface hits at one end of the rod with a velocity $u$ in a direction perpendicular to the rod. The collision is completely elastic. After collision, particle comes to rest. The ratio of masses ( $\frac { m } { M }$ ) is $\frac { 1 } { x }$. The value of $x$ will be

The position of the centre of mass of a uniform semi-circular wire of radius $R$ placed in $x - y$ plane with its centre at the origin and the line joining its ends as $x$-axis is given by, $\left( 0 , \frac { x R } { \pi } \right)$. Then, the value of $| x |$ is $\_\_\_\_$ .

The water is filled up to a height of 12 m in a tank having vertical sidewalls. A hole is made in one of the walls at a depth $h$ below the water level. The value of $h$ for which the emerging stream of water strikes the ground at the maximum range is $\_\_\_\_$ m.

When a body slides down from rest along a smooth inclined plane making an angle of $30 ^ { \circ }$ with the horizontal, it takes time $T$. When the same body slides down from the rest along a rough inclined plane making the same angle and through the same distance, it takes time $\alpha T$, where $\alpha$ is a constant greater than 1. The co-efficient of friction between the body and the rough plane is $\frac { 1 } { \sqrt { x } } \frac { \alpha ^ { 2 } - 1 } { \alpha ^ { 2 } }$ where $x =$ $\_\_\_\_$.

As shown in the figure, a block of mass $\sqrt { 3 } \mathrm {~kg}$ is kept on a horizontal rough surface of coefficient of friction $\frac { 1 } { 3 \sqrt { 3 } }$. The critical force to be applied on the vertical surface as shown at an angle $60 ^ { \circ }$ with horizontal such that it does not move, will be $3x$. The value of $x$ will $\left[ g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 } ; \sin 60 ^ { \circ } = \frac { \sqrt { 3 } } { 2 } ; \cos 60 ^ { \circ } = \frac { 1 } { 2 } \right]$ $\mu = \frac { 1 } { 3 \sqrt { 3 } }$

In the reported figure, two bodies $A$ and $B$ of masses 200 g and 800 g are attached with the system of springs. Springs are kept in a stretched position with some extension when the system is released. The horizontal surface is assumed to be frictionless. The angular frequency will be \_\_\_\_ rad $\mathrm{s}^{-1}$ when $k = 20\mathrm{~N~m}^{-1}$.

Two simple harmonic motions are represented by the equations $x_1 = 5\sin\left(2\pi t + \frac{\pi}{4}\right)$ and $x_2 = 5\sqrt{2}(\sin 2\pi t + \cos 2\pi t)$. The amplitude of the second motion is $\_\_\_\_$ times the amplitude in the first motion.

A body of mass $(2M)$ splits into four masses $\{m, M-m, m, M-m\}$, which are rearranged to form a square as shown in the figure. The ratio of $\frac{M}{m}$ for which, the gravitational potential energy of the system becomes maximum is $x:1$. The value of $x$ is $\_\_\_\_$.

A particle executes S.H.M. with amplitude $A$ and time period $T$. The displacement of the particle when its speed is half of maximum speed is $\frac { \sqrt { x } A } { 2 }$. The value of $x$ is

A solid disc of radius $a$ and mass $m$ rolls down without slipping on an inclined plane making an angle $\theta$ with the horizontal. The acceleration of the disc will be $\frac { 2 } { b } g \sin \theta$, where $b$ is $\_\_\_\_$. (Round off to the Nearest Integer) ( $g =$ acceleration due to gravity) ( $\theta =$ angle as shown in figure)

The radius in kilometer to which the present radius of earth ($R = 6400 \mathrm {~km}$) to be compressed so that the escape velocity is increased 10 times is $\_\_\_\_$.

A ball of mass 10 kg moving with a velocity $10 \sqrt { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along the $x$-axis, hits another ball of mass 20 kg which is at rest. After the collision, first ball comes to rest while the second ball disintegrates into two equal pieces. One piece starts moving along $y$-axis with a speed of $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The second piece starts moving at an angle of $30 ^ { \circ }$ with respect to the $x$-axis. The velocity of the ball moving at $30 ^ { \circ }$ with $x$-axis is $x \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The configuration of pieces after the collision is shown in the figure below. The value of $x$ to the nearest integer is [Figure]

A circular disc reaches from top to bottom of an inclined plane of length $L$. When it slips down the plane, it takes time $t _ { 1 }$. When it rolls down the plane, it takes time $t _ { 2 }$. The value of $\frac { t _ { 2 } } { t _ { 1 } }$ is $\sqrt { \frac { 3 } { x } }$. The value of $x$ will be

The centre of a wheel rolling on a plane surface moves with a speed $v _ { 0 }$. A particle on the rim of the wheel at the same level as the centre will be moving at a speed $\sqrt { x } v _ { 0 }$. Then the value of $x$ is $\_\_\_\_$ .

In the given figure, two wheels $P$ and $Q$ are connected by a belt $B$. The radius of $P$ is three times that of $Q$. In the case of the same rotational kinetic energy, the ratio of rotational inertias $\left( \frac { I _ { 1 } } { I _ { 2 } } \right)$ will be $x : 1$. The value of $x$ will be $\_\_\_\_$.

A 2 kg steel rod of length 0.6 m is clamped on a table vertically at its lower end and is free to rotate in the vertical plane. The upper end is pushed so that the rod falls under gravity. Ignoring the friction due to clamping at its lower end, the speed of the free end of the rod when it passes through its lowest position is $\_\_\_\_$ $\mathrm { m } \mathrm { s } ^ { - 1 }$. (Take $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$)

A container is divided into two chambers by a partition. The volume of first chamber is 4.5 litre and second chamber is 5.5 litre. The first chamber contain 3.0 moles of gas at pressure 2.0 atm and second chamber contain 4.0 moles of gas at pressure 3.0 atm. After the partition is removed and the mixture attains equilibrium, then, the common equilibrium pressure existing in the mixture is $x \times 10 ^ { - 1 } \mathrm {~atm}$. Value of $x$ (nearest integer) is $\_\_\_\_$

A pendulum bob has a speed of $3\mathrm{~m~s}^{-1}$ at its lowest position. The pendulum is 50 cm long. The speed of bob, when the length makes an angle of $60^{\circ}$ to the vertical will be ($g = 10\mathrm{~m~s}^{-2}$) \_\_\_\_ $\mathrm{m~s}^{-1}$.

Two waves are simultaneously passing through a string and their equations are: $y_1 = A_1 \sin k(x - vt)$, $y_2 = A_2 \sin k(x - vt + x_0)$. Given amplitudes $A_1 = 12\text{ mm}$ and $A_2 = 5\text{ mm}$, $x_0 = 3.5\text{ cm}$ and wave number $k = 6.28\text{ cm}^{-1}$. The amplitude of resulting wave will be $\_\_\_\_$ mm.

A rod $CD$ of thermal resistance $10.0\mathrm{~KW}^{-1}$ is joined at the middle of an identical rod $AB$ as shown in figure. The ends $A$, $B$ and $D$ are maintained at $200^{\circ}\mathrm{C}$, $100^{\circ}\mathrm{C}$ and $125^{\circ}\mathrm{C}$ respectively. The heat current in $CD$ is $P\mathrm{~W}$. The value of $P$ is $\_\_\_\_$.

Time period of a simple pendulum is $T$. The time taken to complete $\frac { 5 } { 8 }$ oscillations starting from mean position is $\frac { \alpha } { 12 } T$. The value of $\alpha$ is $\_\_\_\_$ .

If one wants to remove all the mass of the earth to infinity in order to break it up completely. The amount of energy that needs to be supplied will be $\frac { x } { 5 } \frac { G M ^ { 2 } } { R }$ where $x$ is $\_\_\_\_$. (Round off to the Nearest Integer) ( $M$ is the mass of earth, $R$ is the radius of earth, $G$ is the gravitational constant)

Consider two identical springs each of spring constant $k$ and negligible mass compared to the mass $M$ as shown. Fig. 1 shows one of them and Fig. 2 shows their series combination. The ratios of time period of oscillation of the two SHM is $\frac { T _ { b } } { T _ { a } } = \sqrt { x }$, where value of $x$ is $\_\_\_\_$. (Round off to the Nearest Integer)

Two separate wires $A$ and $B$ are stretched by 2 mm and 4 mm respectively, when they are subjected to a force of 2 N. Assume that both the wires are made up of same material and the radius of wire $B$ is 4 times that of the radius of wire $A$. The length of the wires $A$ and $B$ are in the ratio of $a : b$. Then $\frac { a } { b }$ can be expressed as $\frac { 1 } { x }$, where $x$ is $\_\_\_\_$.

In the reported figure, heat energy absorbed by a system in going through a cyclic process is $\_\_\_\_$ $\pi \mathrm { J }$.