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Two particles move parallel to the $x-$axis about the origin with the same amplitude $A$ and angular frequency $\omega$. At a certain instance they are found at a distance $\frac{A}{3}$ from the origin on opposite sides but their velocities are in the same direction. What is the phase difference between the two?

Two concentric shells A and B of radii a and b are given charged $q_A$ and $A_B$.

(a) Find the potential difference between the shells.

(b) What happens when they are connected by means of a conducting wire.

In the given figure, the initial and final states of a gas are shown by points $i$ and $f$. The internal energy of the gas at $i$ is $10\text{J}$.

For the path $iaf$ : $\Delta Q=50\text{J}$ and $\Delta W=20\text{J}$.

For the path $ibf$, if $\Delta Q=36\text{J},$ the value of $\Delta W$ will be equal to

A tank is filled with water upto a height H. Water is allowed to come out of a hole P in one of the walls at a depth h below the surface of water as shown in figure. Express the horizontal distance X in terms of H and h

Find the distance covered by a particle during the time interval $t=0$ and $t=6\text{s}$ for which the speed-time graph is shown in figure

A $500\text{kg}$ car, moving with a velocity of $36{\text{kmh}}^{-1}$ on a straight road unidirectionally, doubles its velocity in one minute. The power delivered by the engine for doubling the velocity is

Two bodies $A$ and $B$ are arranged as shown in figure. Which of the following options is correct regarding body $A$?

(Assume that the bodies are placed in an adibatic container)

The principle of conservation of linear momentum can be strictly applied during a collision between two particles provided the time of impact is

Component of $3\hat{i}+4\hat{j}$ perpendicular to $\hat{i}+\hat{j}$ and in the same plane as that of $3\hat{i}+4\hat{j}$ is

From a circular disk of radius ($R=50\text{cm}$), a square is cut with one of its radii as the diagonal of the square (as shown in figure). The distance of the center of mass of the remaining part from the geometrical center of the disc is

The potential energy of $1\text{kg}$ particle moving along the $X-$ axis is given by $U=(\frac{x^4}{4}-\frac{x^2}{2})\text{J}$. The total mechanical energy of the particle is $2\text{J}$. Then maximum speed of the particle is (in $\text{m/s}$)

The radius of curvature of the path of the charged particle in a uniform magnetic field is directly proportional to

[MNR 1995; UPSEAT 1999, 2000]

A plane is inclined at an angle $\alpha ={30}^{\circ }$ with respect to the horizontal. A particle is projected with a speed $u=2\text{m/s}$, from the base of the plane, making an angle $\theta ={15}^{\circ }$ with respect to the plane as shown in the figure. The distance from the base, at which the particle hits the plane is close to [Take $g=10\text{m}/s^2]$

A piece of wood is taken deep inside a long column of water and released it will move up

A ring of radius R, made of an insulating material carries a charge Q uniformly distributed on it. If the ring rotates about the axis passing through its centre and normal to plane of the ring with constant angular speed $\omega$, then the magnitude of the magnetic moment of the ring will be?

The peak value of an alternating e.m.f. E is given by $E=E_{0}cos\omega t$ is 10 volts and its frequency is 50 Hz. At time $t=\frac{1}{600}sec,$ the instantaneous e.m.f. is

Find the tension in the string connected to the block of mass $4\text{kg}$. Assume the strings and pulleys to be ideal. (Take $g=10m/s^2)$

A ball dropped from a point $A$ falls down vertically to $C$, through the midpoint $B$. The descending time from $A$ to $B$ and that from $A$ to $C$ are in the ratio

A ball of mass $4\text{kg}$ moving with velocity $4\text{m/s}$ collides with another identical ball at rest. The kinetic energy of the balls after collision is $\frac{7}{8}$ times of the original. Find the coefficient of restituton $\left(e\right)$.

The tension in a string holding a solid block $({\rho }_b=700{\text{kg/m}}^3)$ below the surface of a liquid $({\rho }_l=1000{\text{kg/m}}^3)$ as shown in figure is $10\text{N}$ when the system is at rest. What will be the tension in the string if the system has an upward acceleration of $5{\text{m/s}}^2$?

$[\text{Take}g=10{\text{m/s}}^2]$

Kirchhoff's second law is an extension of:

The energy in monochromatic X-rays of wavelength 1 angstrom is roughly equal to

A rope of mass $2m$ is used to pull a block of mass $m$ as shown in the figure. Find the force exerted by the rope on the block. Assume all surfaces as frictionless.

In given figure, all surfaces are smooth and frictionless. Spring is compressed by $2\text{m}$ and then released. Maximum distance travelled by the block over the inclined plane is

(Take $g=10{\text{m/s}}^2$)

A system consists of two identical small balls of mass $2\text{kg}$ each connected to the two ends of a $1\text{m}$ long light rod. The system is rotating about a fixed axis through the centre of the rod and perpendicular to it at an angular speed of $9\text{rad/s}$. An impulsive force of average magnitude $10\text{N}$ acts on one of the masses in the direction of its velocity for $0.20\text{s}$. Calculate the new angular velocity of the system.

A block of mass $100\text{g}$ attached to a spring of spring constant $100\text{N/m}$. The block is moved to compress the spring by $10\text{cm}$ and then released on a frictionless floor as shown below. If the collisions with the wall in the front are elastic in nature, then the time period of the motion is

A ceiling fan is rotating with an angular velocity of $4rev/s$. It takes $40s$ to stop when it is switched off. The angular retardation during this interval is

The magnetic field in a travelling electromagnetic wave has a peak value of 20 nT. The peak value of electric field strength is

Two people $A$ and $B$ of mass $m$ and $2m$ are standing at the two ends of the wooden plank of mass $8m$. Both people start walking towards each other. Find the ratio of the velocities of $A$ and $B$ so that the plank will remain at rest.

A body is moving down into a well through a rope passing over a fixed pulley of radius $20\text{cm}$. Assume that there is no slipping between the rope and pulley. Calculate the angular velocity and angular acceleration of the pulley at an instant when the body is going down at a speed of $40\text{cm/s}$ and has an acceleration of $4{\text{m/s}}^2$?

A particle of mass m moves with a variable velocity v, which changes with distance covered x along a straight line as $v=k\sqrt{x}$,where k is a positive constant . The work done by all the forces acting on the particle, during the first t second is

A rod of length $l$ made of material of thermal conductivity $k$ can transfer a given amount of heat in $12\text{seconds}$. If the length of the rod is doubled keeping the volume constant, it is noted that rod can transfer same heat in $t\text{seconds}$. Find the value of ${}^{\prime }{t}^{\prime }$.

Two wires $A$ and $B$ of the same cross-sectional areas, Young's modulii $Y_1$ and $Y_2$ and coefficients of linear expansion ${\alpha }_1$ and ${\alpha }_2$ respectively, are joined together and fixed between rigid supports at either ends at $0{}^{\circ }\text{C}$. The tension in the compound wire when wire $A$ is heated to temperature $T_1$ and wire $B$ is cooled to temperature $T_2$ is equal to tension in the compund wire when only wire $A$ is cooled to temperature $T_2$. Find the correct option.

The power factor of the circuit in below figure is $\frac{1}{\sqrt{2}}$. The capacitance of the circuit is equal to

Length of the hour hand of a clock is $\frac{60}{\pi }\text{cm}$. Find the magnitude of average acceleration (in ${\text{rad/hr}}^2$) between the positions shown in the figure.

The root mean square speed of a gas molecule is $300\text{m/s}$. What will be the root mean square speed of the molecules if the atomic mass is doubled and absolute temperature is halved?

Two balls $A$ and $B$ are thrown with speed $u$ and $u/2$ respectively. Both the balls cover the same horizontal distance before returning to the horizontal plane of projection. If the angle of projection of ball of $B$ is ${15}^{\circ }$ with the horizontal, then the angle of projection of $A$ is

An inductance of 1 mH a condenser of $10\mu F$ and a resistance of $50\omega$ are connected in series. The reactance of inductor and condensers are same. The reactance of either of them will be

When an object is shot from the bottom of a long smooth inclined plane kept at an angle $60°$with horizontal, it can travel a distance $x_1$ along the plane. But, when the inclination is decreased to $30°$ and the same object is shot with the same velocity, it can travel $x_2$ distance. Then $x_1:x_2$ will be

A body is moving from rest under constant acceleration and let $S_1$ be the displacement in the first $(p-1)$ sec and $S_2$ be the displacement in the first p sec. The displacement in $(p^2-p+1{)}^{th}$ sec. will be

A metal stick of mass $1$ kg and length 1 metre is held vertically with one end on the floor and is then allowed to fall. All values are in SI units. Take approximations.

$\begin{array}{cc}\text{Column-I}& \text{Column-II}\\ \text{(a) Initial mechanical energy}& \text{(p)}5.425\\ {\text{(b)(Angular velocity)}}^2\text{just before hitting the ground}& \text{(q) 29.429}\\ \text{(c) Linear velocity of the free end just before hitting the ground}& \text{(r) 4.9}\\ \text{(d) Moment of inertia}& \text{(s) 0.333}\end{array}$

The retarding acceleration of $4.9{\text{ms}}^{-2}$ due to frictional force stops the car of mass $400\text{kg}$ travelling on a road. The coefficient of friction between the tyre of the car and the road is

A bomb is dropped on an enemy post by an aeroplane flying horizontally with a velocity of $60{\text{km/h}}^{-1}$ and at a height of $490\text{m}$. At the time of dropping the bomb, how far should the aeroplane be from the enemy post so that the bomb may directly hit the target?

If a unit vector is represented by $0.5\hat{i}+0.8\hat{j}+c\hat{k}$, then the value of ‘c’ is

Determine height $h$ of oil in the U tube as shown in figure. Density of oil $=0.9\text{g/cc}$; density of liquid is $1.6\text{g/cc}$ and density of mercury $=13.6\text{g/cc}$.