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In an ideal pulley particle system, mass $m_2(>m_1)$ is connected with a vertical spring of stiffness $k$. If mass $m_2$ is released from rest, when the spring is undeformed, find the maximum compression of the spring.

A large open tank has two holes in the wall. One is a square hole of side $L$ at a depth y from the top and the other is a circular hole of radius $R$ at a depth $4y$ from the top. When the tank is completely filled with water, if the quantity of water flowing out per second from both the holes is same, then $R$ is equal to

A ring of mass $7.2\text{kg}$ and of radius $25\text{cm}$ is experiencing a tension of $880\text{N}$ while making rotation about its own axis is such a way that each part of the ring has uniform speed. The frequency of rotation of the ring (in $\text{rev/min}$) is (assume surfaces of contact are frictionless)

A small particle of mass m is projected at an angle $\theta$ with the x-axis with an initial velocity $v_0$ in the x-y plane as shown in the figure. At a time $t<\left(v_{0}sin\theta /g\right),$ the angular momentum of the particle is

A body of mass $2\text{kg}$ starts from rest, uniformly accelerates from the point $(1,2,3)$ to point $(2,3,5)$ in $3$ seconds. The net work done by the body in $3$ seconds is

The position $\overrightarrow{r}=(2.00t^3-5.00t)\hat{i}+(6.00-7.00t^4)\hat{j},with\overrightarrow{r}$ in meters and t in seconds.

What is the angle between the positive direction of the x-axis and a line tangent to the particle's path at t=2.00 s?

A wire of mass $M$ and length $L$ is bent in form of a square. Calculate the MI of the square about the central axis perpendicular to the plane.

A laser beam $\lambda =633nm$ has a power of $3mW$. What will be the pressure exerted on a surface by this beam if the cross-section area is $3m{m}^2$? (Assume perfect reflection and normal incidence).

A body covers one-third of the distance with a velocity $v_1$, the second one-third of the distance with a velocity $v_2$ and the remaining distance with a velocity $v_3$. The average velocity is

Instantaneous temperature difference between a cooling body and the surroundings, obeying Newton's law of cooling, is $\theta$. Which of the following represents the variation of ln $\theta$ with time $t$?

Apparent depth of $X$ in all three cases are $H_1{H}_2,H_3$ respectively when seen from aboove, then

(given $H=30cm,n=1.5$ and $R=3m$

Two blocks A and B each of mass m are connected by a massless spring of natural length L and spring constant K. The blocks are initially resting on a smooth horizontal floor with spring at its natural length as shown in the figure. A third identical block C, also of mass m, moves on the floor with a speed v along the line joining A and B and collides with A. Then,

Two forces $\overrightarrow{F_1}=\hat{i}-2\hat{j}+2\hat{k}$ $\text{N}$ and $\overrightarrow{F_2}=-3\hat{i}+\hat{j}-3\hat{k}$ $\text{N}$ acts on a body and causes it to displace from position $\overrightarrow{s_1}=\hat{i}-\hat{j}$ $\text{m}$ to $\overrightarrow{s_2}=2\hat{j}-a\hat{k}$ $\text{m}$. Find the value of ${}^{\prime }{a}^{\prime }$ if the amount of net work done by the forces is $5\text{J}$ during the displacement.

A fully charged capacitor has a capacitance C. It is discharged through a small coil of resistance wire embedded in a thermally insulated block of specific heat capacity s and mass m. If the temperature of the block is raised by $\Delta$ T, the potential difference V across the capacitance was?

Two equal and opposite charges of $4\times {10}^{-8}$ C when placed $2\times {10}^{-2}cm$ away, form a dipole. If this dipole is placed in an external electric field of $4\times {10}^8$ newton/coulomb, the value of maximum torque and the work done in rotating it through ${180}^{\circ }$ will be

In the figure shown, the minimum value of the acceleration of wedge ($a$) at which the cylinder starts rising up the inclined surface is

A rain drop falls near the surface of the Earth with almost uniform velocity because

If the velocity time graph looks like this

A cycle wheel of radius 0.4 m completes one revolution in one second then the acceleration of a point on the cycle wheel will be

A jar is filled with two non - mixing liquid 1 and 2 having densities ${\rho }_1$ and ${\rho }_2$ respectively. A solid ball, made of a material of density ${\rho }_3$, is dropped in the jar. It comes to equilibrium in the position shown in the figure. Which of the following is true for ${\rho }_1,{\rho }_2$ and ${\rho }_3$?

If $\overrightarrow{A}=2\hat{i}+3\hat{j}-\hat{k}$ and $\overrightarrow{B}=-\hat{i}+3\hat{j}+4\hat{k}$, then projection of $\overrightarrow{A}$ on $\overrightarrow{B}$ will be

A block of mass 'M' is placed on the top of a bigger inclined plane of mass '10M' as shown in figure. All the surfaces are frictionless. The system is released from rest. Find the distance moved by the bigger block at the instant the smaller block reaches the ground.

A stone is allowed to fall from the top of a tower $100\text{m}$ high and at the same time, another stone is projected vertically upwards from the ground with a velocity of $25{\text{ms}}^{-1}$. The two stones will meet after

Two particles A and B initially at rest move towards each other under mutual force of attraction. At the instant when the velocity of A is $v$ and that of B is $2v$, the velocity of the center of mass of the system will be

A particle is moving in a circle with uniform speed v. In moving from a point to another diametrically opposite point

The path of projectile is represented by $y=Px-Q{x}^2$

$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{(a)}& \text{Range}& \text{(p)}& \frac{P}{Q}\\ \text{(b)}& \text{Maximum height}& \text{(q)}& P\\ \text{(c)}& \text{Time of flight}& \text{(r)}& \frac{P^2}{4Q}\\ \text{(d)}& \text{Tangent of angle of projection}& \text{(s)}& \sqrt{\frac{2}{Qg}}P\end{array}$

A Carnot engine works between temperatures ${127}^{\circ }\text{C}$ and $-{73}^{\circ }\text{C}$ . In process $1$, by increasing the temperature of source by ${50}^{\circ }\text{C}$, efficiency of engine becomes ${\eta }_1$. In process $2$, by decreasing the temperature of sink by ${50}^{\circ }\text{C}$, efficiency of engine becomes ${\eta }_2$. Then which of the following option is correct?

A particle is observed from two frames $S_{1}and{S}_2$. The frame $S_2$ moves with respect to $S_1$ with an acceleration a. Let $F_{1}and{F}_2$ be the pseudo forces on the particle when seen from $S_{1}and{S}_2$ respectively. Which of the following are not possible?

A force $\overrightarrow{F}=3\hat{i}+c\hat{j}+2\hat{k}$ acting on a particle causes a displacement $\overrightarrow{S}=-4\hat{i}+2\hat{j}-3\hat{k}$ in its own direction. If the work done by the force is $6\text{J}$, then the value of $c$ will be

A horizontal wind is blowing with a velocity $v$ towards north-east. A man starts running towards north with acceleration $a$. The time after which man will feel the wind blowing towards east is

Two identical ladders are arranged as shown in figure. Mass of each ladder is $M$ and length is $L$. A load of mass $m$ has been attached at the touching point $\left(\text{O}\right)$ of the ladders, as shown in figure. The system is in equilibrium. Find the magnitude of static frictional force acting at either point $A$ and $B$.

A ball of mass $m=1\text{kg}$ has an initial velocity $10\text{m/s}$. A variable force $F=20t+30t^2\text{N}$ is applied on the ball in the direction of motion of the ball for $1$ second. Find the final momentum of the ball.

Suppose the distance between the atoms of a diatomic gas remains constant. Its specific heat at constant volume per mole is

Oxygen boils at $-{183}^{\circ }C$ . This temperature is approximately

A ball falls on the ground from a height of $2.0\text{m}$ and rebounds up to a height of $1.5\text{m}$. Find the coefficient of restitution for the collision of the ball with the ground. Take $g=10{\text{m/s}}^2$.

A uniform rod of mass 'm' and length 'L' is tied to a vertical shaft. It rotates in horizontal plane about the vertical axis at angular velocity $\omega$. How much horizontal force does the shaft excert on the rod?

A cabin rotates on a smooth horizontal table with a uniform angular velocity $\omega$ in a circular path of radius $R$ (cabin is in a horizontal plane). A smooth path $\text{AB}$ is made inside the cabin as shown in figure. If a particle is released from $\text{A}$ to reach $\text{B}$ along the path $\text{AB}$, then find the time taken by the particle to reach point $\text{B}$, considering length of path $\text{AB}$ $=L$.

Derivative of $f\left(x\right)=4\sqrt{x}+\frac{1}{\sqrt{x}}$ is

Four metal plates numbered 1, 2, 3 and 4 are arranged as shown in Fig. The area of each plate is A and the separation between the plates is d. The capacitance of the arrangement is

A system consists of particles of mass $4\text{kg}$ placed at $(4,0),6\text{kg}$ placed at $(0,-4)$ and $3\text{kg}$ placed at $(-3,4)$. How far from the origin must a particle of mass $10\text{kg}$ be placed so that the moment of inertia of the system about the origin becomes $300{\text{kgm}}^2$?

A body is being moved along a straight line by a machine delivering a constant power. The distance covered by the body in time t is proportional to

The same progressive wave is represented by two graphs I and II. Graph I shows how the displacement y varies with the distance x along the wave at a given time. Graph II shows how y varies with time t at a given point on the wave. The ratio of measurements AB to CD, marked on the curves, represents:

A block of mass $m$ is at rest on a smooth inclined plane, which is rotated with a constant angular velocity ${}^{\prime }{\omega }^{\prime }$ about a vertical axis as shown in figure. Then, $\omega$ will be:

At what angle should a ball be projected up an inclined plane with a velocity so that it may hit the incline normally. The angle of the inclined plane with the horizontal is $\alpha$.

A $0.098\text{kg}$ block slides down a frictionless track as shown (Assume no energy loss on the entire track)

The time taken by the block to move from A to C is

A weight thermometer contains 51 gm of mercury at $0^{\circ }C$. When placed in an oil bath 1.0 gm is found to overflow. Find the temperature of oil bath. Coefficient of cubical expansion of mercury and glass are ${0.00018}^{\circ }C$ and ${0.000026}^{\circ }C$ respectively.

A projectile is fired from the surface of the earth with a velocity of $5\text{m/s}$ at an angle $\theta$ with the horizontal. Another projectile, fired from another planet with a velocity of $3\text{m/s}$ at the same angle, follows a trajectory which is identical with the trajectory of the projectile fired from the earth. The value of acceleration due to gravity on the planet (in $m{s}^{-2}$) is $\left({\text{Given g = 9.8 ms}}^{-2}\right)$