Work, Energy and Power

A 0.5 kg block moving at a speed of $12\ \mathrm{ms^{-1}}$ compresses a spring through a distance 30 cm when its speed is halved. The spring constant of the spring will be ______ $\mathrm{Nm^{-1}}$.

A ball is released from rest from point P of a smooth semi-spherical vessel as shown in figure. The ratio of the centripetal force and normal reaction on the ball at point Q is A while angular position of point Q is $\alpha$ with respect to point P. Which of the following graphs represent the correct relation between A and $\alpha$ when ball goes from Q to R ?

Arrange the four graphs in descending order of total work done; where $W_1$, $W_2$, $W_3$ and $W_4$ are the work done corresponding to figure a, b, c and d respectively.

A pendulum of length 2 m consists of a wooden bob of mass 50 g. A bullet of mass 75 g is fired towards the stationary bob with a speed $v$. The bullet emerges out of the bob with a speed $\dfrac v3$ and the bob just completes the vertical circle. The value of $v$ is ______ ms$^{-1}$. (if $g=10$ m/s$^2$).

One end of a massless spring of spring constant $k$ and natural length $l_0$ is fixed while the other end is connected to a small object of mass $m$ lying on a frictionless table. The spring remains horizontal on the table. If the object is made to rotate at an angular velocity $\omega$ about an axis passing trough fixed end, then the elongation of the spring will be :

A stone tide to a string of length L is whirled in a vertical circle with the other end of the string at the centre. At a certain instant of time, the stone is at its lowest position and has a speed $u$. The magnitude of change in its velocity, as it reaches a position where the string is horizontal, is $\sqrt{x\left(u^2-gL\right)}$. The value of $x$ is-

A rolling wheel of 12 kg is on an inclined plane at position P and connected to a mass of 3 kg through a string of fixed length and pulley as shown in figure. Consider PR as friction free surface.

The velocity of centre of mass of the wheel when it reaches at the bottom Q of the inclined plane PQ will be $\dfrac12\sqrt{xgh}$ m/s. The value of $x$ is ______.

A particle of mass $m$ is moving in a circular path of constant radius $r$ such that its centripetal acceleration ($a$) is varying with time $t$ as $a=k^2rt^2$, where $k$ is a constant. The power delivered to the particle by the force acting on it is given as

A pendulum is suspended by a string of length 250 cm. The mass of the bob of the pendulum is 200 g. The bob is pulled aside until the string is at 60° with vertical as shown in the figure. After releasing the bob, the maximum velocity attained by the bob will be ______ ms$^{-1}$. (if $g=10$ m/s$^2$)

A block of mass 2 kg moving on a horizontal surface with speed of 4 ms$^{-1}$ enters a rough surface ranging from $x=0.5$ m to $x=1.5$ m. The retarding force in this range of rough surface is related to distance by $F=-kx$ where $k=12$ Nm$^{-1}$. The speed of the block as it just crosses the rough surface will be :

Water falls from a 40 m high dam at the rate of $9\times10^4$ kg per hour. Fifty percentage of gravitational potential energy can be converted into electrical energy. Using this hydro electric energy number of 100 W lamps, that can be lit, is :

(Take $g=10$ ms$^{-2}$)

A particle of mass 500 gm is moving in a straight line with velocity $v=bx^{5/2}$. The work done by the net force during its displacement from $x=0$ to $x=4$ m is : (Take $b=0.25$ m$^{-3/2}$ s$^{-1}$).

In the given figure, the block of mass $m$ is dropped from the point 'A'. The expression for kinetic energy of block when it reaches point 'B' is

Two bodies A and B of masses 5 kg and 8 kg are moving such that the momentum of body B is twice that of the body A. The ratio of their kinetic energies will be :

A body of mass 0.5 kg travels on straight line path with velocity $v=\left(3x^2+4\right)$ m/s. The net workdone by the force during its displacement from $x=0$ to $x=2$ m is :

A disc of mass 1 kg and radius R is free to rotate about a horizontal axis passing through its centre and perpendicular to the plane of disc. A body of same mass as that of disc is fixed at the highest point of the disc. Now the system is released, when the body comes to the lowest position, its angular speed will be $4\sqrt{\dfrac x{3R}}$ rad s$^{-1}$ where $x=$ ______.

($g=10$ ms$^{-2}$)

A body of mass 8 kg and another of mass 2 kg are moving with equal kinetic energy. The ratio of their respective momenta will be :

Sand is being dropped from a stationary dropper at a rate of 0.5 kgs$^{-1}$ on a conveyor belt moving with a velocity of 5 ms$^{-1}$. The power needed to keep the belt moving with the same velocity will be :

Two cylindrical vessels of equal cross-sectional area 16 cm$^2$ contain water upto heights 100 cm and 150 cm respectively. The vessels are interconnected so that the water levels in them become equal. The work done by the force of gravity during the process, is [Take, density of water = $10^3$ kg/m$^3$ and g = 10 ms$^{-2}$] :

The velocity of the bullet becomes one third after it penetrates 4 cm in a wooden block. Assuming that bullet is facing a constant resistance during its motion in the block. The bullet stops completely after travelling at $(4+x)$ cm inside the block. The value of $x$ is :