Conservation of Energy

For a particle in SHM, the K.E. at any instant is given by$K=K_0{\cos }^2\omega t$ .The total energy of SHM is 

 An object is projected with a velocity $\sqrt{\frac{8gR}{3}}$ from earth. The velocity of the object, at the maximum height reached by it will be 

Find the height at which the string becomes stack for the following diagram.

If a ball is allowed to fall freely under the influence of gravity, its

A spring is held vertically and if a load is attached to it, it elongates by a distance $d$. If the body attched to the spring is allowed to fall freely, the maximum compression produced in the spring is

Consider the situation shown in the figure. The system is released rest and the block $A$ travels a distance $5 \mathrm{m}$ in downward direction $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)
Whe work done by tension on the block $B$

A mass of $1 \mathrm{kg}$ falls from a height of $1 \mathrm{m}$ and lands on a mass less platform supported by a spring having spring constant $15 \mathrm{N} {\mathrm{m}}^{-1}$ as shown in the figure. The maximum compression of the spring is.
(acceleration due to gravity $=10 \mathrm{m} {\mathrm{s}}^{-2}$)

A block of mass $\sqrt{2} \mathrm{kg}$ is released from the top of an inclined smooth surface as shown in figure. If spring constant of spring is $100 \mathrm{N} {\mathrm{m}}^{-1}$ and block comes to rest after compressing the spring by $1 \mathrm{m}$, then the distance travelled by block before it comes to rest is

$\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$

A man is supplying a power of $500 \mathrm{W}$ at an instant on a massless string as shown in figure. If at that instant speed of block is $10 \mathrm{m} {\mathrm{s}}^{-1}$, and increasing at the rate of $10 \mathrm{m} {\mathrm{s}}^{-2}$. Then the mass of block is

A sphere of mass $m$ makes SHM in a hemispherical bowl $ABC$ and it moves from $A$ to $C$ and back to $A$ via $ABC$, so that $PB=h$. If acceleration due to gravity is $g$, the speed of the ball when it just crosses the point $B$ is

On a smooth curved track shown in the diagram, an small particle is projected from the point $O$ such that it just manages to reach the point $P$. Assume that particle always remain contact with track. The velocity of the particle on reaching the point $P$ will be equal to :

A particle of charge $Q$ moves with speed $v$ directly towards another particle of charge $Q^{\text{'}}$ which is connected to a light spring. The far end of the spring is fixed to a smooth horizontal table. Let $P$ and $E$ be the total momentum and energy of the charge-spring system respectively. Ignore gravity. Then

The bob of a simple pendulum of length $4 \mathrm{m}$ is held so that the string is horizontal and then let go. The velocity of the bob when it reaches the lowest position is (Take $g=9.8 \mathrm{m} {\mathrm{s}}^{-2}$)

Total energy of an isolated system-

A ball is thrown vertically up with a velocity $20 \raisebox{1ex}{$\mathrm{m}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{s}$}\right.$. At what height will its kinetic energy be half its original value?

An object is dropped from a certain distance when she is at half distance she is ______ in it?

A particle slides down along an inclined plane $AC.$ If the plane is frictionless, kinetic energy of particle at $C$ is

A body of mass $0.15 \mathrm{kg}$ moving with a velocity of $15 \mathrm{m} {\mathrm{s}}^{-1}$ comes to rest when it hits a spring that is fixed at another end. If the force constant of the spring is $1500 \mathrm{N} {\mathrm{m}}^{-1},$ the compression in the spring is ______

A block of mass $M$ in the  Fig. moving on the frictionless horizontal surface collides with the spring of spring constant

$k$ and compresses it by length $L$. The maximum momentum of the block after collision is

The potential energy of a $1 \mathrm{kg}$ particle is free to move along $x$-axis is given by $V\left(x\right)=\left[\frac{x^4}{4}-\frac{x^2}{2}\right] \mathrm{J}$ . The total mechanical energy of the particle is $2 \mathrm{J}$ Then the maximum speed (in ${\mathrm{ms}}^{-1}$) is