On a particle placed at origin, a variable force $F=-ax$ (where $a$ is a positive constant) is applied. If $U\left(0\right)=0$, then the graph between $PE$ of particle $U\left(x\right)$ and $x$ is best represented by

A rectangular block is moving along a frictionless path, when it encounters the circular loop as shown. The block passes points $1,2,3,4,1$ before returning to the horizontal tracks. At point $3$

Figure shows a body $A$ at the top of a frictionless hemispherical inverted bowl of radius $R.$ If the body starts slipping from the highest point, then the horizontal distance between the point where it leaves contact with sphere and the point at which the body was placed is

A particle is moving in a vertical circle such that the tension in the string at the toproust point $B$ is zero. The acceleration of the particle at point $A$ is $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$

The bob of pendulum is projected with horizontal velocity of $\sqrt{3gl},l$ is the length of string. Find the angular displacement of string before it becomes slack.

A particle of mass $m$, attached to the end of a string of length $l$, is released from the initial position $A$, as shown in the figure. The particle moves in a vertical circular path $O.$ When it is vertically below $O,$ the string makes contact with nail $N$ placed directly below $O$ at a distance $h$ and rotates around it. If the particle just completes the vertical circle about $N,$ then 

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