A particle located in one dimensional potential field has potential energy function $U\left(x\right)=\frac{a}{x^2}-\frac{b}{x^3},$ where $a$ and $b$ are positive constants. The position of equilibrium corresponds to $x=$

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

In the figure shown, a particle is released from the position Aon a smooth track. When the particle reaches at $B\mathit{,}$ then normal reaction on it by the track is

A particle is moving along a vertical circle of radius $R.$ At $P,$ what will be the velocity of particle (assume critical condition at $C) ?$

A particle of mass $m$ attached to the end of string of length $l$ is released from the horizontal position. The particle rotates in a circle about $O$ as shown. When it is vertically below $O,$ the string makes contact with a nail $N$ placed directly below $O$ at a distance $h$ and rotates around it. For the particle to swing completely around the nail in a circle,

A ball is released from the top of vertical circular pipe. Find angle $\mathrm{\theta }$ with vertical where the ball will lose contact with inner side wall of pipe and start moving with outer side wall. (Thickness of pipe is small as compared to radius of circle)