A particle travels along the path $\mathrm{y}\mathrm{ }=\mathrm{ }\mathrm{a}\mathrm{ }+\mathrm{ }\mathrm{b}\times \mathrm{x}\mathrm{ }+\mathrm{ }\mathrm{c}\times {\mathrm{x}}^2\mathrm{ }$ where, a, b, c are positive constants. If $v_0$ is the speed of the particle and it is constant. If at the given instant particle is at x = 0. Radius of curvature of the path is :

In the given figure, $a=15 \mathrm{m} {\mathrm{s}}^{-2}$ represents the total acceleration of a particle moving in the clockwise direction in a circle of the radius $R=2.5 \mathrm{m}$at a given instant of time. The speed of the particle is

A mass $\text{m}$ moves in a circle on a smooth horizontal plane with velocity ${\mathrm{v}}_0$ at a radius ${\mathrm{R}}_0$ . The mass is attached to a string which passes through a smooth hole in plane as shown.

The tension in the string is increased gradually and finally $\text{m}$ moves in a circle of radius $\frac{{\mathrm{R}}_0}{2}$. The final value of the kinetic energy is:

If the kinetic energy of a particle of mass $m,$ performing uniform circular motion in a circle of radius $r,$ is $E,$ find the acceleration of the particle.