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

A body of mass $$5 \times 10^{-3} \mathrm{~kg}$$ is launched upon a rough inclined plane making an angle of $$30^{\circ}$$ with the horizontal. Obtain the coefficient of friction between the body and the plane if the time of ascent is half of the time of descent.

A boy is pushing a ring of mass $$3 \mathrm{~kg}$$ and radius $$0.6 \mathrm{~m}$$ with a stick as shown in figure. The stick applies a force of $$3 \mathrm{~N}$$ on the ring and rolls it without slipping with an acceleration of 0.4 m/s$$^2$$. The coefficient of friction between the ground and the ring is large enough that rolling always occurs and the coefficient of friction between the stick and the ring is $$\frac{F}{10}$$. The value of $$F$$ is

A load of mass $m$ falls from a height $$h$$ on the scale pan hung from a spring as shown. If the spring constant is $$k$$ and mass of the scale pan is zero and the mass $$m$$ does not bounce relative to the pan, then the amplitude of vibration is