A piece of wire is bent in the shape of a parabola $y=k{x}^2$ ($y$-axis vertical) with a bead of mass $m$ on it. The bead can slide on the wire without friction. It stays at the lowest point of the parabola when the wire is at rest. The wire is now accelerated parallel to the $x$-axis with a constant acceleration $a$. The distance of the new equilibrium position of the bead, where the bead can stay at rest with respect to the wire from the $y$-axis is

Two fixed frictionless inclined planes making an angle $30°$ and $60°$ with the vertical are shown in the figure. Two blocks $A$ and $B$ are placed on the two planes respectively. What is the relative vertical acceleration of $A$ with respect to $B$?

(Use $\sin 30°=\cos 60°=\frac{1}{2} \text{and} \sin 60°=\cos 30°=\frac{\sqrt{3}}{2}$)

A particle of mass $m$ is at rest at the origin at time $t=0$. It is subjected to a force $F\left(t\right)=F_0{e}^{–\mathrm{bt}}$ in the $x$-direction. Its speed $v\left(t\right)$ is depicted by which of the following curves?

A mass$m$supported by a massless string wound around a uniform hollow cylinder of mass $m$ and radius $R\mathit{.}$ If the string does not slip on the cylinder, with what acceleration will the mass fall on release ? 

In the arrangement shown in figure the masses of the wedge $M$ and the body $m$ are known. The appreciable friction exists only between the wedge and the body $m$, the friction coefficient being equal to $k$. The masses of the pulley and the thread are negligible. Find the acceleration of the body $m$ relative to the horizontal surface on which the wedge slides.