A small object of uniform density rolls up a curved surface with an initial velocity $v$. It reaches upto a maximum height of $3v^2/4g$ with respect to the initial position. The object is

$A$ ring of mass $m$ is rolling without slipping with linear velocity $v$ as shown is figure. $A$ rod of identical mass is fixed along one of its diameter. The total kinetic energy of the system is :-

A meter stick is held vertically with one end on the floor and is allowed to fall. The speed of the other end when it hits the floor assuming that the end at the floor does not slip is ......... $m / s$ $\left(g=9.8 \,m / s ^2\right)$

A thin uniform rod oflength $l$ and mass $m$ is swinging freely about a horizontal axis passing through its end . Its maximum angular speed is $\omega$. Its centre of mass rises to a maximum height of:

Two coaxial discs, having moments of inertia $I_1$ and $\frac{I_1}{2}$ are a rotating with respectively angular velocities $\omega_1$ and $\frac{\omega_1}{2}$, about their common axes. They are brought in contact with each other and thereafter they rotate with a common angular velocity. If $E_f$ and $E_i$ are the final and initial total energies, then $(E_f -E_i)$ is

A solid sphere and a hollow cylinder roll up without slipping on same inclined plane with same initial speed $v$. The sphere and the cylinder reaches upto maximum heights $h_1$ and $h_2$, respectively, above the initial level. The ratio $h_1: h_2$ is $\frac{n}{10}$. The value of $\mathrm{n}$ is__________.