A thin circular ring of mass $M$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega .$ Two objects, each of mass $m,$ are attached gently to the opposite ends of a diameter of the ring. The ring rotates now with an angular velocity
$\frac{{\omega M}}{{M + m}}$
$\frac{{\omega (M - 2m)}}{{M + 2m}}$
$\frac{{\omega M}}{{M + 2m}}$
$\frac{{\omega (M + m)}}{M}$
A circular disk of moment of inertia $I_t$ is rotating in a horizontal plane, about its symmetry axis, with a constant angular speed $\omega _i$. Another disk of moment of inertia $I_b$ is dropped coaxially onto the rotating disk. Initially the second disk has zero angular speed. Eventually both the disks rotate with a constant angular speed $\omega _f$. The energy lost by the initially rotating disc to friction is
A solid sphere is rolling on a frictionless surface, shown in figure with a translational velocity $v\,\,m/s.$ If it is to climb the inclined surface then $v$ should be
The centre of mass of two masses $m$ and $m'$ moves by distance $\frac {x}{5}$ when mass $m$ is moved by distance $x$ and $m'$ is kept fixed. The ratio $\frac {m'}{m}$ is
A wheel of radius $r$ rolls without slipping with a speed $v$ on a horizontal road. When it is at a point $A$ on the road, a small jump of mud separates from the wheel at its highest point $B$ and drops at point $C$ on the road. The distance $AC$ will be
Two spheres are rolling with same velocity (for their $C. M.$) their ratio of kinetic energy is $2 : 1$ & radius ratio is $2 : 1$, their mass ratio will be :