A solid sphere rolls without slipping on a rough surface and the centre of mass has a constant speed $v_0$. If the mass of the sphere is $m$ and its radius is $R$, then find the angular momentum of the sphere about the point of contact
$\frac{3}{5}\,Mv_0R$
$\frac{4}{5}\,Mv_0R$
$\frac{7}{5}\,Mv_0R$
$\frac{7}{2}\,Mv_0R$
A binary star consists of two stars $\mathrm{A}$ (mass $2.2 \mathrm{M}_5$ ) and $\mathrm{B}$ (mass $11 \mathrm{M}_5$ ), where $\mathrm{M}_5$ is the mass of the sun. They are separated by distance $\mathrm{d}$ and are rotating about their centre of mass, which is stationary. The ratio of the total angular momentum of the binary star $\mathrm{A}$ to the angular momentum of star $\mathrm{B}$ about the centre of mass is
A thin rod of mass $M$ and length $a$ is free to rotate in horizontal plane about a fixed vertical axis passing through point $O$. A thin circular disc of mass $M$ and of radius $a / 4$ is pivoted on this rod with its center at a distance $a / 4$ from the free end so that it can rotate freely about its vertical axis, as shown in the figure. Assume that both the rod and the disc have uniform density and they remain horizontal during the motion. An outside stationary observer finds the rod rotating with an angular velocity $\Omega$ and the disc rotating about its vertical axis with angular velocity $4 \Omega$. The total angular momentum of the system about the point $O$ is $\left(\frac{ M a^2 \Omega}{48}\right) n$. The value of $n$ is. . . . .
A flywheel can rotate in order to store kinetic energy. The flywheel is a uniform disk made of a material with a density $\rho $ and tensile strength $\sigma $ (measured in Pascals), a radius $r$ , and a thickness $h$ . The flywheel is rotating at the maximum possible angular velocity so that it does not break. Which of the following expression correctly gives the maximum kinetic energy per kilogram that can be stored in the flywheel ? Assume that $\alpha $ is a dimensionless constant
Explain Cartesian components of angular momentum of a particle.
A ring of mass $M$ and radius $R$ is rotating with angular speed $\omega$ about a fixed vertical axis passing through its centre $O$ with two point masses each of mass $\frac{ M }{8}$ at rest at $O$. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant the angular speed of the system is $\frac{8}{9} \omega$ and one of the masses is at a distance of $\frac{3}{5} R$ from $O$. At this instant the distance of the other mass from $O$ is