A spherical shell of 1 kg mass and radius $$R$$ is rolling with angular speed $$\omega$$ on horizontal plane (as shown in figure). The magnitude of angular momentum of the shell about the origin $$O$$ is $$\frac{a}{3}R^2\omega$$. The value of $$a$$ will be

Given that it is an spherical shell

We know that moment of inertia of a spherical shell with respect to its centre is $$I=\frac{2}{3}MR^2$$

But with respect to origin it is $$I+MR^2$$ (Here R is the perpendicular distance between origin and centre of the sphere

So moment of inertia with respect to origin is $$I=\frac{5}{3}MR^2$$

We also know that Angular momentum $$L=I\omega\ $$

substitute I

$$L=\frac{5}{3}MR^2\omega\ $$

Given that M=1

$$L=\frac{5}{3}R^2\omega\ $$

So a will be 5