# Rolling of a disk and sphere

I am confused regarding the fact that when a disk is rolling on an inclined plane without slipping and similarly a solid sphere is rolling on an inclined plane without slipping then the sphere has more angular velocity. the moment of inertia of a sphere is less than the disk so it that if the moment of inertia is less then the angular velocity is more? Also the Mass and shape specifications of both objects is same.

Also if a sphere's angular velocity is more then why aren't tyres of cars,etc spherical?

I am confused regarding the fact that when a disk is rolling on an inclined plane without slipping and similarly a solid sphere is rolling on an inclined plane without slipping then the sphere has more angular velocity. the moment of inertia of a sphere is less than the disk so it that if the moment of inertia is less then the angular velocity is more?

Assuming that they have the same mass and same dimensions (except that the disk is hollow), then yes, the sphere has more angular velocity because it has a lower moment of inertia.

Also if a sphere's angular velocity is more then why aren't tyres of cars,etc spherical?

Other than the fact that spherical tyres are quite impossible to implement, look bad, and probably unsafe, you'll probably want the higher moment of inertia, as it will make your ride smoother due to its higher resistance to change in motion. This is the reason why your engine has a flywheel - without it your ride will be jerky and unsmooth.

You need to consider the particular circumstances of a body rolling down an incline.

Consider a sphere and a disc, with the same mass, M, and radius, R, rolled one after the other down the same slope. They'll both speed up as they roll down. Say you pick the moment when they're each going $V$ m/s. So they both have the same translational kinetic energy, $\frac12MV^2$ and they are both rotating at the same angular velocity, $\omega=\frac{V}{R}$.

But the sphere has a lower moment of inertia, so at that angular velocity, it has less rotational kinetic energy. Since the sphere has less total kinetic energy, it must have that velocity farther up the slope. At any point on the slope, the sphere is going faster.

It's a quirk of the math that all "rollers" of any shape will roll down together. All solid spheres of any size will roll together to beat all discs: all discs of any size will roll together to beat all rings.