Angular Momentum of a Rolling Sphere

When a solid sphere of radius R and mass M rolls purely on a rough horizontal surface and strikes a wall elastically, its angular momentum just after the strike can be determined.

Angular Momentum Definition

Angular momentum is a property of rotating objects and is defined as the product of the moment of inertia and the angular velocity of the object. It is given by the formula:

L = Iω

where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.

Moment of Inertia of a Sphere

The moment of inertia of a solid sphere about an axis passing through its center of mass is given by the formula:

I = (2/5)MR²

where I is the moment of inertia, M is the mass of the sphere, and R is the radius of the sphere.

Rolling Motion

When the sphere rolls purely on a rough horizontal surface, it experiences both translational and rotational motion. The relationship between the linear velocity (v) and angular velocity (ω) of the sphere is given by:

v = Rω

where v is the linear velocity and R is the radius of the sphere.

Angular Momentum of a Rolling Sphere

The angular momentum of a rolling sphere can be calculated by substituting the moment of inertia and the angular velocity into the angular momentum formula. In this case, the sphere strikes the wall elastically, meaning there is no loss of energy during the collision.

After the strike, the linear velocity of the sphere changes direction, but the angular velocity remains the same. Therefore, the angular momentum just after the strike is the same as it was before the strike.

Conclusion

The angular momentum of the sphere just after striking the wall elastically is equal to the angular momentum before the strike. This is because there is no external torque acting on the sphere during the collision, and the angular velocity remains unchanged.