Rolling Sphere on an Inclined Plane — Inertia, Rotation Equation, Energy, and Timing
A solid sphere rolls without slipping down an inclined plane. Its tangent axis moment of inertia uses the parallel axis theorem from the center. The rotation equation about the contact point balances gravity torque and spin inertia. This yields a constant angular acceleration proportional to gravity and the slope sine. Kinetic energy combines translation and rotation and scales with the square of angular speed. Rolling without slipping links center speed and spin and its time derivative links accelerations
From constant angular acceleration the travel time over a given distance follows a square root law
Linear momentum is not conserved because external forces act along the plane and normal to it
Angular momentum is not conserved in general since external torques about fixed points are nonzero
Choosing the contact point eliminates unknown friction in the torque balance during pure rolling
