Several rigid bodies rolls down without slip on an incline with height $H$, their final velocity $v$ on the bottom of the incline are as follow, where initial velocity $v_0 = 0$.
| Form | $I$ | $v$ | Motion |
|---|---|---|---|
| hoop | $mr^2$ | $\sqrt{gH}$ | roll |
| disk | $\tfrac12 mr^2$ | $\sqrt{\tfrac{4}{3}gH}$ | roll |
| solid sphere | $\tfrac25 mr^2$ | $\sqrt{\tfrac{10}{7}gH}$ | roll |
| box* | $0$ | $\sqrt{2gH}$ | slide |
*box is as point mass particle, slides down on a frictionless incline.
Notice that the last row is not for rigid body and the incline is frictionless, while for other cases the incline is with friction. It is given for comparison only.
- The President and Fellows of Harvard College, “Rolling Down an Incline”, Harvard Natural Sciences Lecture Demonstrations, Harvard University, 2022, url https://sciencedemonstrations.fas.harvard.edu/presentations/rolling-down-incline [20221129].