14. a ball initially at rest rolls without slipping
an incline of height 'h'. what is the linear
speed of the ball when it rolls onto the
horizontal surface?
solution:

v = √(10gh/7)

Explanation:

Initial gravitational energy = final rotational energy + kinetic energy

PE = RE + KE

mgh = ½ Iω² + ½ mv²

For a solid sphere, I = ⅖ mr².

For rolling without slipping, ω = v/r.

mgh = ½ (⅖ mr²) (v/r)² + ½ mv²

mgh = ⅕ mv² + ½ mv²

mgh = 7/10 mv²

10/7 gh = v²

v = √(10gh/7)