Question

A marble and a cube are placed at the top of a ramp. Starting from rest at the same height, the marble rolls without slipping and the cube slides (no kinetic friction) down the ramp. Determine the ratio of the center-of-mass speed of the cube to the center-of-mass speed of the marble at the bottom of the ramp.

Answer 1

The cube. From the conservation of energy:

$$mgh = 0.5mv^2$$

$$v=\sqrt{2gh}$$

The marble. From the conservation of energy:

$$0.5MV^2 + 0.5 Iω^2 = Mgh$$

For a sphere:

$$I=0.4 Mr^2$$

$$\omega=\frac{V}{r}$$

Thus,

$$0.5MV^2 + 0.5 (0.4 Mr^2)(\dfrac{V}{r})^2 = Mgh$$

$$V=\sqrt{\frac{10gh}{7}}$$

So,

$$\frac{v}{V}=\sqrt{\frac{7}{5}}$$