Question

A solid disk and a hallow sphere roll down a frictionless ramp. Which object will reach the bottom of the ramp first if they both start from rest? Use the conservation of energy (including rotational kinetic energy) to explain.

Answer 1

Given that a solid disk and a hallow sphere roll down a frictionless ramp. Let's determine the object that will reach the bottom of the ramp first if they both start from rest.

Now, apply the formulas for the Inertia of the solid disk and hallow sphere:

For the solid disk, we have:

`I_1=(mR^2)/(2)`

For the hallow sphere, we have:

`I_2=(2)/(3)mR^2`

Now, apply the conservation of energy

`(1)/(2)mv_1^2+(1)/(2)mv_2^2=mgh`

The solid disk will have a lesser rotational inertia while the rotational inertia of the hallow sphere will be greater. Since the solid disk will have a lower rotational inertia the angular momentum of the solid sphere will be greater than the angular momentum of the hallow sphere.

Since the angular momentum of the solid is greater, the solid disk will first reach the bottom of the ramp.

ANSWER:

The solid disk will first reach the bottom of the ramp.