Chapter 8, Problem 1OQ

A solid ball and a solid cylinder roll down a ramp. They both start from rest at the same time and place. Which gets to the bottom first?

1. They get there at the same time.
2. They get there at almost exactly the same time except for frictional differences.
3. The ball gets there first.
4. The cylinder gets there first.
5. Can’t tell without knowing the mass and radius of each.

To determine

The body reaching the bottom of the ramp first.

Answer to Problem 1OQ

Solution:

(E). The solid ball, assuming both bodies have the same the radius and the same mass.

Given:

We have a solid sphere and a solid cylinder, but we are not told the radius or the mass. They start to move from rest at the same time and place.

Explanation of Solution

The object with the lower moment of inertia will reach the bottom of the ramp first. The moment of inertia depends on the mass distribution; the closer the mass to the axis, the lower the moment of inertia.

The moment of inertia is calculated with the next equation:

  $I=\int r^{2}dm$

Where r is the distance from the axis to the dm . Solving this equation for the mass distribution of a solid sphere:

  $I_s=\frac{2}{5}m.r^2$

Where m is the mass and r is the radius. The moment of inertia for a solid cylinder with an axis passing through its centre:

  $I_c=\frac{1}{2}m.r^2$

If we assume, they have the same mass and the same radius, the sphere (the ball) has the lower moment of inertia and it will reach the bottom of the ramp first.

So, correct option is E.

Conclusion:

On two objects that have the same mass and radius, the moment of inertia depends on the mass distribution.