Question

Two wheels having the same radius and mass rotate at the same angular velocity (Fig. 8–38). One wheel is made with spokes so nearly all the mass is at the rim. The other is a solid disk. How do their rotational kinetic energies compare?

(a) They are nearly the same.

(b) The wheel with spokes has about twice the KE.

(c) The wheel with spokes has higher KE, but not twice as high.

(d) The solid wheel has about twice the KE.

(e) The solid wheel has higher KE, but not twice as high.

FIGURE 8-38

MisConceptual Question 7.

Short answer

The correct option is (b).

Step-by-step solution

Rotational kinetic energy

Rotational kinetic energy is proportional to the moment of inertia of a body for constant angular velocity.For this problem, the angular velocity is the same for both objects; so you have to find out which one has a large moment of inertia.

Both wheels have the same mass and radius.

Let m be the mass and r be the radius of the wheels.

Explanation 

You can consider the wheel with spokes as a circular ring since, approximately, all of the mass is distributed over the rim.

Therefore, the moment of inertia of the wheel is \({I_1} = m{r^2}\) when the rotational axis is through the center.

Now, you know the moment of inertia of a circular solid disk is \({I_2} = \frac{1}{2}m{r^2}\) when the rotation axis is through the disk's center.

As both have the same angular velocity, the wheel with spokes has about twice the kinetic energy of the wheel as a solid disk.

Hence, option (b) is correct.