Chapter 11: Q11Q (page 320)

A cannonball and a marble roll smoothly from rest down an incline. Is the cannonball’s (a) time to the bottom and (b) translational kinetic energy at the bottom more than, less than, or the same as the marble’s?

Short Answer

Expert verified

a) Both cannonball and marble will reach the bottom at the same time.

b) Translational kinetic energy of the cannonball at the bottom will be more than that of marble.

Step by step solution

Step 1: Given information

A cannonball and a marble roll smoothly from rest down an incline.

Understanding the concept

The larger the radius, the distance traveled by the cannonball in one revolution is much greater than the marble. The larger the mass of the cannonball, the more translational kinetic energy of the cannonball will be more than that of marble

(a) Determining if the cannonball’s time to the bottom is more than, less than, or the same as the marble’s

Both cannonball and marble are at rest.

The radius of the marble will be smaller than that of the cannonball; therefore marble will have less rotational inertia than the cannonball. Therefore, it will have to makemore rotationsper second.

Because of the larger radius, the distance traveled by the cannonball in one revolution is much greater than the marble. Hence, the motion of the balls is such that the larger distance per rotation by the cannonball is compensated by the more rotations of marble.

Therefore, the cannonball and marble will reach the bottom at the same time.

Also, using the kinematic equation,

$Δ y = v_{0} t + \frac{1}{2} a t^{2}$

The initial velocity of both objects is the same, also they would be covering the same distance with the same acceleration, and this implies that the time would be the same.

(b) Determining if the cannonball’s translational kinetic energy at the bottom is more than, less than, or the same as the marble’s

Becauseof the largermass of the cannonball,the translationalkinetic energy of the cannonball will be more than that of marble.

Now, see that the velocity of both masses wouldbe the sameat the bottom,

$v_{f}^{2} = v_{0}^{2} + 2 a Δ y$

Again, the initial velocity, height, and distance arethe same, so the final velocity would be the same for both objects. But sincea cannonballhas greater mass, it would have higher kinetic energy.

Therefore, both cannonball and marble will reach the bottom at the same time and the translational kinetic energy of the cannonball at the bottom will be more than that of marble.