Math Snap

STEP 1

1. The ball rolls down the ramp without slipping, meaning there is a pure rolling motion.
2. The ball has a uniform circular shape.
3. The speed of the center of mass is denoted as $v_{\text{cm}}$.
4. The radius of the ball is $R$.

STEP 2

1. Understand the motion of the ball.
2. Determine the speed of each point on the ball.
3. Rank the speeds of the points.

STEP 3

In pure rolling motion, the ball rotates about its center of mass while its center of mass moves linearly down the ramp.

STEP 4

The speed of the center of mass, $v_{\text{cm}}$, is the linear speed of the ball's center.

STEP 5

The speed of the top of the ball is the sum of the translational speed of the center of mass and the rotational speed at the top of the ball. This speed is $v_{\text{top}} = v_{\text{cm}} + R\omega$, where $\omega$ is the angular speed.

STEP 6

The speed of the bottom of the ball is the difference between the translational speed of the center of mass and the rotational speed at the bottom of the ball. This speed is $v_{\text{bottom}} = v_{\text{cm}} - R\omega$.

STEP 7

Since the ball rolls without slipping, $v_{\text{cm}} = R\omega$. Therefore, the speeds are:
- $v_{\text{top}} = v_{\text{cm}} + R\omega = 2v_{\text{cm}}$
- $v_{\text{bottom}} = v_{\text{cm}} - R\omega = 0$

Rank the speeds:
1. Top of the ball: $2v_{\text{cm}}$
2. Center of mass: $v_{\text{cm}}$
3. Bottom of the ball: $0$
The ranking of the speeds is:
1. Top of the ball
2. Center of mass
3. Bottom of the ball