Question2. Explanation Tas! (6 points)
A ball rolls down ramp without slipping.
At any given instant, rank the following points on the ball by their speed: - The center of mass - The bottom of the ball - The top of the ball

Studdy Solution

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: - $v_{\text{top}} = v_{\text{cm}} + R\omega = 2v_{\text{cm}}$ - $v_{\text{bottom}} = v_{\text{cm}} - R\omega = 0$

STEP 8

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

Was this helpful?

Question2. Explanation Tas! (6 points)
A ball rolls down ramp without slipping.
At any given instant, rank the following points on the ball by their speed: - The center of mass - The bottom of the ball - The top of the ball

Studdy Solution

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: - $v_{\text{top}} = v_{\text{cm}} + R\omega = 2v_{\text{cm}}$ - $v_{\text{bottom}} = v_{\text{cm}} - R\omega = 0$

STEP 8

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

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Question2. Explanation Tas! (6 points)A ball rolls down ramp without slipping.At any given instant, rank the following points on the ball by their speed: - The center of mass - The bottom of the ball - The top of the ball

Studdy Solution

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 vcm v_{\text{cm}} vcm​.4. The radius of the ball is R R 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, vcm v_{\text{cm}} vcm​, 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 vtop=vcm+Rω v_{\text{top}} = v_{\text{cm}} + R\omega vtop​=vcm​+Rω, where ω \omega ω 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 vbottom=vcm−Rω v_{\text{bottom}} = v_{\text{cm}} - R\omega vbottom​=vcm​−Rω.

STEP 7

STEP 8

Rank the speeds:1. Top of the ball: 2vcm 2v_{\text{cm}} 2vcm​2. Center of mass: vcm v_{\text{cm}} vcm​3. Bottom of the ball: 0 0 0 The ranking of the speeds is:1. Top of the ball2. Center of mass3. Bottom of the ball

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Question2. Explanation Tas! (6 points)
A ball rolls down ramp without slipping.
At any given instant, rank the following points on the ball by their speed: - The center of mass - The bottom of the ball - The top of the ball

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 $R$ .

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

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

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.

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 $v_{\text{bottom}} = v_{\text{cm}} - R\omega$ .

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