# How Does Moving Towards the Center Affect a Merry-Go-Round's Kinetic Energy?

• carsof
In summary, a child on a revolving merry-go-round moves towards the center, causing the angular momentum (L) and rotational kinetic energy (K) of the system to remain constant. The statements "L is conserved" and "K is conserved" are true in this scenario. The net torque on the system is zero, allowing L to be conserved. The child's movement towards the center does not affect the rotational kinetic energy of the system since the moment of inertia (I) decreases at the same rate as the increase in tangential velocity, resulting in a constant K.
carsof

## Homework Statement

A child is initially sitting near the outer rim of a revolving merry-go-round. Suddenly, the child moves towards the center of the merry-go-round (while it is still revolving). For the merry-go-round+child system, let the symbols L and K refer to the magnitude of the angular momentum (about the center of the merry-go-round) and rotational kinetic energy, respectively.

Consider the following statements:

Ia. L is conserved Ib. L increases Ic. L decreases

IIa. K is conserved IIb. K increases IIc. K decreases

Which of these statements are true? (The explanation is for the choice of ’II’)

## Homework Equations

So, I know L is conserved/constant when dl/dt=0. And I know dl/dt=0 when net torque =0. But, how can I tell from reading this problem that the net torque is zero?
when I draw a diagram of the merry go round and the child on it and make my axis the center of the merry go round, I get Net torque = -mgR (where m is mass of child and R is radius of merry go round). I just don't get how to tell that the net torque in certain problems =0 (and when it doesn't).

Also, I thought for this problem that initially, the merry go round is rotating (has Krot), and the child is not moving w respect to merry go round (K=0). Then, the child is moving (has K trans). So, I don't see how K rotational of the system increases ?
I'm just so confused :(

## The Attempt at a Solution

Listed above[/B]

carsof said:
Net torque = -mgR
The torque on the child is not mgR. It is generated by friction not by gravity. Regardless of that, your system is child + merry go round. There is no net torque acting on that system because it is isolated (assuming no friction, air resistance, etc.)

From L = Iω and Krot = (1/2)Iω2, you can easily show that Krot = L2/(2I). What happens to I when the child moves towards the center while L stays constant?

## 1. When is the net torque equal to zero?

The net torque is equal to zero when the sum of all torques acting on an object is balanced and there is no rotational acceleration. This means that the clockwise and counterclockwise torques cancel each other out, resulting in a net torque of zero.

## 2. How can you determine when the net torque is zero?

To determine when the net torque is zero, you need to calculate the torque produced by each force acting on the object. If the sum of these torques is equal to zero, then the net torque is also zero. This can be represented by the equation ∑T = 0, where ∑T is the sum of all torques.

## 3. What is the significance of having a net torque of zero?

Having a net torque of zero means that the object is in rotational equilibrium. This means that the object will maintain a constant rotational speed or remain at rest, depending on its initial state. Objects in rotational equilibrium do not experience rotational acceleration.

## 4. Can the net torque ever be zero if there are forces acting on the object?

Yes, the net torque can be zero even if there are forces acting on the object. This is because torque is dependent on both the magnitude and direction of the force, as well as the distance from the pivot point. If the forces are balanced and act at equal distances from the pivot point in opposite directions, the net torque will be zero.

## 5. How does the net torque affect an object's rotational motion?

The net torque determines whether an object will experience rotational acceleration or remain in rotational equilibrium. If the net torque is not zero, the object will experience rotational acceleration and its rotational motion will change. If the net torque is zero, the object will maintain a constant rotational speed or remain at rest.

• Introductory Physics Homework Help
Replies
5
Views
253
• Introductory Physics Homework Help
Replies
8
Views
2K
• Introductory Physics Homework Help
Replies
18
Views
5K
• Introductory Physics Homework Help
Replies
3
Views
2K
• Introductory Physics Homework Help
Replies
2
Views
1K
• Introductory Physics Homework Help
Replies
4
Views
6K
• Introductory Physics Homework Help
Replies
7
Views
370
• Introductory Physics Homework Help
Replies
9
Views
6K
• Introductory Physics Homework Help
Replies
2
Views
2K
• Introductory Physics Homework Help
Replies
16
Views
1K