Angular momentum changing as mass moves to center

In summary: You are right that the child's moment of inertia is now zero, but this does not mean that the total moment of inertia is now only that of the disk. In summary, the total initial moment of inertia is 400 kgm2 and the total initial angular momentum is 800pi kgm2. When the child moves to the center of the merry-go-round, their moment of inertia becomes zero and the total final moment of inertia is now only 0.5MR2. However, the total final angular momentum remains the same at 800pi kgm2, since there is no external torque acting on the system.
  • #1
jybe
41
1

Homework Statement


A 25kg child is spinning on a merry-go-round of mass 150kg and radius 2m at a constant angular velocity of 1rev/s. The child slowly walks to the center of the merry-go-round. Treat the child as a point mass and the merry-go-round as a uniform solid disk, and neglect friction of the merry-go-round's axle.

1) What is the total initial moment of inertia (Ii) and angular momentum (Li)?

2) What is the total final moment of inertia (If) and angular momentum (Lf)?

Homework Equations


L = IW (W for angular velocity)
Ichild = mr2
Idisk = 0.5MR2

The Attempt at a Solution



1)
Ichild = mr2
Idisk = 0.5MR2

Itotal = R2(m + 0.5M)
Itotal = 22(25 + 0.5*150)
Itotal = 400 kgm2 (total initial moment of inertia)

Linitial = Ii*W
Linitial = 400*(1 rev/s)*(2 pi)
Linitial = 800pi kgm2 (total initial angular momentum)

2)
If = mr2 + 0.5MR2

But this is where I get confused. Is it correct that once the child has moved to the center of the merry-go-round, the radius is now zero? So If is now equal to only the moment of inertia of the merry-go-round, 0.5MR2?
 
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  • #2
jybe said:
But this is where I get confused. Is it correct that once the child has moved to the center of the merry-go-round, the radius is now zero? So If is now equal to only the moment of inertia of the merry-go-round, 0.5MR2?
That is correct for a point child. Unfortunately, most children are not point children. :smile:
 
  • #3
jybe said:
800pi kgm2
Check the units.
 

1. How does angular momentum change as mass moves towards the center?

As mass moves towards the center, the distance from the axis of rotation decreases, resulting in a decrease in the moment of inertia. This decrease in moment of inertia causes an increase in angular velocity, resulting in a change in angular momentum. This is known as the conservation of angular momentum, which states that the total angular momentum of a system remains constant unless acted upon by an external torque.

2. What factors can affect the change in angular momentum as mass moves towards the center?

The change in angular momentum as mass moves towards the center is affected by the moment of inertia, angular velocity, and external torque. As the moment of inertia decreases, the angular velocity must increase to maintain the same amount of angular momentum. Additionally, if an external torque is applied, it will cause a change in the system's angular momentum.

3. Is the change in angular momentum always proportional to the change in mass as it moves towards the center?

No, the change in angular momentum is not always proportional to the change in mass as it moves towards the center. The change in angular momentum also depends on the moment of inertia, angular velocity, and external torque. While the change in mass will affect the change in moment of inertia, it may not have a direct effect on the change in angular velocity.

4. Can angular momentum increase as mass moves towards the center?

Yes, angular momentum can increase as mass moves towards the center. This can occur when there is a decrease in the moment of inertia, but the angular velocity increases at a greater rate. This increase in angular velocity results in a greater change in angular momentum, despite the decrease in moment of inertia.

5. How does the change in angular momentum as mass moves towards the center affect rotational motion?

The change in angular momentum as mass moves towards the center affects rotational motion by causing a change in the system's angular velocity. This change in angular velocity can result in changes in the direction and speed of rotational motion. It also plays a crucial role in maintaining the stability and balance of rotating objects, such as planets and galaxies.

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