Angular Momentum of Slotted Homogeneous Disk Help

[johnchen25]

Homework Statement

The slotted homogeneous disk of mass m freely turns about the vertical axis with angular velocity ωi. A small marble of mass 0.05m is placed near the center of the disk. The marble travels outward along the slot due to centrifugal force. Determine the angular velocity of the disk after the marble has left the slot. Neglect the effect of the slot on the moment of inertia of the disk.

Homework Equations

H = Iω
I = (1/2) mR^2

The Attempt at a Solution

I'm not sure how to attack this problem. Any suggestions?

 

[singular]

Try thinking about it in terms of conservation of angular momentum. So, the angular momentum of the system with mass m+0.05m with angular velocity ωi will be the same when the marble has left the slot and the system has mass m and angular velocity ___.

 

[johnchen25]

Thanks for the reply. The problem says that the disk has an angular velocity of ωi before adding the marble ball in the center. Would adding the marble change the angular velocity since the disk would hence contain more mass?

 

[singular]

Thanks for the reply. The problem says that the disk has an angular velocity of ωi before adding the marble ball in the center. Would adding the marble change the angular velocity since the disk would hence contain more mass?

Yes, but I am not sure how the motion of the marble changes the angular velocity. This is probably crucial to solving the problem. You should be able to calculate the speed of the marble with the centrifugal force, but I am not sure how that would look in terms of the discs momentum. My intuition tells me that you don't need to consider the centrifugal force. Since we are looking at the problem in an inertial frame of reference, we don't need to consider pseudo-forces.

 

[singular]

Sorry I am not being much help. I am in the middle of finals. If I get some spare time, I will work out the problem and help you further if nobody beats me to it.

 

[singular]

Here is something to help you start.

In order to calculate the angular velocity of a system like this, you want to look at the conservation of angular momentum. Let d refer to disc, m refer to marble, and c refer to composite system.

Applying conservation of angular momentum to the system where i refers to the disc rotating without the marble (this should be the exact same as calculating the angular momentum of the composite system when the marble is at a radial distance of r=0) and f refers to the final state of the system that the problem wants you to look at we get

$$I_{di}\omega_{di}+I_{mi}\omega_{mi}=I_{cf}\omega_{cf}$$

You know the angular momentum of the disc before the marble is placed in the slot and the angular momentum of the marble before it is placed in the slot should be easy enough to figure out (what is the angular momentum of the marble when it is sitting in your hand?).

This should be enough to get you started.

Now, let me ask you this. Will the angular momentum of the system be equal to the initial angular momentum when the marble is at r = 0.5R? What about r=.6935647747R?

 

[Delphi51]

I don't understand the question; looks like something missing.
The marble "travels along the slot" . . . how far?
During this travel angular momentum is conserved so
I*ω1 = I*ω + mr²ω
so that the ω could be found when the marble has reached any given radius.
What happens as "the marble leaves the slot"? Perhaps this happens at the radius R of the disk and it is thrown off into space. If so, the solution would be
Iω1 = Iω + mr²ω with r replaced by R and the whole thing solved for ω.

 

[singular]

Perhaps this happens at the radius R of the disk and it is thrown off into space. If so, the solution would be
Iω1 = Iω + mr²ω with r replaced by R and the whole thing solved for ω.

This was my assumption.

I was able to find the problem http://books.google.com/books?id=pq...esnum=2&ved=0CAwQ6AEwAQ#v=onepage&q=&f=false" if you want to see the whole problem. The only difference is notation. I wrote I_cω_c, which is the same as your RHS and I included the angular momentum of the marble before it is placed in the slot (where it's ω=0), which is probably unnecessary.