Solving a Spinning Disk Physics Exam Problem

In summary, the conversation discusses a problem involving a spinning disk on a level surface and how it reaches a new angular velocity without sliding. The final solution is shown to be W = Wo/3, although there were some discrepancies in the initial calculations. The conversation also includes a discussion about the use of moment of inertia for a solid sphere versus a disk, the role of friction, and different approaches to solving the problem. The conversation concludes with a suggestion to seek help in the Mathematics forum for tips on formatting.
  • #1
discoverer02
138
1
Here's the problem and I'm close to the answer, but I guess close isn't good enough on a Physics exam.

A spinning solid disk, rotating with angular velocity Wo, is put down on a level surface. It slides and rolls until it reaches an angular velocity W at which it rolls without sliding. Show that W = Wo/3

I place my origin on the ground so that angular momentum is conserved.

The disk spins clockwise
1 = right when the disk is placed on the floor.
2 = right when the disk begins rolling without spinning.
L = angular momentum.
cm = center of mass.
I = moment of inertia about cm
R = radius
M = Mass
V = Velocity of cm

L1spin + L1cm = L2spin + L2cm

IWo + 0 = IW - RMV ==> with no slip V = RW
(2/5)MWoR^2 = (2/5)MWR^2 - MWR^2
-2Wo/3 = W

Besides probably messing up the signs, why do I end up with twice what I need?
 
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  • #2
hmmm...

I've never done anything like this one & I don't have an answer, but I have some questions that may help...

It looks like you're using the moment of inertia for a solid sphere rather than a cylinder. Is that part of the problem?

Clearly friction is involved here, otherwise the disk would just keep spinning. Doesn't that mean that neither energy nor momentum are conserved?

Can this be solved without knowing μs? Can it be that regardless of what the initial angular velocity is, it will always scrub off exactly 2/3 of its speed before it stops slipping, independent of the coefficient of static friction? That's certainly counter-intuitive.

On the other hand, maybe it's μk that's relevant, since the disk is sliding initially. But once it stops slipping, static friction must take over, right?
 
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  • #3
This is amazing!

Let
Δt = time elapsed until disk stops slipping
f = force of friction while slipping (assumed this to be constant)

Then the linear momentum gained by the disk is equal to the impulse provided by the friction so
fΔt = Mvcm ....Equation 1

and the change in angular momentum is equal to the angular impulse which (I hope :smile: ) is given by
TΔt = Iω0 - Iω

and T = fR so
fRΔt = (1/2)MR2(ω0 - ω)
fΔt = (1/2)MR(ω0 - ω)

Now, substuting from Equation 1:
Mvcm = (1/2)MR(ω0 - ω)
vcm = (1/2)R(ω0 - ω)

and vcm = Rω so
Rω = (1/2)R(ω0 - ω)
ω = (1/2)ω0 - (1/2)ω
(1/2)ω0 = (3/2)ω
ω0 = 3ω

Whoda thunk that?

...

By the way, go to the Mathematics forum and see Greg Bernhardt's announcement about Making math symbols and the thread Additional math notation for tips on formatting.
 
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  • #4
You're right. I was taking the moment of inertial of a sphere rather than a disk and I also messed up this signs in the equation.

Yours is a very interesting approach. I'm impressed. I tend to stay within the mechanics of what was presented in class and not think about the problem enough. It obviously pays to think about the problem.

Thanks very much for the help.
 

1. How do I approach solving a spinning disk physics exam problem?

First, identify the variables given in the problem, such as the mass, radius, and angular velocity of the disk. Then, use the relevant equations for rotational motion, such as the moment of inertia and angular momentum equations, to solve for the unknown variable.

2. What are the key concepts involved in solving a spinning disk physics exam problem?

The key concepts involved in solving a spinning disk physics exam problem include rotational motion, moment of inertia, angular velocity, and angular momentum. It is also important to understand the relationship between these concepts and how they apply to a spinning disk.

3. How do I calculate the moment of inertia of a spinning disk?

The moment of inertia of a spinning disk can be calculated by using the formula I = 1/2 * m * r^2, where m is the mass of the disk and r is the radius from the axis of rotation to the edge of the disk. Alternatively, if the disk has a known density, the moment of inertia can be calculated using the formula I = (1/2) * m * (r^2 + h^2), where h is the height of the disk.

4. Can you provide an example of a spinning disk physics exam problem and how to solve it?

Sure! Let's say we have a disk with a radius of 0.5 meters and an angular velocity of 5 radians per second. The mass of the disk is 2 kilograms. We are given the moment of inertia of the disk to be 0.25 kg.m^2. To solve for the angular momentum of the disk, we can use the equation L = I * ω. Substituting in our values, we get L = 0.25 * 5 = 1.25 kg.m^2/s.

5. What are some common mistakes to avoid when solving a spinning disk physics exam problem?

One common mistake is forgetting to convert units. Make sure all units are consistent and convert if necessary. Another mistake is not properly labeling diagrams or not considering the direction of the rotational motion. Lastly, it is important to always double check your calculations and make sure they are accurate.

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