Angular Momentum: Flywheel & Sphere Homework

In summary, a flywheel with a cavity in its upper surface, rotating about a fixed vertical axis with a moment of inertia of I and a distance a from the axis, is initially rotating with angular velocity w1. A sphere of mass M and moment of inertia i is dropped into the cavity with no horizontal motion, spinning with angular velocity w2 about a vertical axis. Using the parallel axis theorem and conservation of angular momentum, an expression can be derived for the angular velocity of the composite system when the sphere and flywheel are at relative rest. This expression takes into account the moment of inertia of the sphere about the flywheel's axis, which is i + Ma^2, and the flywheel's moment of inertia also changes due to the
  • #1
Zell2
28
0

Homework Statement


A flywheek with a cavity in its upper surface is constrained to rotate about a fixed vertical axis. Its moment of intertia about the axis is I and the cvity is a distance a from the axis. It is initially rotating with angular velocity w1. Into the cavity is dropped a sphere of mass M and moment of inertia i which is spinning with angular velocity w2 about a vertical axis. At the moment it is dropped into the cavity it has no horizontal motion. Derive an expression for the angular velocity of the composite system when the sphere and the flywheel are at relative rest.

Homework Equations


Parallel axis theorem.
L=Iw

The Attempt at a Solution


I think I'm struggling with this problem because the sphere and the flywheel are rotating around different axis.

The moment of inertia of the sphere about the flwheel's axis of rotation= i + Ma^2, so using the conservation of angular momentum if the sphere wasn't rotating initially the systems angular velocity would be straightforward to find.

However the sphere is rotating, and also the flywheel is constrained so I'm not sure how this affects the system.

A push in the right direction would be appreciated.
Thanks
 
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  • #2
Zell2 said:

The Attempt at a Solution


I think I'm struggling with this problem because the sphere and the flywheel are rotating around different axis.

The moment of inertia of the sphere about the flwheel's axis of rotation= i + Ma^2, so using the conservation of angular momentum if the sphere wasn't rotating initially the systems angular velocity would be straightforward to find.

That's correct, unless I'm missing something huge here. The sphere's angular velocity relative to the flywheel's is given, you found the moment of inertia of the sphere relative to the flywheel's axis, and you should, as mentioned, use conservation of momentum now. Note that the flywheel's moment of inertia also changes.
 
  • #3
The sphere's angular velocity relative to the flywheel's is given[\quote]
I think it's this bit I'm struggling with. If the sphere's axis of rotation is different to the axis of rotation of the flywheel, how do the two combine? If the system was moving freely I'd expect the angular momentum about each axis to be conserved, but the flywheel is constrained so does an external torque act to prevent rotation about the sphere's axis.

Thanks
 
  • #4
Yes, as the flywheel is constrained it will continue to rotate about its own axis. The sphere is spinning about it's own central axis and rotating about the same axis as the fly wheel.
 
  • #5
Does the phrase "the sphere and the flywheel are at relative rest" just mean the centre of mass of the sphere is at rest relative to the flywheel?

Also I'm not completely sure what radou meant when he said " Note that the flywheel's moment of inertia also changes." Does this mean the moment of inertia of the system about the flywheels axis isn't I+i+Ma^2?

Thanks for your help.
 
  • #6
Zell2 said:
Does the phrase "the sphere and the flywheel are at relative rest" just mean the centre of mass of the sphere is at rest relative to the flywheel?
Yes
Zell2 said:
Also I'm not completely sure what radou meant when he said " Note that the flywheel's moment of inertia also changes." Does this mean the moment of inertia of the system about the flywheels axis isn't I+i+Ma^2?
I think what he means is you now have to include the sphere in your moment of inertia for the fly wheel. However, this
Zell2 said:
I+i+Ma^2
is not quite correct (almost there). Note that the sphere is spinning about its own axis but is rotating about the fly wheel's axis (think of a planet orbiting the sun in a circular orbit). Note also, that the sphere need not be spinning with the same angular velocity as it is rotating about the fly wheel's axis. Does that make sense?
 
  • #7
I understand what is going on qualitatively, but I'm not sure how it quantativly changes the system.

Sorry for being a bit slow.
 
  • #8
Zell2 said:
I understand what is going on qualitatively, but I'm not sure how it quantativly changes the system.

Sorry for being a bit slow.
No problem, that's what we're here for :smile: . Okay ignore the fact that there sphere is spinning for the moment. Now, if the sphere is just dropped into the fly wheel (and is not spinning), what is the new moment of inertia of the fly wheel and sphere system?
 
  • #9
So then total moment of inertia= I + i + Ma^2
using the parallel axis theorem
 
  • #10
You can treat the sphere as a point mass hence the new moment of inertia of the fly wheel and non-rotating sphere is I + Ma^2
 
  • #11
Oh ok, how would this change if the sphere is spinning?
Thanks
 
  • #12
Okay so, now let's write an expression for the conservation of angular momentum;

[tex]I\omega_{1} + i\omega_{2} = \left(I+Ma^2\right)\omega_{1'} + i\omega_{2'}[/tex]

Now, after some consideration I believe that the phrase "the sphere and the flywheel are at relative rest" means the sphere stops spinning (my apologies :redface:), thus negated the final term on the RHS.
 
  • #13
So even though the angular momentum vectors before are through different points you can add them like scalars to give the total angular momentum (presumably because they are parallel) ?
thanks
 
  • #14
Zell2 said:
So even though the angular momentum vectors before are through different points you can add them like scalars to give the total angular momentum (presumably because they are parallel) ?
thanks
Yes, as I understand it.
 
  • #15
Thank you,
 
  • #16
Zell2 said:
Thank you,
Twas a pleasure :smile:
 

What is angular momentum?

Angular momentum is a property of a rotating object that describes its tendency to continue rotating in a particular direction. It is a vector quantity that is dependent on an object's mass, velocity, and distance from the axis of rotation.

How is angular momentum calculated?

Angular momentum is calculated by multiplying an object's moment of inertia, which describes its resistance to rotation, by its angular velocity, which is its rate of rotation. The resulting value is a vector quantity with units of kilogram meters squared per second.

What is a flywheel?

A flywheel is a mechanical device that stores rotational energy by spinning on an axis. It is often used in machines and vehicles to smooth out fluctuations in power and maintain a consistent speed.

What is the purpose of a flywheel in angular momentum?

In angular momentum, a flywheel is used to demonstrate the principle of conservation of angular momentum. This principle states that the total angular momentum of a system remains constant, regardless of any internal or external forces acting on it. The flywheel's rotating motion helps to visualize this concept.

What is a sphere's moment of inertia?

The moment of inertia of a sphere is a measure of its resistance to rotation about a specific axis. For a solid sphere, it is calculated as (2/5) * m * r^2, where m is the mass of the sphere and r is its radius. This value increases as the mass or radius of the sphere increases.

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