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Rotation of a Spherical Top

  • #1

Homework Statement


A solid sphere of mass M and radius R rotates freely in space with an angular velocity ω about a fixed diameter. A particle of mass m, initially at one pole, moves with constant velocity v along a great circle of the sphere. Show that, when the particle has reached the other pole, the rotation of the sphere will have been retarded by an angle

α=ωT(1-√[2M/(2M+5m)])

Homework Equations




The Attempt at a Solution


Apologies for the ugly formatting, can someone please link me an article on how to type equations?
For the problem, I can find the solution if I set the sphere's rotation about the z axis and I assume that angular momentum about the z axis is constant. My question lies with the assumption that angular momentum about the z axis is constant. Angular momentum is constant if there is no external torque. However, the particle's movement from the top pole to the bottom pole implies that there is a torque.

What am I missing?
 

Answers and Replies

  • #2
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Angular momentum is the sum of what for this problem?
 
  • #3
Angular momentum of the sphere as it rotates about the z axis in the space frame and and a particle of mass m moves along the circumference of the sphere in the xz plane as this plane rotates in the body frame.

## L_z = I_z \omega_z ##
## L_{e2} = I_{e2} \omega_{e2} ##
 
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  • #4
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Write the angular moments in terms of their masses and R.
 
  • #5
The moment of inertia about the e3 axis at t=0 is the moment of inertia of the sphere (since the particle lies on the e3 axis).

## I = \frac{2R^2M}{5} ##

At time t the moment of inertia about the e3 axis is the moment of the sphere plus a contribution from the particle, a distance d from the z axis.

## I = \frac{2R^2M}{5} + d^2 ##
It can be shown that the distance d is given by
## d = mR^2 sin^2 \theta ##
## \theta = \frac{vt}{R} ##

Substituting into equation for I
## I = \frac{2R^2M}{5} + mR^2 sin^2 \theta ##


The e3 axis should precess, should it not? I can't see how the e3 axis remains parallel to the z axis after t=0.
 
  • #6
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... and θ (t)?
 
  • #7
By trigonometry,
## \theta (t) = \frac{vt}{R} ##

I can solve the problem correctly if I assume that e3 does not precess, my question is why does this axis not precess while there is torque about e2 from the movement of the particle?
 
  • #8
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mass m, initially at one pole, moves with constant velocity v
No torque. And, no, I'm not exactly happy with the appeal to the original problem statement myself.
 
  • #9
Constant linear velocity v, constant angular velocity
## \omega_{e2} = \dot{\theta}(t) ##
## \omega_{e2} = \frac{v}{R} ##

In the body frame, the angular velocity introduces a centrifugal force which must be balanced by a centripetal force. The centrifugal force is given by
## F_{cf} = m ( \omega_{e2} \times r) \times \omega_{e2} ##
## F_{cf} = m R \omega_{e2}^2 \hat{r} ##

The balancing centripetal force is in the negative ## \hat{r} ## direction
## F_{cp} = - m R \omega_{e2}^2 \hat{r} ##

Taking torque to be
## \Gamma = r \times F ##
## \Gamma_{e2} = 0 ##
since r and F both lie in the ## \hat{r} ## direction.

Is this the reason torque is zero?
The reason I thought torque was not zero is somewhere in my notes I have written
## \Gamma = r \times \omega ##
Using this logic,
## \Gamma_{e2} = R \hat{r} \times \omega \hat{e_2} ##
which is a nonzero quanitity.
 
  • #10
TSny
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The e3 axis should precess, should it not? I can't see how the e3 axis remains parallel to the z axis after t=0.
Hello, Ben.

The problem states that "A solid sphere of mass M and radius R rotates freely in space with an angular velocity ω about a fixed diameter."

It seems to me that this could be interpreted to mean that the sphere is mounted on a fixed axle like a globe but is otherwise free to rotate about this axle. If so, then there would be an external torque on the axle to keep it always aligned parallel to the z axis. As you noted, in this case you will get the answer stated in the problem.
 

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