A homemade mechanics problem: A beetle on a globe

[wrobel]

I composed a problem and initially thought it could be solved by purely analytical means, but it turns out it cannot.

The problem is as follows: a homogeneous ball of radius $R$ can rotate freely about its fixed center $O$. Let $J$ denote its moment of inertia relative to an axis passing through the point $O$. There is no gravity in this problem.

A beetle of mass $m$ sits on the ball. Initially, the system is at rest. Then, the beetle begins to crawl such that its trajectory draws a circle of radius $b$ on the ball. When the beetle returns to its initial position, it stops.

By what angle has the ball rotated when it reaches its final position?

I mean that a rigid body with a fixed point can be moved from any orientation to another by a single rotation. The question asks for the angle of this rotation between the initial and final positions of the ball once the beetle has stopped.
Interestingly, the answer does not depend on the beetle's law of motion along the circle and the ball stops as the beetle stops.

 

[anuttarasammyak]

Conservation of angular momentum
$$I_G\omega_G+I_B\omega_B=0$$
The bug returns to the start mark with period
$$T=\frac{2\pi}{\omega_B-\omega_G}$$

Thus rotation angle of the Globe during T is
$$T\omega_G= \frac{2\pi}{\frac{\omega_B}{\omega_G}-1}=-\frac{2\pi}{\frac{I_G}{I_B}+1}$$

 

[wrobel]

The ball has three degrees of freedom

 

[anuttarasammyak]

The rotation axis is the line on which the center of the ball and the center of beetle trajectory circle are. Do I mistreat degree of freedom ?

 

[Hornbein]

Since the center of the ball is fixed all the action is confined to a 2-D surface, so I would say the ball and beetle each have two degrees of freedom while embedded in a 3D space.

The beetle's motion is restricted to a circle. Is that required to be a great circle?

 

[wrobel]

The rotation axis is the line on which the center of the ball and the center of beetle trajectory circle are.

This does not follow from anything. The ball is a rigid body with a fixed point, and thus it has three degrees of freedom. It is not obliged a priori to rotate about a fixed axis.

 

[wrobel]

The beetle's motion is restricted to a circle. Is that required to be a great circle?

No!

 

[Hornbein]

This does not follow from anything. The ball is a rigid body with a fixed point, and thus it has three degrees of freedom. It is not obliged a priori to rotate about a fixed axis.

If it were obliged to rotate around a fixed axis then one number would be enough to describe it's motion so it would have one degree of freedom. If the center were not fixed then it would have three degrees of freedom.

 

[DaveC426913]

This does not follow from anything. The ball is a rigid body with a fixed point, and thus it has three degrees of freedom. It is not obliged a priori to rotate about a fixed axis.

It is not obliged to rotate about a fixed axis, but it does, because there is no lateral forces involved. It becomes irrelevant that it's a sphere; it might as well be a cylinder.

Thus, the problem is simplified to 2 dimensions.

 

[DaveC426913]

the beetle begins to crawl such that its trajectory draws a circle of radius b on the ball

Why do you say "radius b"? b is equal to R - the radius of the sphere.

The beetle can't make to sphere spin any way except on a plane that passes through the ball's centre.

 

[Baluncore]

Why do you say "radius b"? b is equal to R - the radius of the sphere.

The beetle can't make to sphere spin any way except on a plane that passes through the ball's centre.

The beetle's circle can be any size, up to a great circle.

The sphere will rotate about an axis that passes through the centre of the beetle's circle, and the centre of the sphere.

 

[wrobel]

It is not obliged to rotate about a fixed axis, but it does

It's a bit funny because I have equations written down and you're just saying words :)

 

[wrobel]

here is my analysis

 

[DaveC426913]

It's a bit funny because I have equations written down and you're just saying words :)

Granted.

I may be talking out my butt.

The beetle's circle can be any size, up to a great circle.

The sphere will rotate about an axis that passes through the centre of the beetle's circle, and the centre of the sphere.

I am not sure how those are both true.

In order for the beetle to walk path p that is not a great circle, it would have to look like diagram A.


View attachment 371769
But why would the sphere rotate along the axis a (as seen in A's top view)

The sphere would always rotate such that the beetle's path is a great circle, no matter what.

In diagram B: beetle's path p causes sphere to rotate on axis a which is perpendicular p. Thus p will always be a great circle.

 

[DaveC426913]

The beetle's circle can be any size, up to a great circle.

If the beetle walks a straight path it must form a great circle, and will pass through its antipodes.
To follow the path in wrobel's diagram, it must continually veer left:

(note: antipodes is on far side of the sphere, so beetle's left is toward top of diagram)

It's a bit funny because I have equations written down and you're just saying words :)

Granted, I may be misunderstanding the problem as-written.




What causes the sphere to rotate? The beetle's movement, right?

So why does the sphere not rotate on the axis perpendicular to the line from beetle to sphere centre?



Let's remove the sphere's rotation for a moment and just examine the beetle's path on a fixed sphere.

On a fixed sphere, every straight path the beetle chooses is part of some great circle. So it goes with out saying that the beetle is always walking a great circle. How could it be otherwise?



As with Baluncore, above: in your diagram the beetle is following a curved path. It must constantly correct, turning continually left, to maintain that path.

If the beetle were to walk a straight line - any straight line - around the sphere, it would pass through the antipodes, right?

But you have your beetle not passing through the antipodes.

You have it veering off a straight path that puts it about 30 degrees off straight by the time it's crossed half the sphere.

 

[wrobel]

The sphere will rotate about an axis that passes through the centre of the beetle's circle, and the centre of the sphere

This contradicts the formulas attached.

 

[BillTre]

Insect neurophysiologists use a cleaver version of this set-up to get recordings from insect nervous systems while the insect is walking.
The insect is immobilized by mounting it on a stiff wire or stick glued or waxed ti its back.
The ball (ping pong or styrofoam) is put under the insect (its feet reflexively grab it) and recording wires going into the insect don't have to move because only the ball moves when it "walks".
Thus you can get recordings from multiple delicate electrodes from a walking insect.

 

[anuttarasammyak]

This does not follow from anything. The ball is a rigid body with a fixed point, and thus it has three degrees of freedom. It is not obliged a priori to rotate about a fixed axis.

But in OP a priori

Then, the beetle begins to crawl such that its trajectory draws a circle of radius b on the ball.

I observe this such that condition decides the axis of rotations both beetle and the globe as posted #4.

Say beetle crawls from point A to point B on the globe in any route, the resulted globe rotation angles $\theta,\phi$ depend on the trajectory.

 

[Hornbein]

If the beetle walks a straight path it must form a great circle, and will pass through its antipodes.
To follow the path in wrobel's diagram, it must continually veer left:

He says it isn't required to be a great circle so the path isn't required to be straight. The idea could be that the insect's path is the same as if one were to have drawn a circle on the sphere and the insect followed that path. In this case I think an outside observer would not see the path as a circle. Or maybe he means that an outside observer would see the path as a circle but an observer fixed to the surface would not. He said the insect returns to it's original location so I guess it is the first condition.

 

[wrobel]

The angular velocity of the ball changes its direction relative to the ball-fixed frame (see the text attached) and thus it changes its direction relative to the lab frame. Thus, there is no fixed axis about which the ball rotates.