# B Dumb question about inertia and rotational inertia

#### Graca

Consider two bodies, A and B, of equal mass set at a short distance. Body A is spinning and body B is at rest. Then, through some kind of electromagnetic device, a strong repulsive force is established between them. Will both be displaced at the same speed?

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#### snoopies622

I don't see why not, since angular momentum and linear momentum are different quantities.

#### Graca

But... Doesn't the spinning body offer resistance to change its axis of rotation?

#### snoopies622

If the force is only repulsive, why would there be such a rotational change?

#### Graca

Let's imagine the spinning body is a cylinder and the repulsive force is applied at one extremity? Then, would there be a rotational change?

#### snoopies622

Are you saying that the force is applied in the same direction as the body's axis of rotation? In that case I would say that there's no torque and therefore no change in the rotational state. On the other hand, if it's applied to an extremity in any other direction, then I would say that there is a torque and therefore a change in the rotational state, and that changes everything.

#### Graca

Like two pipes laying side by side (one of them is spinning) and the repulsive force being applied at the end. Would they get apart at the same speed then?

#### FactChecker

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Even if the repulsive force is applied at one extremity, its effect on the linear motion of the center of mass is the same. The calculations of the rotational acceleration and the linear acceleration of the center of mass can be done separately.

#### Graca

But if the force is applied at the end, wouldn't that change the plane of rotation and therefore wouldn't the spinning pipe offer more resistance to movement?

#### snoopies622

If they're lying parallel to one another and one is spinning such that its angular momentum points along the direction of the pipe, and both pipes are pushed at end points away from each other, then both pipes will take on different rotational states, and the one that was initially at rest will be spinning faster, while the one that was initially spinning will be spinning in a different direction altogether due to precession. (Now having trouble about the linear acceleration part. For me this is no longer so simple a question!)

#### FactChecker

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But if the force is applied at the end, wouldn't that change the plane of rotation and therefore wouldn't the spinning pipe offer more resistance to movement?
If the center of rotation is held in place, that will supply a force that opposes the linear motion and the result will be only the rotation. But if there is nothing holding the object in place, the center of mass will move linearly exactly as though the force was applied at the center.

#### FactChecker

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Notice that the equation F=mA does not specify that the force is applied at the center of mass. The equation is correct even if the force is off-center.

#### Graca

Yes, two lying parallel pipes, but only one is spinning. The repulsive force being applied between them at the extremity of the pipes. I can imagine the pipes getting apart, but the pipe at rest moving with a greater speed that the pipe that is spinning. Is this correct?

#### FactChecker

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Yes, two lying parallel pipes, but only one is spinning.
How can one be spinning, but remain parallel? [EDIT: I understand this if the axis of rotation is along the length of the pipe. Thanks @snoopies622 ]
The repulsive force being applied between them at the extremity of the pipes. I can imagine the pipes getting apart, but the pipe at rest moving with a greater speed that the pipe that is spinning. Is this correct?
No. I am not completely sure what you mean, but the spin should not matter when the linear acceleration at the center of mass is calculated.

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#### snoopies622

If the same force is applied to both pipes for the same distance, then the energy invested in both is the same. I suppose one could therefore calculate the linear accelerations by doing the rotational parts first and then subtracting the energies. For me the rotational part is more intuitive.

#### snoopies622

How can one be spinning, but remain parallel?
I guess he means that the axis of rotation is along the same direction as the pipe itself.

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#### Graca

Let's say each pipe is 1 meter long. They are lying parallel. One pipe is spinning around the direction of its length. At the end of each pipe there is some kind of switchable magnet. So, a repulsive force is switched on. The pipes' ends would get apart, but for the spinning one, that means changing its axis of rotation. Then, would it move away at a different speed than the one not spinning?

#### snoopies622

You have stated the question clearly, but the more I think about it, the more I don't know the answer! :) Must go to bed anyway, will sleep on this. Good night and good luck, will check in in the morning. Good question.

#### Graca

Thanks, good night!

#### FactChecker

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Let's say each pipe is 1 meter long. They are lying parallel. One pipe is spinning around the direction of its length. At the end of each pipe there is some kind of switchable magnet. So, a repulsive force is switched on. The pipes' ends would get apart, but for the spinning one, that means changing its axis of rotation. Then, would it move away at a different speed than the one not spinning?
No. The spin can give some stability of orientation, but it does not change the linear acceleration at the center of mass. The equation $F = m A$ is beautifully simple. It applies in all cases, regardless of complications.

#### Graca

I understand that. But let me give another example: two parallel bycicle wheels (only one spinning, the same repulsive force as in the previous example). What means then stability if they move away at the same speed?

#### FactChecker

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The stability is regarding the orientation. Suppose the rotational momentum vector is very large in one wheel. Then it will take a lot of torque to change its angle much. That can be called stability since it appears to resist a change. But the center of mass of the wheel is a completely different issue. While resisting a rotation, it can still be pushed away as though it was not spinning at all.

#### Graca

Even if i push far away off the center of mass? Wouldn't that be a rotation of the axis?

#### FactChecker

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Yes. Suppose you push on one end instead of at the center. That end will move more than the center would, but the center will move exactly according to $F = mA$. So there will be a move away of the center and also a rotation, with the end you pushed moving away faster, the center moving exactly at $F=mA$, and the other end moving less. The spin may reduce the amount of rotation due to the force (and make it rotate in a strange direction), but it will not change the linear motion of the center.

#### Graca

Then, in the case of the two wheels (the spinning one and the not spinning one), with the centers moving according to F=mA , would each of the ends (on the same side) move at the same speed?

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