# Is the continued rotation of the ball due to inertia?

I think the rotation of the ball is due to the inertia of the ball, because when there is no external force, the ball will continue to rotate. But some people think that the rotation of the ball is not due to inertia, they say it needs centripetal force. And I think the centripetal force is the internal force, not the external force. So the rotation of the ball should be due to inertia.

• A force does not necessarily cause rotation. Is it the net torque about the center of mass that matters. – ja72 Nov 10 '19 at 22:32

It's hard to say whether or not what you talk about is "due to inertia". Inertia is essentially just mass, but I think the term is usually either misused or just unclear when it comes up, so I try to stay away from it.

We we can say is that if a ball is floating in space and rotating (as viewed from a non-accelerating reference frame), then it will continue to keep rotating as there are no net external forces or torques acting on the ball. I suppose you can link this to Newton's first law that "an object will stay in motion unless a net force acts on it", and if you want to call this being due to inertia then that's fine.

However, I would not discount the counterpoint you raise. If you consider the particles that make up the ball, then it's a different story. Each individual particle not on the axis of rotation does have a net force acting on it, which is the centripetal force you mention. So, these particles are undergoing acceleration. But keep in mind that "inertia" it's still at play here, as their reaction to this force depends on their "inertia" (mass).

So, attribute what you want to inertia. However, I would primarily focus on whether or not net forces and/or net torques are present, and if so, what effect they will have on your object.

• Newton's law is that when there is no external force, the rigid body moves in a straight line. But I think that when there is no external force, the rotation will be maintained. Is this in conflict with Newton's law? – enbin zheng Nov 12 '19 at 1:44
• @enbinzheng No. That version of Newton's law is for point particles. For extended bodies and rotation then you say "when there is no external torque the body maintains its angular velocity" – Aaron Stevens Nov 12 '19 at 1:46
• Do you mean that the rotation of the ball can also be inertial motion? – enbin zheng Nov 12 '19 at 12:40
• @enbinzheng If you are moving along with a particle on the ball, then no. Inertial motion means not accelerating. – Aaron Stevens Nov 12 '19 at 14:51
• @enbinzheng Well based on how you defined your scenario, then yes. You said there are no forces. – Aaron Stevens Nov 13 '19 at 13:05

A rigid body rotating about its center of mass has angular momentum. You need torque to change this angular momentum just as you need force to change linear momentum.

These statements say nothing about mass or inertia on their own \begin{aligned} \boldsymbol{F} & = \tfrac{{\rm d}}{{\rm d}t} \boldsymbol{p} \\ \boldsymbol{\tau} & = \tfrac{{\rm d}}{{\rm d}t} \boldsymbol{L} \end{aligned}

It is the definition of momentum and angular momentum where inertia pops out. As defined at the center of mass, linear and angular momentum are:

\begin{aligned} \boldsymbol{p} & = m\, \boldsymbol{v}_{\rm cm} \\ \boldsymbol{L}_{\rm cm} & = \mathbf{I}_{\rm cm} \boldsymbol{\omega} \end{aligned}

So inertia is the quantity that transforms motion into momentum for both linear and angular motion.

In conclusion, the rotational motion of a free rigid body is maintained due to the conservation of angular momentum. It is not centrifugal forces (which act internally in this case, unless the center of mass tracks a path). If inertia is changing (ice skater twisting with his/her hands pulled in) then rotational motion will change also.

In your example, rotational inertia is constant (a sphere has uniform mass moment of inertia) so the resistance to change in motion seems to be due to inertia, but it is actually still due to angular momentum.

• What is your conclusion? When there is no external force, the object will rotate all the time. Is this also inertia? – enbin zheng Nov 11 '19 at 2:11
• Note that the OP might be mixing up inertia (noun) with inertial (adjective)? At least both are being used in the question in a seemingly (and incorrectly) interchangabe manner. That could be leading to some misunderstandings. @enbinzheng do you recognize the difference between inertia and inertial? Which word do you mean to use? – Aaron Stevens Nov 11 '19 at 4:44
• @AaronStevens inertia – enbin zheng Nov 11 '19 at 4:57
• @enbinzheng So then what do you mean by asking if the rotation is inertia (i.e. title of your question)? – Aaron Stevens Nov 11 '19 at 4:58
• @enbinzheng It would be more clear then if you said "is the continued rotation due to inertia". Inertia is not a form of motion. – Aaron Stevens Nov 11 '19 at 5:16