# Conceptual question about a rotating ring

#### issacnewton

Hi

I was thinking about this conceptual problem. Consider a thin ring of radius R, which is rotating about the axis passing through its center of mass. Now lets says there is no gravity and the ring is rotating at some constant angular velocity $$\omega$$ , so the angular momentum is conserved. Now if we look at some infinitesimal element of mass dm, on the circumference
then it is rotating about the axis too. So there must be a centripetal force acting on this mass element. First I couldn't think of any force which can play the role of centripetal force here.
Then I thought about a model. Lets say that some 10 beads are connected in circle with springs. So we have a polygon with 10 vertices (which are beads)and 10 sides (which are springs). Now lets set this in circular motion with some constant angular velocity. Now if we look at one of the beads, then we can imagine that two springs which connect it to the neighboring beads will get stretched. If we increase the angular velocity, then the springs will stretch even more. So we can model the thin ring as the beads connected by the springs and then its easy to see which forces act as the centripetal force. As we have seen here, its the force exerted by the two springs on a bead acts as the centripetal force. So the net force acting on a bead is directed towards the center of mass of the whole system. So this is how I resolved the question. I want to know if the reasoning is correct.

Thanks

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#### tiny-tim

Homework Helper
hi issacnewton! … if we look at some infinitesimal element of mass dm, on the circumference then it is rotating about the axis too. So there must be a centripetal force acting on this mass element.

As we have seen here, its the force exerted by the two springs on a bead acts as the centripetal force. So the net force acting on a bead is directed towards the center of mass of the whole system.
yes, that looks fine in a solid spinning ring, each tiny element rdθ has tension forces at an angle of (i think) dθ/2 to the element, so there's a small resultant force radially inward, much like your springs #### issacnewton

thanks tiny-tim. My reasoning was right...

#### tiny-tim

Homework Helper
it's bound to happen sometimes! keep eating the apples! #### pallidin

hi issacnewton! yes, that looks fine in a solid spinning ring, each tiny element rdθ has tension forces at an angle of (i think) dθ/2 to the element, so there's a small resultant force radially inward, much like your springs Not saying your wrong, but I do not see how a rotating ring can generate inward force vectors. I would think the force vectors would be tangent and out; not tangent, out and inwards.

#### tiny-tim

Homework Helper
… I would think the force vectors would be tangent and out; not tangent, out and inwards.
sorry, pallidin, i'm confused radial can be either in or out, but tangential has to be out …

poor ol' tangents never get to come in! but the two (slightly angled) tangent forces combine to have an inward radial component #### pallidin

...but the two (slightly angled) tangent forces combine to have an inward radial component OK, but what is the source, in this scenario, for a second tangential force which results in a small, downward vector? I do not see it.
A rotating mass ring expands outwardly.

This would be like saying that the ring expands AND compresses at the same time.
Fine, that's a balloon. No problem there.
But, I thought we are talking just about a rotating solid mass ring, not a ring made of, say, expansive balloon material.

#### issacnewton

pallidin, is my model correct? I don't see how tangential vectors come into the picture.

#### dgOnPhys

pallidin, is my model correct? I don't see how tangential vectors come into the picture.
tiny-tim model is like yours when you have an infinite number of beads and springs: each bead will be pulled in two (almost) opposite tangential directions by its springs; this generates a (small) centripetal force

#### pallidin

Small? Is there ANY evidence of it's existence at all?
Centripetal is a fictitious force. It does not exist.

Perhaps I am wrong, but please show me any evidence otherwise.

#### dgOnPhys

Small? Is there ANY evidence of it's existence at all?
Centripetal is a fictitious force. It does not exist.

Perhaps I am wrong, but please show me any evidence otherwise.
well let's take an element ds of the rotating ring circumference; in the inertial frame it rotates with (say) constant angular velocity in a circular trajectory; for this to happen, kinematics requires a constant centripetal acceleration. Newton equation requires then a force causing this acceleration: a constraint reaction (or tension), not a fictitious force at all.

Fictitious forces would appear in the non-inertial frame where the element ds is at rest. The fictitious force will be the centrifugal force (the opposite of the product of mass and centripetal acceleration) which will balance the constraint force.

In the inertial frame, its free-body diagram will show two tension forces on its extremities tangent to the arc ds (if ds subtends an angle $d \theta$ the angle between the two forces will be $\pi-d \theta$). Since the angular velocity of the ring is constant, the two tensions will have the same modulus F so the resulting force will be directed along the radial direction towards the circle center with modulus $2 F sin(d \theta / 2) = F d \theta$, which is a small force accelerating the small mass dm of the arc element ds to a finite centripetal acceleration $\omega^2 R$.

#### issacnewton

This problem signifies why simple models are important in physics. It helps understanding.

#### tiny-tim

Homework Helper
Centripetal is a fictitious force. It does not exist.
no, https://www.physicsforums.com/library.php?do=view_item&itemid=84" is a fictitious force that does not exist

centripetal force is simply a name for whatever existing force (such as tension or friction) is providing the https://www.physicsforums.com/library.php?do=view_item&itemid=27" that keeps something in a circle

in this case, the centripetal force is provided by the radial components of the nearby tangential tensions

as issacnewton said …
… if we look at some infinitesimal element of mass dm, on the circumference then it is rotating about the axis too. So there must be a centripetal force acting on this mass element.

As we have seen here, its the force exerted by the two springs on a bead acts as the centripetal force. So the net force acting on a bead is directed towards the center of mass of the whole system.

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#### mikelepore

The original post is a very clear and helpful example. I believe that, once this principle is accepted, we can generalize it to say the following:

Consider any rigid body, arbitrary shape, initially at rest. Any one atom or molecule of the body, which is not accelerating (neglecting thermal jiggling), is being pulled in many directions simultaneously by atomic or molecular bonds, and these bond forces add up to a net force of zero. But now have the object rotate because of its conserved angular momentum -- now all of the bonds pulling on any one atom or molecule will add up to a net force directed toward the center of rotation.

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