Friction and Rolling Homework: Understand v=rω

[andyrk]

Homework Statement

This is a conceptual doubt.
"A sphere rolls without slipping moving with a constant speed on the fixed rough surface. Friction between the surface and the sphere is sufficient to prevent slipping."

What does it mean that friction is enough to prevent slipping? From what I understood friction only comes into play in rolling motion when the lowermost point of contact has a tendency to either go forward or backward, i.e v≠rω So either v>rω (forward slipping of the lowermost point in contact) so friction acts backwards which is static friction. Or v<rω (backward slipping of the lowermost point in contact) so friction acts forward which is static friction. Here 'v' is the translational velocity of the centre of mass of the sphere and 'ω' is the angular velocity of the sphere. So when the case of v=rω
comes up, which is the case of rolling without slipping or pure rolling, then the point in contact doesn't have an tendency to either go backward or forward. It is instantaneously at rest. So it is not sliding/slipping. So why should frictional force arise in such a case as given "friction is sufficient to prevent slipping". This is a v=rω case and not v>rω or v<rω. This is when the rolling is "Uniform Pure Rolling". And also it should be Uniform Pure Rolling since it is given in the question that "moving with a constant speed"
Also please explain the terms Uniform Pure Rolling and Accelerated Pure Rolling. In the former, I think that there is no net torque present, since ω=constant so α=0 (Angular Acceleration). So it means that when a sphere is in Uniform Pure Rolling, it is given a force(for Δt time) initially which results in a torque and the sphere starts to roll(such that the condition v=rω is met after Δt time from the time of application of the force). That gave the sphere a constant velocity after Δt time. But then if the surface is frictionless it would keep in Uniform Pure Rolling forever, so there is no need for friction to be "present" or to be "sufficient" to prevent rolling without slipping. Friction should only be present if the sphere has a tendency to slip at its lowermost point of contact with the surface which will be only possible when v≠rω after Δt time. So is the reason behind mention of friction in this question that because we don't know whether the condition v=rω was met initially or not? Had we been sure that the force that we applied to the sphere is applied such that v=rω is met just after Δt seconds pass, then we wouldn't have considered friction, is that correct? Also, the frictional force acts such that to oppose motion. If there is an excess of either angular velocity (friction acts forward) or an excess of translational velocity(friction acts backwards), then what force is the frictional force opposing and cancelling off so that the point in contact comes at rest instantaneously? I mean we say that frictional force acts forward since the v<rω. So it reduces the angular velocity at the lowermost point until it becomes equal to v. So how can frictional force reduce velocity which are two different quantities?

 

[CWatters]

"A sphere rolls without slipping moving with a constant speed on the fixed rough surface. Friction between the surface and the sphere is sufficient to prevent slipping."

That sounds like the preamble to a problem. What does the rest of it say? It sounds like they are just being sure you rule out slipping as an issue for this problem.

Also please explain the terms Uniform Pure Rolling and Accelerated Pure Rolling

For Uniform Pure Rolling there is no skidding..

v = rω

if not equal you have skidding not pure rolling...

Accelerated Pure Rolling is same thing but with cylinder/ball accelerating.

dv/dt = r dω/dt
eg
acceleration = radius * angular acceleration

More..

 

[namanjain]

actually a more appropriate definition for pure rolling or rolling without slipping is
∂(relative)=0 → accelerated pure rolling
v(relative)=0 → pure rolling


When V=ωR, the static friction occurs like that in case of circular motion, friction providing centripetal force.

 

[haruspex]

"A sphere rolls without slipping moving with a constant speed on the fixed rough surface. Friction between the surface and the sphere is sufficient to prevent slipping."

What does it mean that friction is enough to prevent slipping? From what I understood friction only comes into play in rolling motion when the lowermost point of contact has a tendency to either go forward or backward, i.e v≠rω

You are quite right that if the ball is moving at constant speed in rolling contact with no forces acting parallel to the surface (e.g., horizontal surface, not a ramp) then there is no need for friction to maintain the rolling state. However, there may be a later part of the question where circumstances change.

Also please explain the terms Uniform Pure Rolling and Accelerated Pure Rolling. In the former, I think that there is no net torque present, since ω=constant so α=0 (Angular Acceleration). So it means that when a sphere is in Uniform Pure Rolling, it is given a force(for Δt time) initially which results in a torque and the sphere starts to roll(such that the condition v=rω is met after Δt time from the time of application of the force). That gave the sphere a constant velocity after Δt time. But then if the surface is frictionless it would keep in Uniform Pure Rolling forever, so there is no need for friction to be "present" or to be "sufficient" to prevent rolling without slipping. Friction should only be present if the sphere has a tendency to slip at its lowermost point of contact with the surface which will be only possible when v≠rω after Δt time. So is the reason behind mention of friction in this question that because we don't know whether the condition v=rω was met initially or not? Had we been sure that the force that we applied to the sphere is applied such that v=rω is met just after Δt seconds pass, then we wouldn't have considered friction, is that correct? Also, the frictional force acts such that to oppose motion.

It opposes relative motion of the surfaces in contact. Not sure if that answers your question below. If not, then I didn't understand it.

If there is an excess of either angular velocity (friction acts forward) or an excess of translational velocity(friction acts backwards), then what force is the frictional force opposing and cancelling off so that the point in contact comes at rest instantaneously? I mean we say that frictional force acts forward since the v<rω. So it reduces the angular velocity at the lowermost point until it becomes equal to v. So how can frictional force reduce velocity which are two different quantities?

 

[Nickie]

First of all, it is important to understand the concept of pure rolling and slipping. In pure rolling, the point of contact between the object and the surface is instantaneously at rest, meaning that there is no relative motion between them. This is achieved when the translational velocity of the object's center of mass is equal to the product of its angular velocity and the radius of the object, as stated in the equation v=rω. On the other hand, slipping occurs when there is a difference in velocity between the object's center of mass and its point of contact with the surface.

Now, in the scenario described in the question, the sphere is rolling without slipping on a rough surface. This means that there is a sufficient amount of friction present between the surface and the sphere to prevent slipping. This does not mean that friction is actively working to prevent slipping, but rather it is just present in case the sphere does start to slip. In other words, the frictional force is there as a safety measure to maintain the condition of pure rolling.

Moreover, in uniform pure rolling, as you correctly mentioned, there is no net torque acting on the object, meaning that there is no change in its angular velocity. This also means that there is no change in the translational velocity of the object's center of mass. However, in the case of accelerated pure rolling, there is a net torque acting on the object, resulting in a change in its angular velocity and therefore its translational velocity. In this case, friction plays a role in maintaining the condition of pure rolling by providing the necessary torque to counteract the external torque.

Additionally, in both cases, the frictional force acts to oppose any tendency of slipping. This means that if there is an excess of either angular or translational velocity, the frictional force will act in the opposite direction to prevent slipping. So in the case of v<rω, the frictional force acts to reduce the angular velocity until it becomes equal to the translational velocity, thus maintaining the condition of pure rolling.

In conclusion, friction plays a crucial role in maintaining the condition of pure rolling and preventing slipping. It is not actively working to prevent slipping, but rather it is present as a safety measure in case the object does start to slip. This is why it is mentioned in the question that friction is sufficient to prevent slipping. I hope this helps clarify your doubts.