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## 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?