Rotational dynamics - direction of friction

jaumzaum

I'm studying rotational dynamics and I've got a couple questions I can't answer. I want to describe de movement of the bodies in the cases below, the coefficient of friction in all the cases is [itex]\mu[/itex] . I will say what I think it would happen, and I would appreciate if you guys judge it right or wrong, as well as answering my other questions.

a = translational acceleration (acceleration of the center of mass)
γ = angular acceleration



In (A) we have a sphere (I = 2/5 MR²) rotating without sliding in a horizontal plane. If we had friction to the right, a would increase and [itex]\gamma[/itex] would decrease, absurd because a = [itex]\gamma[/itex] R.
If we had a frictions to the left, a would decrease and [itex]\gamma[/itex] increase. Absurd too. So will the sphere stay rotating forever?

In (B) we have a sphere initially at rest at an inclined plane. If the sphere does not rotate, where is the friction? I would say the friction is upwards, as the sphere needs to increase both a and γ . Is it right?

In (C) we have a sphere initially moving up an inclined plane, without sliding. Is it possible? I mean; if there was a friction upwards, a would be increasing and γ decreasing, but if there was a friction downwards, a would be decreasing and γ increasing, both things are imposible. So is there impossible to be a sphere rolling up an inclined plane without sliding?

And a general question: How is the direction of the static friction determined generally? I've learned friction is opposite to the displacement of the body in relation to the plane of the friction but if we have a body rolling without sliding, the contact point have instantaneous velocity = 0, so there is no instantaneous displacement from the ground, where should the friction be?

ehild

The static friction is zero when the resultant of the other external forces is zero, and the ball rolls with constant velocity. In real life, there are other forces (rolling resistance, air resistance) which will make it stop sooner or later.

Friction acts against relative motion of the surfaces in contact. When kept in rest, the static friction is zero. If you release the ball, gravity acts downward along the slope, accelerating the CM of the ball. The ball would slide but then the surfaces in contact would move with respect to each other. Static friction prevents it, so instantaneously, the point of contact stays in rest. The static friction is opposite to the force of gravity along the slope.But gravity exerts torque with respect to the point of contact, so the ball starts to roll downward.

Yes, the ball can roll upward the slope. Have you tried to roll up a real ball along a real slope? What happened?

Gravity would decelerate the CM of the ball. To keep it rolling, (so keeping the contact surfaces in rest with respect each other) the angular speed has to slow down, too. So the static friction provides a torque with respect to the CM which decelerates rotation. So it points upward.

The static friction prevents relative motion of the surfaces in contact. Assume there is no friction first and figure out in what direction would the surface of the body in contact with the ground accelerate. Friction is opposite to it.

ehild

jaumzaum

Thanks!
One last question:

The static friction if opposite to the resultant acceleration (if we consider no friction) so? I think it makes more sense now. I thought it was opposite to velocity (if we consider no friction). I was confused because if it was opposite to velocity, the case C, the velocity is upwards, so the static friction should be downwards. But if we consider the acceleration rule, the acceleration is downwards, so the static friction can be upwards. Is it right?

ehild

Well, the static friction is against the possible relative motion of the surfaces in contact, which would happen without the force of friction. So it is against relative acceleration. Think of the case when a block sits on the plateau of a truck. The truck is accelerating and the block stays in rest on the plateau. What is the direction of the static friction?
First, static friction acts both on the block and on the truck. The two forces are opposite to each other.

The static friction hinders relative motion. Without it, the block would slide backwards. So the static friction exerted on the block points into the direction of the acceleration of the truck. The static friction provides the force that accelerates the block together with the truck. At the same time, the static friction exerted on the truck opposes the acceleration of the truck.

ehild