Why does static friction point up the ramp for an object that is rolling without slipping up an inclined plane?

Question

A solid sphere rolls without slipping up and then down an incline.

Why does the static friction force point up the incline both when the object is rolling up and also when it's rolling down? I know it's necessary for the force to point up when the object is rolling up, because then it's the only source of torque on the object, and for the object to roll back down the incline, there must be a torque which acts in the opposite direction of the initial angular velocity. However, I don't understand why this is the case. Why does the force need to point up the incline to create the necessary torque, particularly when the object is rolling back down the ramp?

Answer 1

If someone could explain this intuitively, . . . . .
I am not sure what exactly you mean by this.

The FBD for a body rolling without slipping on an incline is as follows.

On first meeting this sort of system the fact that the frictional force is in the same direction whether the body is rolling up an incline or down the incline is "counterintuitive".

I will go through the usual reasons given for this behavior later but consider this.

You set up an experiment in which the motion of a body rolling down an incline is videoed.
You then process the video and produce another video in which time is reversed and the body appears to be rolling up the incline.
So you now have two videos.

If you the give the two videos to a friend who was not part of the experiment and the processing and ask them which is the original one and which is the processed one, would they be able to tell you?

Assuming that the processing is perfect etc, the answer is, "no" and that is because the system of forces acting on the body does not change if the direction of motion of the body changes.

The no slipping condition, $v_{\rm CoM} = r\,\omega$, requires that the linear velocity of the centre of mass, $v_{\rm CoM}$, of the body, and the angular velocity, $\omega$, change in tandem.

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Friction is in such a direction to try and oppose relative motion or to try and prevent relative motion.

In the rolling down the slope just suppose that the ball was not rolling.
There would be a friction force up the slope so that the final state is the ball rolling down the slope without slipping.
You could say that the frictional force does it two ways:

• The frictional force opposes the force down the slope thus reducing the linear acceleration so that the linear speed down the slope is not as great as it would have been without the frictional force.
• The frictional force provides a torque about the centre of mass of the ball which produces an angular acceleration of the ball and increases its angular speed.

These two effects produce a convergent result - the no slip condition.

Once that non slip condition is reached the ball must undergo just the right amount of linear acceleration down the slope and angular acceleration about the centre of mass of the ball to maintain the no slip condition.
The frictional force does that.

Going up the hill the situation is the same with the frictional force decreasing the linear acceleration along the slope whilst providing a torque to change the angular speed to maintain the no slip condition.

To appreciate what is happening in this case just imagine what would happen if a ball spinning counter-clockwise was placed on the track.
The frictional force would have to accelerate the ball up the slope whilst slowing down its speed of rotation.
Eventually the no slip condition is reached and then the frictional force is there to maintain that condition of no relative movement between the slope and the ball.

So in each case what the frictional force does is to try and get to a no slip condition (no relative movement between the ball and the slope) and then maintain that condition.

Answer 2

An item rolling down a ramp is accelerating; that means that its rotation rate is increasing. Only friction on the ramp acts as a torque-inducing force, so friction on the ramp is increasing the rotation (and angular momentum) by applying up-the-ramp force at the point of contact.

The item rolling UP a ramp is decelerating; that means that its rotation rate is decreasing, but also that its rotation is in the opposite direction of the down-rolling item.

The decrease in the rate, means torque acts in the same direction as in the down-rolling case, because this is a decrease in the opposite direction of rotation from the upward rolling. The torque is thus twice sign-reversed: the torque direction is the same in both cases.

The force diagram is the same whether the initial velocity is up-the-ramp or down-the-ramp, so having the same torque direction implies the force direction of friction is also the same.