Static friction for a ball rolling without slipping

[Lola Luck]

Homework Statement

A bowling ball rolls without slipping up a ramp that slopes upward at an angle β to the horizontal. Treat the ball as a uniform solid sphere, ignoring the finger holes. Explain why the friction force must be directed uphill.

Homework Equations

F=ma, torque=I(alpha), a=r(alpha) (I don't think I need any of these for this question)

The Attempt at a Solution

I thought that because the instantaneous velocity for the contact point is in the downhill direction, the friction force must act uphill. However there's an example in my textbook in which the ball rolls downhill and the force of static friction is still uphill. Apparently, the force of friction acts uphill wether the ball is rolling uphill or downhill.

 

[Nathanael]

I thought that because the instantaneous velocity for the contact point is in the downhill direction, the friction force must act uphill.

This would be correct if the friction was kinetic friction. But, if the ball rolls, then the contact point does not slide (the instantaneous velocity of the contact point is zero!) and so the friction is static, which means it is free to act in either direction.

 

[Lola Luck]

Ok, so how do I know it's uphill in this case? I think it's because gravity acts downhill and the force of static friction has to oppose gravity.

 

[Nathanael]

I think it's because gravity acts downhill and the force of static friction has to oppose gravity.

It can act in either direction that it needs to. The key is that the ball is rolling the whole time. What does it mean for the ball to be rolling?

 

[Lola Luck]

That it's accelerating? I'm not sure what you mean.

 

[Nathanael]

Rolling means there is a special relationship between the angular velocity and the linear velocity.
Look at your "relevant equation"

a=r(alpha)

or equivalently, $v=r\omega$ where v is the linear velocity and ω is the angular velocity.

 

[Lola Luck]

If the linear velocity is decreasing (it should be due to gravity), the rotational velocity also decreases. So static friction needs to act uphill to slow the ball down?

 

[Nathanael]

If the linear velocity is decreasing (it should be due to gravity), the rotational velocity also decreases. So static friction needs to act uphill to slow the ball down?

Exactly :)
If the linear velocity is decreasing, then the torque (from friction) must slow the ball's rotation.

 

[Lola Luck]

This topic makes so much more sense now. Thank you for your help! :)

 

[haruspex]

To clarify one point:

the instantaneous velocity for the contact point is in the downhill direction

In rolling contact with a stationary surface, the point of contact of the wheel/ball is instantaneously stationary.