free-body-diagramfrictionnewtonian-mechanicsrotational-dynamics
Consider this situation:
A ball is moving forward and undergoing rotation. Assume that it is not slipping. Eventually, the velocity and rate of rotation of the ball decrease, and it comes to a halt.
But if you observe the direction of friction (when the ball is rotating clockwise), you will see that the friction should have provided a clockwise torque to the ball and the angular velocity of the body should have increased. But this doesn't happen. Why?
I think that your skepticism comes about because you intuitively think that the force of kinetic friction should change gradually to static friction as the ball speeds up, since the relative motion between the ball's spinning surface and the ground decreases to zero. Your textbook assumes that this transition is actually instantaneous, and that the kinetic friction force is exactly the same until there is no relative motion at all, at which point the friction is entirely static. It seems counter-intuitive, but that's actually how it is.
If you've ever seen someone play a cello or any stringed instrument with a bow, you can see a little better how this works. A cellist steadily moves his bow across the string, which vibrates as it slips and catches on the sticky bow, switching between kinetic and static friction. If friction had a gradual transition, I imagine that the bow would just push the string to a certain displacement and reach equilibrium, and we'd be robbed of a great instrument, and a lot of excellent music.
Does your problem make more sense now? The angular acceleration and the torque are constant, because the kinetic friction force is constant until it disappears. Once the ball spins with the floor, there's no torque on the ball because it's just rolling—to the floor it seems to be still. But as the ball is spinning up (and the slipping is slowing down), the force on it remains the same. Friction is just weird like that.
If $F_f=mg\sin\theta$ then there would be no net force down the slope and hence no linear acceleration of the ball down the slope.
Note that that the FBD is incorrectly drawn.
As the ball rolls down the slope without slipping the centre of mass of the ball undergoes a linear acceleration and there is also an angular acceleration of the ball.
As drawn there is no torque about the centre of mass of the ball and so there can be no angular acceleration of the ball.
The point of application of the frictional force $f$ must be moved as shown below.
In this case it is fairly obvious as to the direction of the frictional force but it is worth a little consideration as for some problems that direction is not quite as obvious eg a ball rolling up a slope.
If the ball slip down without rolling its acceleration would be greater than if the ball was rolling with no slipping.
In terms of energy the ball now converts its loss of gravitational potential energy into both linear and rotational kinetic energy so its final linear speed would be less when rolling.
That means when rolling the net force down the slope acting at the centre of mass must be smaller than when there is no rolling.
Thus the frictional force must act the slope.
The angular acceleration of the ball is in the clockwise direction hence the torque about the centre of mass must be clockwise again indicating that the frictional force is up the slope.
One can now set up two equations $F=ma$ and $\tau = I \alpha$ and with the no slipping condition $a=r \alpha$ solve a problem.
It is true that if the slope is too steep and/or the coefficient of static friction is too small the ball will roll and slip under the action of a kinetic frictional force ie the no slipping condition cannot be satisfied but usually this is not the case.
Perhaps you misheard your teacher?
Best Answer
First let us clear up some definitions of the terms "reaction force", "normal force", and "frictional force".
Whenever there is a contact between two bodies, there is a reaction force on each body at every point of contact. A reaction force can be split into a normal component (sometimes called the "normal contact force"), and a tangential component (sometimes called the "force due to friction"). The direction of the force due to friction - the tangential component - is such that it opposes relative motion due to sliding / slipping.
This means that a perfectly rigid cylinder can roll on a perfectly rigid and rough surface forever since there is no sliding so no friction to provide a torque.
So why do balls we observe normally slow down?
The answer lies in something called "rolling resistance" (sometimes confusingly referred to as "rolling friction", or just "friction"), and entirely explains why a football comes to a stop after rolling it along the ground.
The key is that footballs and dirt are both compressible - they are not rigid bodies. On contact, the weight of the ball deforms both it and the dirt. This means that there are many points of contact between the ball and the dirt. Due to our definition, there are now many normal forces - to be precise, one per point of contact. We will ignore the tangential frictional components for now.
The deformation of the ball and the surface means that the line of action of these normal components is not through the center of the ball (see diagram).
As a result, the ball experiences two torques: a counter-clockwise one from the normal components to the right of the centerline, and a clockwise torque from the normal components to the left.
Since the normal forces are larger on the right side, the counter-clockwise torques are greater, and thus there is a net counter-clockwise torque and the ball slows to a stop.
Notice how we did not even consider any tangential frictional components at all. Just due to the normal components the ball is slowed.
There are a couple of points I have not covered, such as what role the frictional components do actually play and what happens in different types of deformations (e.g. the surface doesn't deform). Also you may be wondering why the normal forces on the right are greater. The answer to all these can be found at: https://lockhaven.edu/~dsimanek/scenario/rolling.htm . This is also where my diagram came from and I credit for these explanations. The exact diagram from your question is also used here and cited as a "naive pictures of friction and a rolling cylinder."