# Acceleration of a ball rolling down incline without slipping

Take the example of a sphere rolling down an incline without slipping. Is it true that for the ball to be not slipping, $mg sin \theta=F_f$? To me that makes no sense because then $F_{netx}$ = 0 and then $a$ would also be zero.

Assume that $F_f$ does equal $mgsin\theta$, what would happen? Doing a force analysis of the situation, it seems like the ball should not be accelerating at all. Is there a distinction that I'm missing here?

Additionally, why do we only include $F_f$ when analyzing torque on the system?

• The force of friction does not act on the center of the ball, it acts on the ball at the contact between the ball and the surface. If you move it there, you will derive some different effects. Commented Feb 10, 2017 at 3:05
• Why do you think that $F_f=mg\sin\theta$? I don't see any reason why you might think that. Commented Feb 10, 2017 at 3:09
• My physics teacher said that in class, saying that if $mgsin\theta$ was bigger than friction the sphere would slip and not roll properly. Commented Feb 10, 2017 at 6:14
• See this question and associated answer for a detailed analysis Commented Feb 10, 2017 at 18:35

## 1 Answer

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?