Rolling ball on a surface with friction

Question

For a point particle moving on a surface under the infuluence of gravity, the equation of motion is very easy to write down - the force on the particle is simply the projection of its weight $mg\mathbb{\hat z}$ onto the surface. Adding friction to the model is also easy - the frictional force depends on the normal force through the coefficient of friction, $F_f = \mu N$.

However, if the particle is a (rigid) rolling ball, the problem seems to be much more complicated. The axis and velocity of rotation will be changing. (E.g., starting from rest, both the velocity and angular velocity will be increasing when the ball rolls down a hill.)

Is there a set of equations that would describe this physical situation (i.e, the effect on the velocity and angular velocity)? How would one derive it?

Answer 1

On an inclined plane, problem would be very similar to the ones without any rotation, the only differences would be static friction and torque. In the case of rolling without slipping (case where the bottom of the wheel or a ball is at rest with respect to the ground).

Translational equation is almost the same: $ma = mg \sin{\theta} - f$

However, this time we are dealing with static friction, not the kinetic. And the friction will act tangential to the body (as always) and exert torque on it.

So we have net torque equation:

$I \alpha = Rf$

By using the no slipping acceleration constraint where tangential acceleration is equal to the translational:

$a = R \alpha$

We obtain this equation (assuming that object is a ball/sphere and its moment if inertia can be found in table):

$ \dfrac{2}{5} ma = f $

By plugging it into the translational equation and rearranging a little, we get:

$ \dfrac{7}{5} ma = mg \sin{\theta}$

Translational acceleration is thus:

$a = \dfrac {5g \sin{\theta}}{7}$

By using our constraint equation, we can easily find the angular acceleration:

$ \alpha = \dfrac {5g \sin{\theta}}{7R}$

And we can simply obtain angular velocity equation:

$ \omega = \dfrac {5g \sin{\theta}}{7R} t$

Translational velocity is easily found too (just multiply equation by $R$):

$ v = \dfrac {5g \sin{\theta}}{7}t$

Answer 2

Adding friction to the model is also easy - the frictional force depends on the normal force through the coefficient of friction, $F_f = \mu N$

You need to be careful to differentiate between static friction and kinetic (sliding) friction.

$F_{f}=\mu N$ can be either the kinetic friction force, where $\mu$ is the coefficient of kinetic friction, $\mu_k$, or it can be the maximum possible static friction force to prevent slipping while rolling where the coefficient is that for static friction, or $\mu_s$.

Is there a set of equations that would describe this physical situation (i.e, the effect on the velocity and angular velocity)? How would one derive it?

You can find the equations and with derivations for the case of a solid cylinder rolling down an incline in the link below. The only difference in your example is you would use the moment of inertia for a solid sphere. See https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/11-1-rolling-motion/

Hope this helps.