# A cylinder rolling down an inclined plane *including slipping*

The problem of a cylinder rolling down an inclined plane has been solved several times on this site: 1 year ago 6 years ago etc.

What I have always seen is that the assumption is that the cylinder is not slipping. This is also the assumption in these excellent explanations of the problem. I will refer to Method 1, as it is the easiest.

Because the cylinder is not slipping, the method is able to equate the acceleration of the cylinder with the angular acceleration times the radius R: $$a = \alpha R$$

But what if the cylinder is slipping? How would I go about solving the same problem?

In this case, the friction of the inclined plane isn't enough to allow the cylinder in question to roll. Then, the free body diagram of the cylinder will yield the following equations:

(Mass of cylinder $$M$$, radius of cylinder $$R$$, moment of inertia of cylinder about the center of mass $$I_{cm}$$, inclination of plane $$\theta$$, force due to friction $$f$$, coefficient of kinetic friction $$\mu_k$$, linear acceleration of cylinder $$a$$, and angular acceleration of the cylinder $$\alpha$$).

$$Mg\sin \theta-f=Ma$$ $$\mu_k Mg\cos \theta=f$$ $$R\times f=I_{cm}\alpha$$

In this case, $$\alpha\neq aR$$, and so you can't use this relation. Instead, you'll get unrelated angular and linear accelerations, which you can use along with kinematics to get the required quanity ($$\omega_f$$, $$v_f$$, $$s$$, and so on).

Also note that in case the wedge has no friction, the cylinder won't rotate (no external torque), and it'll slide down similar to a point particle.

If there is no friction at all, the problem is easily solvable because there will be no rolling and only slipping, therefore the shape of the object doesn't matter.

With finite friction the whole thing becomes a lot more complicated, as you will need to know surface properties of both the cylinder and the inclined plane. If the cylinder starts at rest, it gets even more complicated, as there is static and dynamic friction.

It is probably possible to do it analytically with the right approximations, or you will have to simulate it numerically. All in all it becomes less of a physics task and more of a engineering task.

• Which material properties of the cylinder would be relevant here? – wavion Apr 15 '20 at 13:08
• I'm not an expert on friction, but I guess it's about surface roughness which somehow translates into static and dynamic friction coefficients. – Paul G. Apr 15 '20 at 13:18
• To me it didn't seem to be a homework question but one concerning general understanding. If it was one, OP should have mentioned, what parameters were given. – Paul G. Apr 15 '20 at 14:36
• Understood. I was merely pointing out that no physical properties were relevant in this case other than the surface. Cheers! – wavion Apr 15 '20 at 14:42
• I changed "material properties" to "surface properties". – Paul G. Apr 15 '20 at 14:47