If, for a body rolling on an incline, the friction coefficient isn't enough to allow pure rolling will it still roll?

Question

More specifically that it won't be pure rolling (obviously) but would it still have some rotational motion along with its translational motion? (if yes how would we write their mechanical equations). Will this also apply to a body which is on a horizontal surface?

Answer 1

Firstly let's look at the case of a horizontal plane first.

In my answer here I derived that the critical friction coefficient is:

$$\large{\mu_c=\frac{FI}{mg(I+mR^2)}}$$

Now we have three scenarios:

a) No friction at all, $\mu=0$:

Assuming no forces or couples act on the object then Newton tells us that the state of motion remains unchanged, or:

$$v=v_0,$$

and:

$$\omega=\omega_0.$$

Specifically, if the object was rotating it will keep doing so (of course this does not constitute 'rolling'), if it wasn't then it won't start doing so.

b) Lots of friction, $\mu>\mu_c$:

Assuming no forces or couples act on the object then $v$ and $\omega$ will adjust themselves until:

$$v=\omega R$$

c) Small amount of friction $0<\mu<\mu_c$:

The object will both roll and slide.

If for simplicity's sake we assume the object was completely at rest at $t=0$ and a constant accelerating force $F$ acts on its centre of gravity, then the equations of motion are:

$$v=\frac{F-\mu mg}{m}t$$

and:

$$\omega=\mu\frac{mgR}{I}t$$

also:

$$v>\omega R$$.

Secondly, the case of the object on an inclined plane. Here we must bear in mind that on an inclined plane a gravitational force $mg\sin\theta$ always acts on the object.

In my answer here I derived that for that case the critical friction coefficient is:

$$\large{\mu_c=\frac{I}{I+mR^2}\tan\theta}.$$

Again we have three scenarios:

a) No friction at all, $\mu=0$:

Assuming no forces other than gravity are acting on the object and that the object was completely stationary at $t=0$ then the equations of motion are:

$$v=mg\sin\theta t$$

and:

$$\omega=0$$

b) Lots of friction, $\mu>\mu_c$:

Assuming no forces other than gravity are acting on the object and that the object was completely stationary at $t=0$ then the equations of motion are:

$$v=(\sin\theta - \mu_c \cos\theta)gt.$$

$$\omega=(\mu_c \frac{mgR}{I}\cos\theta)t.$$

In addition:

$$v=\omega R$$

c) Small amount of friction $0<\mu<\mu_c$:

The object will both roll and slide.

If for simplicity's sake we assume the object was completely at rest at $t=0$, then the equations of motion are:

$$v=(\sin\theta - \mu \cos\theta)gt.$$

$$\omega=(\mu \frac{mgR}{I}\cos\theta)t.$$

In addition:

$$v>\omega R$$