rotational dynamics - For a solid sphere rolling (pure roll) up a slope (with friction) does friction play a role in slowing it down?

A solid sphere rolling up a slope is slowed down - is this only due to gravity, or is it also because of friction? I need to know this, to calculate the final translational and angular velocity of a solid sphere, at the top of the slope, initially rolling (pure roll) with velocity v on a plane surface and then comes onto this slope.. I'll do this using work - energy theorem, but I need to know the total work done on the sphere first.

I think that friction is the only force that can affect the rotational motion of the sphere, as only it exerts a torque on the sphere..

Answer

In the case of pure rolling (no slipping or sliding) enough friction is provided so that at all times:

$$v=\omega R$$

where $R$ is the radius of the sphere.

Does friction play a part in the energy balance? Yes.

Let $T$ be the total energy of the sphere, $U$ be the potential energy of the sphere and $K$ its kinetic energy, then because the friction force does no work and we assume no other external forces:

$$\Delta T=\Delta U+\Delta K=0$$

when rolling up the incline.

Of course $\Delta U=mg\Delta h$ (for smallish $\Delta h$).

But for $\Delta K$ the situation is slightly more complicated because the sphere is translating and rotating at once, so that:

$$K=K_{trans}+K_{rot}$$

You'll need to find the expression for $K_{rot}$ and use $v=\omega R$ to make everything about $v$.

Without friction, $\omega$ would remain constant and thus also $K_{rot}$.