Friction and a sphere on an inclined plane

If I have a stationary sphere on an inclined plane such that friction is sufficient to prevent the sphere from sliding down the incline. Then about the centre of sphere frictional force is the only force that will create a torque. So shouldn't the sphere keep rotating at one place instead of being still?

First, we assume that all bodies are rigid and rolling friction is absent.

The type of friction most important here is static friction. Static friction between two surfaces acts only when no relative motion is present, and has variable direction and a maximum possible magnitude. Both can be determined by a simple rule:

Static friction opposes the direction of impending relative motion between two surfaces and has the least possible magnitude which will prevent said relative motion.

According to the question, the value of coefficient of friction is sufficiently high, so we will not worry about it.

When the sphere is placed at some height on the incline and then released, the following will happen:

1. Component of gravity normal to the incline will get balanced by normal force.
2. Component of gravity along the incline would induce downward sliding in the absence of static friction.

This might make us think that the frictional force should act in a direction upward the incline to balance gravity with a magnitude equal to $$mg\sin\theta$$ where $$m$$ is the mass of the sphere, and $$\theta$$ is the angle of the incline. But if that were the case it would induce rotation of the sphere at the same spot, which would lead to relative motion between the two surfaces. That does not agree with the earlier rule.

So, static friction will in fact act in the direction upward the incline with a magnitude equal to $$(2/7)mg\sin\theta$$ (this is the value is particular for a solid rigid sphere).

We can easily arrive at this value using: 1. the value of moment of inertia of a solid sphere, 2. The condition for pure rolling angular acceleration = linear acceleration divided by radius. Pure rolling will occur because this is the only case where the surfaces have no relative motion between them.

If you want, then you may make the friction coefficient ($$\mu$$) as large as you want, but still the sphere will roll. Why? Because the frictional force will not exceed the force of gravity acting on the sphere.

See, the friction arises to stop any relative motion between two surfaces. But when the ball is rolling, there is no relative motion between the sphere and the inclined plane. So the friction would be just enough to stop slipping, and this value comes out to be lesser than the force due to gravity on the sphere. If you want then you can rigorously prove this by drawing a free body diagram and finding the torque about the COM to find the anguoar acceleration. I leave it up to you as an exercise.

Also, the claim that the sphere will rotate at a single place can be rejected by energy considerations. If a sphere keeps on rotating at the same place, then the friction is surely going to do some negative work on it. Also there is no source of energy for the sphere. So eventually the rotation must stop. In fact, the rotation shouldn't even start because you require energy to rotate something. Where will you get that from?

• So it means a ball can never stay at rest on any inclined plane of nonzero angle of inclination? – Emiley S Jan 28 '20 at 14:06
• @EmileyS. It can, if you put a fixed obstacles on the inclined plane. – user252699 Jan 28 '20 at 14:08
• @EmileyS See this. This kind of arrangement can also help keep a hollow sphere static. For a solid sphere, you might have to change the density of the sphere and make it a non uniform sphere such that its COM lies exactly above the point of contact of the sphere with the plane. – user252699 Jan 28 '20 at 14:11
• Understood. Thank you. – Emiley S Jan 28 '20 at 14:12

Assume that sphere is in equilibrium then the net forces and net torque should be zero.

1. Net forces are equal then mgsinθ=friction
2. Net torque about centre should be zero then torque of mgsinθ is zero ,also torque of friction should be zero for equilibrium.

From second Friction equal to zero but you say friction is sufficient. Both statement are contradict each other. Hence, sphere should roll over the incline plane.