Work done by static friction on an incline

The total work done by the static friction force acting on a ball rolling down an incline is zero, despite the fact that work is done by the friction force to increase the rotational kinetic energy of the ball. It is shown that an equal and opposite amount of work is done by the friction force to decrease the translational kinetic energy of the ball.


Introduction
When a ball of radius R rolls without sliding down an incline, the angular velocity of the ball, ω, increases with time according to the relation v = Rω where v is the instantaneous linear speed of the ball down the incline.Since v increases with time down the incline, ω also increases with time.
The contact point at the bottom of the ball comes to rest on the incline, otherwise the ball would be sliding down the incline.The ball rotates as a result of a static friction force, F, acting at the bottom of the ball, in a direction up the incline.The friction force therefore has three effects: it prevents sliding, it opposes the gravitational force down the incline, so it reduces the acceleration down the incline compared with frictionless sliding, and it exerts a torque on the ball that increases its angular velocity [1].
Since F is a static friction force, it does not result in a change in the kinetic energy of the ball.If the ball and the surface are both very smooth then rolling friction can be ignored, and the total energy of the ball will be conserved.If the ball starts from rest at the top of the incline, then the kinetic energy of the ball at the bottom of the incline will be equal to its gravitational potential energy, mgh, at the top of the incline.That is, mgh =1 2 mv 2 + 1 2 I cm ω 2 , where m is the mass of the ball and I cm is the moment of inertia of the ball for rotation about an axis through its centre of mass.For a uniform solid ball, I cm = (2/5)mR 2 .

Work done by F
The above relations are described in almost all introductory physics textbooks.What is generally missing is a discussion of the work done by F. The usual assumption is that no work is done by F since the point at which F is applied remains at rest and since the work done is Fx where x is the displacement of the point of application of F. However, something seems to be wrong with this assumption since F is directly responsible for an R Cross increase in rotational kinetic energy of the ball, which means that work was done to increase its rotational kinetic energy.If a torque, τ , is applied to an object and the object rotates through a small angle, dθ, then the work done by τ is given by τ dθ.In the present case, the torque exerted by F is given by FR = I cm dω/dt.If the ball rotates through an angle θ then ω = dθ/dt and the work, W, done by F is given by ( F is also responsible for a decrease in the translational kinetic energy of the ball since it opposes the gravitational force down the incline.The linear acceleration, a, of the ball down an incline with slope β is given by Starting from rest, the linear velocity of the ball after travelling a distance s down the incline is given by so the translational kinetic energy, KE, is given by The change in KE due to F is therefore −Fs which means that the work done by F to change the translational kinetic energy is −Fs.During the same time, the ball rotates through an angle θ = s/R so the work done by the torque FR is FR θ = Fs which is equal and opposite the work done to decrease the translational kinetic energy.The total work done by F is therefore zero, consistent with the fact that the displacement of the point of application of F is zero.
The effect of the static friction force is to convert some of the translational kinetic energy of the ball into rotational kinetic energy in order to maintain the rolling condition v = Rω, without subtracting from or adding to the energy of the ball.Further discussion of this effect is given in [2,3].A similar result is obtained if the ball rolls without sliding up an incline.Since v decreases up the incline, ω must also decrease to maintain the rolling condition.In that case, the static friction force also acts up the incline, and the effect is to decrease the rotational kinetic energy of the ball and to increase its translational kinetic energy by the same amount.