A small disc of mass m slides down a smooth hill of height h without initial velocity and gets onto a plank of mass M lying on the horizontal plane at the base of the hill. (figure). Due to friction between the disc and the plank the disc slows down and, beginning with a certain moment, moves in one piece with the plank.
(1) Find the total work performed by the friction forces in this process.
(2) Can it be stated that the result of obtained does not depend on the choice of the reference frame?

Explanation

(a) When the disc slides and comes to a plank, it has a velocity equal to `v=sqrt(2gh)`. Due to friction between the disc and the plank the disc slows down and after some time the disc moves in one piece with the plank with velocity `v^'` (say).
From the momentum conservation for the system (disc+plank) along horizontal towards right:
`mv=(m+M)v^'` or `v^'=(mv)/(m+M)`
Now from the equation of the increment of total mechanical energy of a system:
`1/2(M+m)v^('^2)-1/2mv^2=A_(f r)`
or, `1/2(M+m)(m^2v^2)/((m+M)^2)-1/2mv^2=A_(f r)`
so, `1/2v^2[(m^2)/(M+m)-m]=A_(f r)`
Hence, `A_(f r)=-((mM)/(m+M))gh=-mugh`
(where `mu=(mM)/(m+M)`=reduced mass)
(b) We look at the problem from a frame in which the hill is moving (together with the disc on it) to the right with speed `u`. Then in this frame the speed of the disc when it just gets onto the plank is, by the law of addition of velocities, `barv=u+sqrt(2gh)`. Similarly the common speed of the plank and the disc when they move together is
`barv=u+(m)/(m+M)sqrt(2gh)`.
Then as above `barA_(f r)=1/2(m+M)barv^2-1/2mbarv^2-1/2M u^2`
`=1/2(m+M){u^2+(2m)/(m+M)usqrt(2gh)+(m^2)/((m+M)^2)2gh}-1/2(m+M)u^2-1/2m2usqrt(2gh)-mgh`
We see that `barA_(f r)` is independent of `u` and is in fact just `0mugh` as in (a). Thus the result obtained does not depend on the choice of reference frame.
Do not however that it will be in correct to apply "conservation of energy" formula in the frame in which the hill is moving. The energy carried by the hill is not negligible in this frame. See also the next problem.