One of the interesting demonstrations of Moment of Inertia includes the "Rolling Race" where objects of same mass and radii but having different Moments of Inertia, are allowed to roll down an incline without slipping, and seeing which one crosses the finish line first. We know that, the object with least moment of inertia wins the race.
The following question is from the book "Concepts of Physics" by Dr.H.C.Verma, from the chapter "Rotational Mechanics", which considers the case where objects roll with slipping:
Page 194, Objective I, Question 24
A solid sphere, a hollow sphere and a disc, all having same mass and radius, are placed at the top of an incline and released. The friction coefficients between the objects and the incline are same and not sufficient to allow pure rolling. Least time will be taken in reaching the bottom by
(a) the solid sphere
(b) the hollow sphere
(c) the disc
(d) all will take same time
I approached the problem as described below:
If it were rolling without slipping in the above question, then the winner would have been the solid sphere (as its Moment of Inertia is $\frac 2 5 mr^2$ whereas for hollow sphere and disc it's $\frac 2 3 mr^2$ and $\frac 1 2 mr^2$ respectively). If it were entirely slipping i.e., if no friction were present then we don't need to worry about rolling or moment of inertia, and all objects reach the bottom at the same time (no one wins).
The above question is an intermediate case where objects roll as well as slip due to insufficient friction. So, I concluded that the solid sphere wins the race (but the margin of winning will be less when compared to rolling without slipping), but the answer is given as - all will reach the bottom at the same time. The outcome is similar to the case of "entirely slipping and no rolling".
The following are my doubts regarding this:
Why is this approach leading to the incorrect answer even though it seems reasonable? Is this an incorrect method?
Why must the outcome be similar to the case where no friction is present? Why should it be biased to one of the extreme cases?
In many sources which I read so far, only the case where the object rolls without slipping is being discussed. If possible, kindly provide useful links for further reading regarding rolling race with slipping, as I could not find one. Kindly clarify my above doubts.
Thank you in advance.
Please Note: Even though this question is based on an exercise problem, I don't think this is off-topic. I am asking about the concept of rolling with slipping which is not covered in most of the sources. So, I believe this question will be helpful for a broader audience. Further, I have showed my own effort in solving this problem. For your kind information, I asked this question after reading this meta page - How do I ask homework questions on Physics Stack Exchange?.
If you still feel this question must be closed, kindly state the reason in the comments, so that I could understand your homework policies and avoid such circumstances in future.