Question

The upper half of an inclined plane with inclination `phi` is perfectly smooth while the lower half is rough. A body starting from rest at the top will again come to rest at the bottom if the coefficient of friction for the lower half is given by
A. `mu = (1)/(tan theta)`
B. `mu = (2)/(tan theta)`
C. `mu = 2 tan theta`
D. `mu = tan theta`

Answer 1

Using work-energy theorem `W_("total") = Delta K`
`W_("gravity") + W_("friction") = (0 - 0) = 0`

`(mg sin theta . L) + (- mu mg cos theta xx (L)/(2)) = 0`
`= mu = (2 sin theta)/(cos theta) = 2 tan theta`