A boy of mass M is applying a horizontal force to slide a box of mass M' on a rough horizontal surface. The coefficient of friction between the shoes of the boy and the floor is μ and that between the box and the floor is μ'. In which of the following cases it is certainly not possible to slide the box?
(a) μ < μ', M < M'
(b) μ > μ', M < M'
(c) μ < μ', M > M'
(d) μ > μ', M > M'

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Solution

(a) μ < μ', M < M'

Let T be the force applied by the boy on the block.

Free body diagram for the box:

The condition for preventing the slide is
f_max > T

μ'M'g > T (i)

Now see the free body diagram of a boy of mass M:

f_max = μmg

The condition for preventing the slide is
f_max > T
μmg > T

The condition for sliding the entire system (block and boy) is
f' > f (block is not slide)
μ'M'g > μmg
μ'M' > μm
μ < μ'
m < M'

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