A plank of mass `M` is placed on a rough horizontal force `F` is applied on it A man of mass m rens on the plank find the acceleration of the man so that the plank does not move on the surface . The coefficent of friction between the plank and the surface is `mu` Assume that the man does not slip on the plank

Explanation

FBD of man
` f_(1) =` force between man and the plank
`rArr f_(1) = ma` ….(i)
`N = mg` …(ii)

`FBD` of plank
Direction of friction force between ground and plank may be in forward or backward direction depending on the value of `f_(1)` which depends on the acceleration of the man

`f_(2)=` force of friction between the plank and surface.
`f_(1)` can have any value in the following range
`:' F - f_(2max) le f_(1) le F + f_(2max)`
`rArr F - mu(m + m)g le ma le f + mu(m + M)g`
`rArr (F)/(m)-(mu(m +M)g)/(m)le a le (F)/(m)+(mu(m+M)g)/(M)`