A plank of mass `M` is placed on a rough horizontal surface and a constant horizontal force `F` is applied on it . A man of mass `m` runs on the plank find the range of acceleration of the man that the plank does not move on the surface. Coefficient 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)`