Mass `m_(2)` placed on a plank of mass `m_(1) `lying on a smooth horizontal plane. A horizontal force `F = alpha_(0) t ` ( `alpha_(0)` is a constant ) is applied to a bar. If acceleratiion of the plank and bar are `a_(1)` and `a_(2)` respectively and the coefficient of friction between `m_(1)` and `m_(2)` is `mu`. Then find acceleration a with time t.

Explanation

If `F lt mu m_(2) g `then both blocks move with common acceleration, i.e., `a_(1) = a_(2)`
When` F gt mu m_(2) g`, then
Equation for block of mass m
`F - mu m_(2) g = m_(2) a_(2)` …(1)
and `mu m_(2) g = m_(1) a_(1)` …..(2)
From equation (1)
`alpha_(0) t - mu m_(2) g = m_(2) a_(2)`
i.e., acceleration `a_(2)` varies with time linearly

its slope positive an intercept negative.
From equation (2) `a_(1)` is independent of time.
So, the graph between a & t is as follow.