A plank of mass `m_1` with a bar of mass `m_2` placed on it lies on a smooth horizontal plane. A horizontal force growing with time t as `F^'=at` (a is constant) is applied to the bar. Find how the accelerations of the plank `w_1` and of the bar `w_2` depend on t, if the coefficient of friction between the plank and the bar is equal to k. Draw the approximate plots of these dependences.

Explanation

Let us write the Newton's second law in projection form along positive x-axis for the plank and the bar
`f r=m_1w_1`, `f r=m_2w_2` (1)
At the initial moment, `f r` represents the static friction, and as the force F grows so does the friction force `f r`, but up to it's limiting value i.e. `f r=f r_s(max)=kN=km_2g`.
Unless this value is reached, both bodies moves as a single body with equal acceleration. But as seen as the force `f r` reaches the limit, the bar starts sliding over the plank i.e. `w_2gew_1`.
Substituting here the values of `w_1` and `w_2` taken from Eq. (1) and taking into account that `f r=km_2g`, we obtain, `(at-km_2g)//m_2ge(km_2)/(m_1)g`, were the sign `"="` corresponds to the moment
`t=t_0` (say)
Hence, `t_0=(kg m_2(m_1+m_2))/(am_1)`
If `tlet_0`, then `w_1=(km_2g)/(m_1)` (constant). and
`w_2=(at-km_2g)//m_2`
On this basis `w_1(t)` adn `w_2(t)`, plots are as shown in the figure of answersheet.