Problem 1.65

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.

Reveal Answer

\begin{aligned} &\text { 1.65. When } t \leqslant t_{0}, \text { the accelerations } w_{1}=w_{2}=a t /\left(m_{1}+m_{2}\right) ;\\ &\text { when } t \geqslant t_{0} \quad w_{1}=\mathrm{kgm}_{2} / m_{1}, \quad w_{2}=\left(a t-k m_{2} g\right) / m_{2} . \text { Here } t_{0}=\\ &=\mathrm{kgm}_{2}\left(m_{1}+m_{2}\right) / a m \text { . See Fig. 4. } \end{aligned}

The Fundamental Equation of Dynamics