Problems

A particle of mass \(m\) is initially at rest on a rough horizontal surface. The particle experiences a force \(mg\sin \pi t\), where \(t\) is time, acting in a fixed horizontal direction. The coefficient of friction between the particle and the surface is \(\mu\). Given that the particle starts to move first at \(t=T_0\), state the relation between \(T_0\) and \(\mu\).

1. For \(\mu = \mu_0\), the particle comes to rest for the first time at \(t=1\). Sketch the acceleration-time graph for \(0\le t \le 1\). Show that \[ 1+\left(1-\mu_0^2\right)^{\frac12} -\mu_0\pi +\mu_0 \arcsin \mu_0 =0\,. \]
2. For \(\mu=\mu_0\) sketch the acceleration-time graph for \(0\le t\le 3\). Describe the motion of the particle in this case and in the case \(\mu=0\).