6.
\begin{figure}[h]
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\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{9e3d76a7-b997-4e46-a5dd-aeeaa5abfa4e-02_211_611_388_1845}
\end{figure}
A rough uniform rod, of mass \(m\) and length \(4 a\), is rod is held on a rough horizontal table. The rod is perpendicular to the edge of the table and a length \(3 a\) projects horizontally over the edge, as shown in Fig. 1.
- Show that the moment of inertia of the rod about the edge of the table is \(\frac { 7 } { 3 } m a ^ { 2 }\).
The rod is released from rest and rotates about the edge of the table. When the rod has turned through an angle \(\theta\), its angular speed is \(\dot { \theta }\). Assuming that the rod has not started to slip,
- show that \(\dot { \theta } ^ { 2 } = \frac { 6 g \sin \theta } { 7 a }\),
- find the angular acceleration of the rod,
- find the normal reaction of the table on the rod.
The coefficient of friction between the rod and the edge of the table is \(\mu\).
- Show that the rod starts to slip when \(\tan \theta = \frac { 4 } { 13 } \mu\)