7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{55024b00-68f0-4f8b-9b08-58479bb291fd-11_570_1045_262_459}
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\caption{Figure 3}
\end{figure}
A particle \(P\) of mass 4 kg is attached to one end of a light inextensible string. A particle \(Q\) of mass \(m \mathrm {~kg}\) is attached to the other end of the string. The string passes over a small smooth pulley which is fixed at a point on the intersection of two fixed inclined planes. The string lies in a vertical plane that contains a line of greatest slope of each of the two inclined planes. The first plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\) and the second plane is inclined to the horizontal at an angle \(\beta\), where \(\tan \beta = \frac { 4 } { 3 }\). Particle \(P\) is on the first plane and particle \(Q\) is on the second plane with the string taut, as shown in Figure 3.
The first plane is rough and the coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\). The second plane is smooth. The system is in limiting equilibrium.
Given that \(P\) is on the point of slipping down the first plane,
- find the value of \(m\),
- find the magnitude of the force exerted on the pulley by the string,
- find the direction of the force exerted on the pulley by the string.