7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2108a1be-0214-42c4-9cb4-8622cc0fa496-5_531_1061_283_468}
\captionsetup{labelformat=empty}
\caption{Fig. 3}
\end{figure}
Figure 3 shows a particle \(X\) of mass 3 kg on a smooth plane inclined at an angle \(30 ^ { \circ }\) to the horizontal, and a particle \(Y\) of mass 2 kg on a smooth plane inclined at an angle \(60 ^ { \circ }\) to the horizontal. The two particles are connected by a light, inextensible string of length 2.5 metres passing over a smooth pulley at \(C\) which is the highest point of the two planes.
Initially, \(Y\) is at a point just below \(C\) touching the pulley with the string taut. When the particles are released from rest they travel along the lines of greatest slope, \(A C\) in the case of \(X\) and \(B C\) in the case of \(Y\), of their respective planes. \(A\) and \(B\) are the points where the planes meet the horizontal ground and \(A B = 4\) metres.
- Show that the initial acceleration of the system is given by \(\frac { g } { 10 } ( 2 \sqrt { 3 } - 3 ) \mathrm { ms } ^ { - 2 }\).
- By finding the tension in the string, or otherwise, find the magnitude of the force exerted on the pulley and the angle that this force makes with the vertical.
- Find, correct to 2 decimal places, the speed with which \(Y\) hits the ground.