4. \(X\) and \(Y\) are two points 1 m apart on a line of greatest slope of a smooth plane inclined at \(60 ^ { \circ }\) to the horizontal. A particle \(P\) of mass 1 kg is released from rest at \(X\).
- Find the speed with which \(P\) reaches \(Y\).
\(P\) is now connected to another particle \(Q\), of mass \(M \mathrm {~kg}\), by a light inextensible string. The system is placed with \(P\) at \(Y\) on the plane and \(Q\) hanging vertically at the other end of the string, which passes over a fixed pulley at the top of the plane.
The system is released from rest and \(P\) moves up the plane with acceleration \(\frac { g } { 5 }\).
\includegraphics[max width=\textwidth, alt={}, center]{cc75a4a5-1c3a-4e36-acfd-21f6246f2a38-1_358_321_2024_1597} - Show that \(M = \frac { 5 \sqrt { } 3 + 2 } { 8 }\).
- State a modelling assumption that you have made about the pulley. Briefly state what would be implied if this assumption were not made.
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