5 A particle \(P\) of mass 0.1 kg is attached to one end of a light inextensible string of length 0.5 m . The other end of the string is attached to a fixed point \(A\). The particle \(P\) moves in a circle which has its centre \(O\) on a smooth horizontal surface 0.3 m below \(A\). The tension in the string has magnitude \(T \mathrm {~N}\) and the magnitude of the force exerted on \(P\) by the surface is \(R \mathrm {~N}\).

1. Given that the speed of \(P\) is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), calculate \(T\) and \(R\).
2. Given instead that \(T = R\), calculate the angular speed of \(P\).
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ef38fda2-230b-431a-8064-82e4a3bff393-12_449_621_260_762} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Fig. 1 shows the cross-section \(A B C D E\) through the centre of mass \(G\) of a uniform prism. The crosssection consists of a rectangle \(A B C F\) from which a triangle \(D E F\) has been removed; \(A B = 0.6 \mathrm {~m}\), \(B C = 0.7 \mathrm {~m}\) and \(D F = E F = 0.3 \mathrm {~m}\).
3. Show that the distance of \(G\) from \(B C\) is 0.276 m , and find the distance of \(G\) from \(A B\).
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ef38fda2-230b-431a-8064-82e4a3bff393-13_494_583_258_781} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The prism is placed with \(C D\) on a rough horizontal surface. A force of magnitude 2 N acting in the plane of the cross-section is applied to the prism. The line of action of the force passes through \(G\) and is perpendicular to \(D E\) (see Fig. 2). The prism is on the point of toppling about the edge through \(D\).
4. Calculate the weight of the prism.
   \includegraphics[max width=\textwidth, alt={}, center]{ef38fda2-230b-431a-8064-82e4a3bff393-14_512_520_258_817} A small object is projected with speed \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) at the foot of a plane inclined at \(45 ^ { \circ }\) to the horizontal. The angle of projection of the object is \(15 ^ { \circ }\) above a line of greatest slope of the plane (see diagram). At time \(t \mathrm {~s}\) after projection, the horizontal and vertically upwards displacements of the object from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
5. Express \(x\) and \(y\) in terms of \(t\), and hence find the value of \(t\) for the instant when the object strikes the plane.
6. Express the vertical height of the object above the plane in terms of \(t\) and hence find the greatest vertical height of the object above the plane.
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