A ball bounces off a smooth plane. The angles its path makes with the plane before and after the impact are \(\alpha\) and \(\beta\), as shown in Fig. 1.2.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-2_317_1082_1468_575}
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\caption{Fig. 1.2}
\end{figure}
The velocity of the ball before the impact is \(u \mathbf { i } - v \mathbf { j }\) and the coefficient of restitution in the impact is \(e\).
Write down an expression in terms of \(u , v , e , \mathbf { i }\) and \(\mathbf { j }\) for the velocity of the ball immediately after the impact. Hence show that \(\tan \beta = e \tan \alpha\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-3_581_486_274_383}
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\caption{Fig. 2.1}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-3_593_392_264_1370}
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\caption{Fig. 2.2}
\end{figure}
A uniform wire is bent to form a bracket OABCD . The sections \(\mathrm { OA } , \mathrm { AB }\) and BC lie on three sides of a square and CD is parallel to AB . This is shown in Fig. 2.1 where the dimensions, in centimetres, are also given.
- Show that, referred to the axes shown in Fig. 2.1, the \(x\)-coordinate of the centre of mass of the bracket is 3.6 . Find also the \(y\)-coordinate of its centre of mass.
- The bracket is now freely suspended from D and hangs in equilibrium.
Draw a diagram showing the position of the centre of mass and calculate the angle of CD to the vertical.
- The bracket is now hung by means of vertical, light strings BP and DQ attached to B and to D , as shown in Fig. 2.2. The bracket has weight 5 N and is in equilibrium with OA horizontal.
Calculate the tensions in the strings BP and DQ .
The original bracket shown in Fig. 2.1 is now changed by adding the section OE, where AOE is a straight line. This section is made of the same type of wire and has length \(L \mathrm {~cm}\), as shown in Fig. 2.3.
\(\begin{array} { l l l l } \begin{array} { l } \text { not to }
\text { scale } \end{array} & 2 & 6 &
\mathrm {~L} \longrightarrow & \mathrm {~L} & &
\mathrm {~L} & \mathrm { O } & 6 & \mathrm {~A} \end{array}\)
Fig. 2.3
The value of \(L\) is chosen so that the centre of mass is now on the \(y\)-axis. - Calculate \(L\).