3 A fish slice consists of a blade and a handle as shown in Fig. 3.1. The rectangular blade ABCD is of mass 250 g and modelled as a lamina; this is 24 cm by 8 cm and is shown in the \(\mathrm { O } x y\) plane. The handle EF is of mass 125 g and is modelled as a thin rod; this is 30 cm long and E is attached to the mid-point of \(\mathrm { CD } . \mathrm { EF }\) is at right angles to CD and inclined at \(\alpha\) to the plane containing ABCD , where \(\sin \alpha = 0.6\) (and \(\cos \alpha = 0.8\) ). Coordinates refer to the axes shown in Fig. 3.1. Lengths are in centimetres. The \(y\) and \(z\)-coordinates of the centre of mass of the fish slice are \(\bar { y }\) and \(\bar { z }\) respectively.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3865b4b3-97c7-412b-aabd-2705a954a847-4_517_1068_573_534}
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\caption{Fig. 3.1}
\end{figure}
- Show that \(\bar { y } = 9 \frac { 1 } { 3 }\) and \(\bar { z } = 3\).
- Suppose that the plane \(\mathrm { O } x y\) in Fig. 3.1 is horizontal and represents a table top and that the fish slice is placed on it as shown. Determine whether the fish slice topples.
The 'superior' version of the fish slice has an extra mass of 125 g uniformly distributed over the existing handle for 10 cm from F towards E , as shown in Fig. 3.2. This section of the handle may still be modelled as a thin rod.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3865b4b3-97c7-412b-aabd-2705a954a847-4_513_1065_1683_539}
\captionsetup{labelformat=empty}
\caption{Fig. 3.2}
\end{figure} - In this new situation show that \(\bar { y } = 14\) and \(\bar { z } = 6\).
A sales feature of the 'superior' version is the ability to suspend it using a very small hole in the blade. This situation is modelled as the fish slice hanging in equilibrium when suspended freely about an axis through O .
- Indicate the position of the centre of mass on a diagram and calculate the angle of the line OE with the vertical.