3. One end of a light elastic string, of natural length 1.5 m and modulus of elasticity 14.7 N ,
3. One end of a light elastic string, of natural length 1.5 m and modulus of elasticity 14.7 N , is attached to a fixed point \(O\) on a ceiling. A particle \(P\) of mass 0.6 kg is attached to the free end of the string. The particle is held at \(O\) and released from rest. The particle comes to instantaneous rest for the first time at the point \(A\).
Find
- the distance \(O A\),
- the magnitude of the instantaneous acceleration of \(P\) at \(A\).
\begin{figure}[h]
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\caption{igure 2}
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A uniform solid \(S\) consists of two right circular cones of base radius \(r\). The smaller cone has height \(2 h\) and the centre of the plane face of this cone is \(O\). The larger cone has height \(k h\) where \(k > 2\). The two cones are joined so that their plane faces coincide, as shown in Figure 2.
- Show that the distance of the centre of mass of \(S\) from \(O\) is
$$\frac { h } { 4 } ( k - 2 )$$
The point \(A\) lies on the circumference of the base of one of the cones. The solid is suspended by a string attached at \(A\) and hangs freely in equilibrium.
Given that \(r = 3 h\) and \(k = 6\)
- find the size of the angle between \(A O\) and the vertical.