Edexcel M3 (Mechanics 3)

2. \begin{figure}[h] \end{figure} A uniform solid right circular cone has base radius \(a\) and semi-vertical angle \(\alpha\), where \(\tan \alpha = \frac { 1 } { 3 }\). The cone is freely suspended by a string attached at a point \(A\) on the rim of its base, and hangs in equilibrium with its axis of symmetry making an angle of \(\theta ^ { \circ }\) with the upward vertical, as shown in Figure 1. Find, to one decimal place, the value of \(\theta\).

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a point \(O\). The point \(A\) is vertically below \(O\), and \(O A = a\). The particle is projected horizontally from \(A\) with speed \(\sqrt { } ( 3 a g )\). When \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\) and the string is still taut, the tension in the string is \(T\) and the speed of \(P\) is \(v\), as shown in Figure 2.

1. Find, in terms of \(a , g\) and \(\theta\), an expression for \(v ^ { 2 }\).
2. Show that \(T = ( 1 - 3 \cos \theta ) m g\). The string becomes slack when \(P\) is at the point \(B\).
3. Find, in terms of \(a\), the vertical height of \(B\) above \(A\). After the string becomes slack, the highest point reached by \(P\) is \(C\).
4. Find, in terms of \(a\), the vertical height of \(C\) above \(B\).

5. \begin{figure}[h] \end{figure} One end of a light inextensible string is attached to a fixed point \(A\). The other end of the string is attached to a fixed point \(B\), vertically below \(A\), where \(A B = h\). A small smooth ring \(R\) of mass \(m\) is threaded on the string. The ring \(R\) moves in a horizontal circle with centre \(B\), as shown in Figure 3. The upper section of the string makes a constant angle \(\theta\) with the downward vertical and \(R\) moves with constant angular speed \(\omega\). The ring is modelled as a particle.

1. Show that \(\omega ^ { 2 } = \frac { g } { h } \left( \frac { 1 + \sin \theta } { \sin \theta } \right)\).
2. Deduce that \(\omega > \sqrt { \frac { 2 g } { h } }\). Given that \(\omega = \sqrt { \frac { 3 g } { h } }\),
3. find, in terms of \(m\) and \(g\), the tension in the string.

6. \begin{figure}[h] \end{figure} The shaded region \(R\) is bounded by the curve with equation \(y = \frac { 1 } { 2 x ^ { 2 } }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\), as shown in Figure 4. The unit of length on each axis is 1 m . A uniform solid \(S\) has the shape made by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis.

1. Show that the centre of mass of \(S\) is \(\frac { 2 } { 7 } \mathrm {~m}\) from its larger plane face. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{ab85ec29-b1fc-45a9-9343-09feb33ab6c5-010_616_431_1420_778}
   \end{figure} A sporting trophy \(T\) is a uniform solid hemisphere \(H\) joined to the solid \(S\). The hemisphere has radius \(\frac { 1 } { 2 } \mathrm {~m}\) and its plane face coincides with the larger plane face of \(S\), as shown in Figure 5. Both \(H\) and \(S\) are made of the same material.
2. Find the distance of the centre of mass of \(T\) from its plane face.