OCR MEI M3 (Mechanics 3) 2011 June

2

1. A particle P of mass 0.2 kg is connected to a fixed point O by a light inextensible string of length 3.2 m , and is moving in a vertical circle with centre O and radius 3.2 m . Air resistance may be neglected. When P is at the highest point of the circle, the tension in the string is 0.6 N .
   1. Find the speed of P when it is at the highest point.
   2. For an instant when OP makes an angle of \(60 ^ { \circ }\) with the downward vertical, find
      (A) the radial and tangential components of the acceleration of P ,
      (B) the tension in the string.
2. A solid cone is fixed with its axis of symmetry vertical and its vertex V uppermost. The semivertical angle of the cone is \(36 ^ { \circ }\), and its surface is smooth. A particle Q of mass 0.2 kg is connected to V by a light inextensible string, and Q moves in a horizontal circle at constant speed, in contact with the surface of the cone, as shown in Fig. 2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5ecb198d-7863-4fc2-81b6-c8b6c37b1859-3_455_609_950_808} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The particle Q makes one complete revolution in 1.8 s , and the normal reaction of the cone on Q has magnitude 0.75 N .
   1. Find the tension in the string.
   2. Find the length of the string.

3 Fixed points A and B are 4.8 m apart on the same horizontal level. The midpoint of AB is M . A light elastic string, with natural length 3.9 m and modulus of elasticity 573.3 N , has one end attached to A and the other end attached to \(\mathbf { B }\).

1. Find the elastic energy stored in the string. A particle P is attached to the midpoint of the string, and is released from rest at M . It comes instantaneously to rest when P is 1.8 m vertically below M .
2. Show that the mass of P is 15 kg .
3. Verify that P can rest in equilibrium when it is 1.0 m vertically below M . In general, a light elastic string, with natural length \(a\) and modulus of elasticity \(\lambda\), has its ends attached to fixed points which are a distance \(d\) apart on the same horizontal level. A particle of mass \(m\) is attached to the midpoint of the string, and in the equilibrium position each half of the string has length \(h\), as shown in Fig. 3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5ecb198d-7863-4fc2-81b6-c8b6c37b1859-4_280_755_1064_696} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} When the particle makes small oscillations in a vertical line, the period of oscillation is given by the formula $$\sqrt { \frac { 8 \pi ^ { 2 } h ^ { 3 } } { 8 h ^ { 3 } - a d ^ { 2 } } } m ^ { \alpha } a ^ { \beta } \lambda ^ { \gamma }$$
4. Show that \(\frac { 8 \pi ^ { 2 } h ^ { 3 } } { 8 h ^ { 3 } - a d ^ { 2 } }\) is dimensionless.
5. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
6. Hence find the period when the particle P makes small oscillations in a vertical line centred on the position of equilibrium given in part (iii).

4 The region \(A\) is bounded by the curve \(y = x ^ { 2 } + 5\) for \(0 \leqslant x \leqslant 3\), the \(x\)-axis, the \(y\)-axis and the line \(x = 3\). The region \(B\) is bounded by the curve \(y = x ^ { 2 } + 5\) for \(0 \leqslant x \leqslant 3\), the \(y\)-axis and the line \(y = 14\). These regions are shown in Fig. 4. \begin{figure}[h] \end{figure}

1. Find the coordinates of the centre of mass of a uniform lamina occupying the region \(A\).
2. The region \(B\) is rotated through \(2 \pi\) radians about the \(y\)-axis to form a uniform solid of revolution \(R\). Find the \(y\)-coordinate of the centre of mass of the solid \(R\).
3. The region \(A\) is rotated through \(2 \pi\) radians about the \(y\)-axis to form a uniform solid of revolution \(S\). Using your answer to part (ii), or otherwise, find the \(y\)-coordinate of the centre of mass of the solid \(S\).