OCR MEI M3 (Mechanics 3) 2009 January

2

1. Fig. 2 shows a light inextensible string of length 3.3 m passing through a small smooth ring R of mass 0.27 kg . The ends of the string are attached to fixed points A and B , where A is vertically above \(B\). The ring \(R\) is moving with constant speed in a horizontal circle of radius \(1.2 \mathrm {~m} , \mathrm { AR } = 2.0 \mathrm {~m}\) and \(\mathrm { BR } = 1.3 \mathrm {~m}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b8573ee2-771c-4a93-88d9-346a9da94494-3_570_659_493_781} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Show that the tension in the string is 6.37 N .
   2. Find the speed of R .
2. One end of a light inextensible string of length 1.25 m is attached to a fixed point O . The other end is attached to a particle P of mass 0.2 kg . The particle P is moving in a vertical circle with centre O and radius 1.25 m , and when P is at the highest point of the circle there is no tension in the string.
   1. Show that when P is at the highest point its speed is \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). For the instant when the string OP makes an angle of \(60 ^ { \circ }\) with the upward vertical, find
   2. the radial and tangential components of the acceleration of P ,
   3. the tension in the string.

3 An elastic rope has natural length 25 m and modulus of elasticity 980 N . One end of the rope is attached to a fixed point O , and a rock of mass 5 kg is attached to the other end; the rock is always vertically below O.

1. Find the extension of the rope when the rock is hanging in equilibrium. When the rock is moving with the rope stretched, its displacement is \(x\) metres below the equilibrium position at time \(t\) seconds.
2. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 7.84 x\). The rock is released from a position where the rope is slack, and when the rope just becomes taut the speed of the rock is \(8.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the distance below the equilibrium position at which the rock first comes instantaneously to rest.
4. Find the maximum speed of the rock.
5. Find the time between the rope becoming taut and the rock first coming to rest.
6. State three modelling assumptions you have made in answering this question.

4

1. The region bounded by the \(x\)-axis and the semicircle \(y = \sqrt { a ^ { 2 } - x ^ { 2 } }\) for \(- a \leqslant x \leqslant a\) is occupied by a uniform lamina with area \(\frac { 1 } { 2 } \pi a ^ { 2 }\). Show by integration that the \(y\)-coordinate of the centre of mass of this lamina is \(\frac { 4 a } { 3 \pi }\).
2. A uniform solid cone is formed by rotating the region between the \(x\)-axis and the line \(y = m x\), for \(0 \leqslant x \leqslant h\), through \(2 \pi\) radians about the \(x\)-axis.
   1. Show that the \(x\)-coordinate of the centre of mass of this cone is \(\frac { 3 } { 4 } h\).
      [0pt] [You may use the formula \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) for the volume of a cone.]
      From such a uniform solid cone with radius 0.7 m and height 2.4 m , a cone of material is removed. The cone removed has radius 0.4 m and height 1.1 m ; the centre of its base coincides with the centre of the base of the original cone, and its axis of symmetry is also the axis of symmetry of the original cone. Fig. 4 shows the resulting object; the vertex of the original cone is V, and A is a point on the circumference of its base. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{b8573ee2-771c-4a93-88d9-346a9da94494-5_716_1228_1027_497} \captionsetup{labelformat=empty} \caption{Fig. 4}
      \end{figure}
   2. Find the distance of the centre of mass of this object from V . This object is suspended by a string attached to a point Q on the line VA, and hangs in equilibrium with VA horizontal.
   3. Find the distance VQ.