OCR MEI M3 (Mechanics 3) 2015 June

2

1. A particle P of mass \(m\) is attached to a fixed point O by a light inextensible string of length \(a\). P is moving without resistance in a complete vertical circle with centre O and radius \(a\). When P is at the highest point of the circle, the tension in the string is \(T _ { 1 }\). When OP makes an angle \(\theta\) with the upward vertical, the tension in the string is \(T _ { 2 }\). Show that $$T _ { 2 } = T _ { 1 } + 3 m g ( 1 - \cos \theta ) .$$
2. The fixed point A is 1.2 m vertically above the fixed point C . A particle Q of mass 0.9 kg is joined to A , to C , and to a particle R of mass 1.5 kg , by three light inextensible strings of lengths \(1.3 \mathrm {~m} , 0.5 \mathrm {~m}\) and 1.8 m respectively. The particle Q moves in a horizontal circle with centre C , and R moves in a horizontal circle at the same constant angular speed as Q , in such a way that \(\mathrm { A } , \mathrm { C } , \mathrm { Q }\) and R are always coplanar. The string QR makes an angle of \(60 ^ { \circ }\) with the downward vertical. This situation is shown in Fig. 2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{70a2c3ce-7bdb-4ddd-92fc-f7dcbdfdcfaf-3_579_1191_881_406} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Find the tensions in the strings QR and AQ .
   2. Find the angular speed of the system.
   3. Find the tension in the string CQ .

3 Fig. 3 shows the fixed points A and F which are 9.5 m apart on a smooth horizontal surface and points B and D on the line AF such that \(\mathrm { AB } = \mathrm { DF } = 3.0 \mathrm {~m}\). A small block of mass 10.5 kg is joined to A by a light elastic string of natural length 3.0 m and stiffness \(12 \mathrm { Nm } ^ { - 1 }\); the block is joined to F by a light elastic string of natural length 3.0 m and stiffness \(30 \mathrm { Nm } ^ { - 1 }\). The block is released from rest at B and then slides along part of the line AF . The block has zero acceleration when it is at a point C , and it comes to instantaneous rest at a point E . \begin{figure}[h] \end{figure}

1. Find the distance BC . At time \(t \mathrm {~s}\) the displacement of the block from C is \(x \mathrm {~m}\), measured in the direction AF .
2. Show that, when the block is between B and \(\mathrm { D } , \frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 4 x\).
3. Find the maximum speed of the block.
4. Find the distance of the block from C when its speed is \(4.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
5. Find the time taken for the block to travel from B to D.
6. Find the distance DE .

4

1. A uniform lamina occupies the region bounded by the \(x\)-axis and the curve \(y = \frac { x ^ { 2 } ( a - x ) } { a ^ { 2 } }\) for \(0 \leqslant x \leqslant a\). Find the coordinates of the centre of mass of this lamina.
2. The region \(A\) is bounded by the \(x\)-axis, the \(y\)-axis, the curve \(y = \sqrt { x ^ { 2 } + 16 }\) and the line \(x = 3\). The region \(B\) is bounded by the \(y\)-axis, the curve \(y = \sqrt { x ^ { 2 } + 16 }\) and the line \(y = 5\). These regions are shown in Fig. 4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{70a2c3ce-7bdb-4ddd-92fc-f7dcbdfdcfaf-5_604_460_605_792} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
   1. Find the \(x\)-coordinate of the centre of mass of the uniform solid of revolution formed when the region \(A\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
   2. Using your answer to part (i), or otherwise, find the \(x\)-coordinate of the centre of mass of the uniform solid of revolution formed when the region \(B\) is rotated through \(2 \pi\) radians about the \(x\)-axis. \section*{END OF QUESTION PAPER}