OCR MEI M3 (Mechanics 3) 2013 June

1

1. A particle P of mass 1.5 kg is connected to a fixed point by a light inextensible string of length 3.2 m . The particle P is moving as a conical pendulum in a horizontal circle at a constant angular speed of \(2.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
   1. Find the tension in the string.
   2. Find the angle that the string makes with the vertical.
2. A particle Q of mass \(m\) moves on a smooth horizontal surface, and is connected to a fixed point on the surface by a light elastic string of natural length \(d\) and stiffness \(k\). With the string at its natural length, Q is set in motion with initial speed \(u\) perpendicular to the string. In the subsequent motion, the maximum length of the string is \(2 d\), and the string first returns to its natural length after time \(t\). You are given that \(u = \sqrt { \frac { 4 k d ^ { 2 } } { 3 m } }\) and \(t = A k ^ { \alpha } d ^ { \beta } m ^ { \gamma }\), where \(A\) is a dimensionless constant.
   1. Show that the dimensions of \(k\) are \(\mathrm { MT } ^ { - 2 }\).
   2. Show that the equation \(u = \sqrt { \frac { 4 k d ^ { 2 } } { 3 m } }\) is dimensionally consistent.
   3. Find \(\alpha , \beta\) and \(\gamma\). You are now given that Q has mass 5 kg , and the string has natural length 0.7 m and stiffness \(60 \mathrm { Nm } ^ { - 1 }\).
   4. Find the initial speed \(u\), and use conservation of energy to find the speed of Q at the instant when the length of the string is double its natural length.

2 A particle P of mass 0.25 kg is connected to a fixed point O by a light inextensible string of length \(a\) metres, and is moving in a vertical circle with centre O and radius \(a\) metres. When P is vertically below O , its speed is \(8.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When OP makes an angle \(\theta\) with the downward vertical, and the string is still taut, P has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the tension in the string is \(T \mathrm {~N}\), as shown in Fig. 2. \begin{figure}[h] \end{figure}

1. Find an expression for \(v ^ { 2 }\) in terms of \(a\) and \(\theta\), and show that $$T = \frac { 17.64 } { a } + 7.35 \cos \theta - 4.9 .$$
2. Given that \(a = 0.9\), show that P moves in a complete circle. Find the maximum and minimum magnitudes of the tension in the string.
3. Find the largest value of \(a\) for which P moves in a complete circle.
4. Given that \(a = 1.6\), find the speed of P at the instant when the string first becomes slack.

3 A light spring, with modulus of elasticity 686 N , has one end attached to a fixed point A . The other end is attached to a particle P of mass 18 kg which hangs in equilibrium when it is 2.2 m vertically below A .

1. Find the natural length of the spring AP . Another light spring has natural length 2.5 m and modulus of elasticity 145 N . One end of this spring is now attached to the particle P , and the other end is attached to a fixed point B which is 2.5 m vertically below P (so leaving the equilibrium position of P unchanged). While in its equilibrium position, P is set in motion with initial velocity \(3.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards, as shown in Fig. 3. It now executes simple harmonic motion along part of the vertical line AB . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-4_721_383_726_831} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} At time \(t\) seconds after it is set in motion, P is \(x\) metres below its equilibrium position.
2. Show that the tension in the spring AP is \(( 176.4 + 392 x ) \mathrm { N }\), and write down an expression for the thrust in the spring BP.
3. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 25 x\).
4. Find the period and the amplitude of the motion.
5. Find the magnitude and direction of the velocity of P when \(t = 2.4\).
6. Find the total distance travelled by P during the first 2.4 seconds of its motion.

4

1. A uniform solid of revolution \(S\) is formed by rotating the region enclosed between the \(x\)-axis and the curve \(y = x \sqrt { 4 - x }\) for \(0 \leqslant x \leqslant 4\) through \(2 \pi\) radians about the \(x\)-axis, as shown in Fig. 4.1. O is the origin and the end A of the solid is at the point \(( 4,0 )\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-5_520_625_408_703} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure}
   1. Find the \(x\)-coordinate of the centre of mass of the solid \(S\). The solid \(S\) has weight \(W\), and it is freely hinged to a fixed point at O . A horizontal force, of magnitude \(W\) acting in the vertical plane containing OA , is applied at the point A , as shown in Fig. 4.2. \(S\) is in equilibrium. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-5_346_512_1361_781} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
      \end{figure}
   2. Find the angle that OA makes with the vertical.
      [0pt] [Question 4(b) is printed overleaf]
2. Fig. 4.3 shows the region bounded by the \(x\)-axis, the \(y\)-axis, the line \(y = 8\) and the curve \(y = ( x - 2 ) ^ { 3 }\) for \(0 \leqslant y \leqslant 8\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-6_631_695_370_683} \captionsetup{labelformat=empty} \caption{Fig. 4.3}
   \end{figure} Find the coordinates of the centre of mass of a uniform lamina occupying this region.