OCR MEI M3 (Mechanics 3) 2013 January

1

1. A particle P is executing simple harmonic motion, and the centre of the oscillations is at the point O . The maximum speed of P during the motion is \(5.1 \mathrm {~ms} ^ { - 1 }\). When P is 6 m from O , its speed is \(4.5 \mathrm {~ms} ^ { - 1 }\). Find the period and the amplitude of the motion.
2. The force \(F\) of gravitational attraction between two objects of masses \(m _ { 1 }\) and \(m _ { 2 }\) at a distance \(d\) apart is given by \(F = \frac { G m _ { 1 } m _ { 2 } } { d ^ { 2 } }\), where \(G\) is the universal gravitational constant.
   1. Find the dimensions of \(G\). Three objects, each of mass \(m\), are moving in deep space under mutual gravitational attraction. They move round a single circle with constant angular speed \(\omega\), and are always at the three vertices of an equilateral triangle of side \(R\). You are given that \(\omega = k G ^ { \alpha } m ^ { \beta } R ^ { \gamma }\), where \(k\) is a dimensionless constant.
   2. Find \(\alpha , \beta\) and \(\gamma\). For three objects of mass 2500 kg at the vertices of an equilateral triangle of side 50 m , the angular speed is \(2.0 \times 10 ^ { - 6 } \mathrm { rad } \mathrm { s } ^ { - 1 }\).
   3. Find the angular speed for three objects of mass \(4.86 \times 10 ^ { 14 } \mathrm {~kg}\) at the vertices of an equilateral triangle of side 30000 m .

2

1. A fixed solid sphere with a smooth surface has centre O and radius 0.8 m . A particle P is given a horizontal velocity of \(1.2 \mathrm {~ms} ^ { - 1 }\) at the highest point on the sphere, and it moves on the surface of the sphere in part of a vertical circle of radius 0.8 m .
   1. Find the radial and tangential components of the acceleration of P at the instant when OP makes an angle \(\frac { 1 } { 6 } \pi\) radians with the upward vertical. (You may assume that P is still in contact with the sphere.)
   2. Find the speed of P at the instant when it leaves the surface of the sphere.
2. Two fixed points R and S are 2.5 m apart with S vertically below R . A particle Q of mass 0.9 kg is connected to R and to S by two light inextensible strings; Q is moving in a horizontal circle at a constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with both strings taut. The radius of the circle is 2.4 m and the centre C of the circle is 0.7 m vertically below S, as shown in Fig. 2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3f674569-7e99-4ba8-84f1-a1eb438e30ed-2_547_720_1946_644} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Find the tension in the string RQ and the tension in the string \(S Q\).

3 Two fixed points X and Y are 14.4 m apart and XY is horizontal. The midpoint of XY is M . A particle P is connected to X and to Y by two light elastic strings. Each string has natural length 6.4 m and modulus of elasticity 728 N . The particle P is in equilibrium when it is 3 m vertically below M, as shown in Fig. 3. \begin{figure}[h] \end{figure}

1. Find the tension in each string when P is in the equilibrium position.
2. Show that the mass of P is 12.5 kg . The particle P is released from rest at M , and moves in a vertical line.
3. Find the acceleration of P when it is 2.1 m vertically below M .
4. Explain why the maximum speed of P occurs at the equilibrium position.
5. Find the maximum speed of P .

4

1. The region enclosed between the curve \(y = x ^ { 4 }\) and the line \(y = h\) (where \(h\) is positive) is rotated about the \(y\)-axis to form a uniform solid of revolution. Find the \(y\)-coordinate of the centre of mass of this solid.
2. The region \(A\) is bounded by the \(x\)-axis, the curve \(y = x + \sqrt { x }\) for \(0 \leqslant x \leqslant 4\), and the line \(x = 4\). The region \(B\) is bounded by the \(y\)-axis, the curve \(y = x + \sqrt { x }\) for \(0 \leqslant x \leqslant 4\), and the line \(y = 6\). These regions are shown in Fig. 4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3f674569-7e99-4ba8-84f1-a1eb438e30ed-3_572_513_1779_778} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
   1. A uniform lamina occupies the region \(A\). Show that the \(x\)-coordinate of the centre of mass of this lamina is 2.56 , and find the \(y\)-coordinate.
   2. Using your answer to part (i), or otherwise, find the coordinates of the centre of mass of a uniform lamina occupying the region \(B\).