OCR MEI M3 (Mechanics 3) 2010 June

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1. Two light elastic strings, each having natural length 2.15 m and stiffness \(70 \mathrm {~N} \mathrm {~m} ^ { - 1 }\), are attached to a particle P of mass 4.8 kg . The other ends of the strings are attached to fixed points A and B , which are 1.4 m apart at the same horizontal level. The particle P is placed 2.4 m vertically below the midpoint of AB , as shown in Fig. 1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c93aed95-f655-45cb-805f-7114a15acccf-2_677_474_482_877} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure}
   1. Show that P is in equilibrium in this position.
   2. Find the energy stored in the string AP . Starting in this equilibrium position, P is set in motion with initial velocity \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically upwards. You are given that P first comes to instantaneous rest at a point C where the strings are slack.
   3. Find the vertical height of C above the initial position of P .
2. 1. Write down the dimensions of force and stiffness (of a spring). A particle of mass \(m\) is performing oscillations with amplitude \(a\) on the end of a spring with stiffness \(k\). The maximum speed \(v\) of the particle is given by \(v = c m ^ { \alpha } k ^ { \beta } a ^ { \gamma }\), where \(c\) is a dimensionless constant.
   2. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).

2 A hollow hemisphere has internal radius 2.5 m and is fixed with its rim horizontal and uppermost. The centre of the hemisphere is O . A small ball B of mass 0.4 kg moves in contact with the smooth inside surface of the hemisphere. At first, B is moving at constant speed in a horizontal circle with radius 1.5 m , as shown in Fig. 2.1. \begin{figure}[h] \end{figure}

1. Find the normal reaction of the hemisphere on \(B\).
2. Find the speed of \(\mathbf { B }\). The ball B is now released from rest on the inside surface at a point on the same horizontal level as O . It then moves in part of a vertical circle with centre O and radius 2.5 m , as shown in Fig. 2.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c93aed95-f655-45cb-805f-7114a15acccf-3_378_663_1427_740} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
3. Show that, when \(B\) is at its lowest point, the normal reaction is three times the weight of \(B\). For an instant when the normal reaction is twice the weight of \(\mathbf { B }\), find
4. the speed of \(\mathbf { B }\),
5. the tangential component of the acceleration of \(\mathbf { B }\).

3 In this question, give your answers in an exact form.
The region \(R _ { 1 }\) (shown in Fig. 3) is bounded by the \(x\)-axis, the lines \(x = 1\) and \(x = 5\), and the curve \(y = \frac { 1 } { x }\) for \(1 \leqslant x \leqslant 5\).

1. A uniform solid of revolution is formed by rotating the region \(R _ { 1 }\) through \(2 \pi\) radians about the \(x\)-axis. Find the \(x\)-coordinate of the centre of mass of this solid.
2. Find the coordinates of the centre of mass of a uniform lamina occupying the region \(R _ { 1 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c93aed95-f655-45cb-805f-7114a15acccf-4_849_841_735_651} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} The region \(R _ { 2 }\) is bounded by the \(y\)-axis, the lines \(y = 1\) and \(y = 5\), and the curve \(y = \frac { 1 } { x }\) for \(\frac { 1 } { 5 } \leqslant x \leqslant 1\). The region \(R _ { 3 }\) is the square with vertices \(( 0,0 ) , ( 1,0 ) , ( 1,1 )\) and \(( 0,1 )\).
3. Write down the coordinates of the centre of mass of a uniform lamina occupying the region \(R _ { 2 }\).
4. Find the coordinates of the centre of mass of a uniform lamina occupying the region consisting of \(R _ { 1 } , R _ { 2 }\) and \(R _ { 3 }\) (shown shaded in Fig. 3).

4 A particle P is performing simple harmonic motion in a vertical line. At time \(t \mathrm {~s}\), its displacement \(x \mathrm {~m}\) above a fixed point O is given by $$x = A \sin \omega t + B \cos \omega t$$ where \(A , B\) and \(\omega\) are constants.

1. Show that the acceleration of P , in \(\mathrm { ms } ^ { - 2 }\), is \(- \omega ^ { 2 } x\). When \(t = 0 , \mathrm { P }\) is 16 m below O , moving with velocity \(7.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) upwards, and has acceleration \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) upwards.
2. Find the values of \(A , B\) and \(\omega\).
3. Find the maximum displacement, the maximum speed, and the maximum acceleration of P .
4. Find the speed and the direction of motion of P when \(t = 15\).
5. Find the distance travelled by P between \(t = 0\) and \(t = 15\).