Edexcel M3 (Mechanics 3) 2007 June

1. The rudder on a ship is modelled as a uniform plane lamina having the same shape as the region \(R\) which is enclosed between the curve with equation \(y = 2 x - x ^ { 2 }\) and the \(x\)-axis.
   1. Show that the area of \(R\) is \(\frac { 4 } { 3 }\).
   2. Find the coordinates of the centre of mass of the lamina.
   3. An open container \(C\) is modelled as a thin uniform hollow cylinder of radius \(h\) and height \(h\) with a base but no lid. The centre of the base is \(O\).
   4. Show that the distance of the centre of mass of \(C\) from \(O\) is \(\frac { 1 } { 3 } h\).
   The container is filled with uniform liquid. Given that the mass of the container is \(M\) and the mass of the liquid is \(M\),
2. find the distance of the centre of mass of the filled container from \(O\).

3. A spacecraft \(S\) of mass \(m\) moves in a straight line towards the centre of the earth. The earth is modelled as a fixed sphere of radius \(R\). When \(S\) is at a distance \(x\) from the centre of the earth, the force exerted by the earth on \(S\) is directed towards the centre of the earth and has magnitude \(\frac { k } { x ^ { 2 } }\), where \(k\) is a constant.

1. Show that \(k = m g R ^ { 2 }\). Given that \(S\) starts from rest when its distance from the centre of the earth is \(2 R\), and that air resistance can be ignored,
2. find the speed of \(S\) as it crashes into the surface of the earth.

4. A light inextensible string of length \(l\) has one end attached to a fixed point \(A\). The other end is attached to a particle \(P\) of mass \(m\). The particle moves with constant speed \(v\) in a horizontal circle with the string taut. The centre of the circle is vertically below \(A\) and the radius of the circle is \(r\). Show that $$g r ^ { 2 } = v ^ { 2 } \sqrt { } \left( l ^ { 2 } - r ^ { 2 } \right) .$$

1. A particle \(P\) moves on the \(x\)-axis with simple harmonic motion about the origin \(O\) as centre. When \(P\) is a distance 0.04 m from \(O\), its speed is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the magnitude of its acceleration is \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   1. Find the period of the motion.
   The amplitude of the motion is \(a\) metres. Find
2. the value of \(a\),
3. the total time, within one complete oscillation, for which the distance \(O P\) is greater than \(\frac { 1 } { 2 } a\) metres.

6. A particle \(P\) is free to move on the smooth inner surface of a fixed thin hollow sphere of internal radius \(a\) and centre \(O\). The particle passes through the lowest point of the spherical surface with speed \(U\). The particle loses contact with the surface when \(O P\) is inclined at an angle \(\alpha\) to the upward vertical.

1. Show that \(\quad U ^ { 2 } = a g ( 2 + 3 \cos \alpha )\). The particle has speed \(W\) as it passes through the level of \(O\). Given that \(\cos \alpha = \frac { 1 } { \sqrt { } 3 }\), (b) show that \(\quad W ^ { 2 } = a g \sqrt { } 3\).

7. \begin{figure}[h] \end{figure} A light elastic string, of natural length \(3 l\) and modulus of elasticity \(\lambda\), has its ends attached to two points \(A\) and \(B\), where \(A B = 3 l\) and \(A B\) is horizontal. A particle \(P\) of mass \(m\) is attached to the mid-point of the string. Given that \(P\) rests in equilibrium at a distance \(2 l\) below \(A B\), as shown in Figure 1,

1. show that \(\lambda = \frac { 15 m g } { 16 }\). The particle is pulled vertically downwards from its equilibrium position until the total length of the elastic string is \(7.8 l\). The particle is released from rest.
2. Show that \(P\) comes to instantaneous rest on the line \(A B\).