Edexcel M3 (Mechanics 3) 2009 January

1. A particle \(P\) of mass 3 kg is moving in a straight line. At time \(t\) seconds, \(0 \leqslant t \leqslant 4\), the only force acting on \(P\) is a resistance to motion of magnitude \(\left( 9 + \frac { 15 } { ( t + 1 ) ^ { 2 } } \right) \mathrm { N }\). At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(t = 4 , v = 0\).

Find the value of \(v\) when \(t = 0\).

2. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(3 m g\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held in equilibrium by a horizontal force of magnitude \(\frac { 4 } { 3 } m g\) applied to \(P\). This force acts in the vertical plane containing the string, as shown in Figure 1. Find (a) the tension in the string,
(b) the elastic energy stored in the string.

3. A rough disc rotates about its centre in a horizontal plane with constant angular speed 80 revolutions per minute. A particle \(P\) lies on the disc at a distance 8 cm from the centre of the disc. The coefficient of friction between \(P\) and the disc is \(\mu\). Given that \(P\) remains at rest relative to the disc, find the least possible value of \(\mu\).

4. A small shellfish is attached to a wall in a harbour. The rise and fall of the water level is modelled as simple harmonic motion and the shellfish as a particle. On a particular day the minimum depth of water occurs at 1000 hours and the next time that this minimum depth occurs is at 2230 hours. The shellfish is fixed in a position 5 m above the level of the minimum depth of the water and 11 m below the level of the maximum depth of the water. Find

1. the speed, in metres per hour, at which the water level is rising when it reaches the shellfish,
2. the earliest time after 1000 hours on this day at which the water reaches the shellfish.

5. \begin{figure}[h] \end{figure} One end \(A\) of a light elastic string, of natural length \(a\) and modulus of elasticity \(6 m g\), is fixed at a point on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. A small ball \(B\) of mass \(m\) is attached to the other end of the string. Initially \(B\) is held at rest with the string lying along a line of greatest slope of the plane, with \(B\) below \(A\) and \(A B = a\). The ball is released and comes to instantaneous rest at a point \(C\) on the plane, as shown in Figure 2. Find

1. the length \(A C\),
2. the greatest speed attained by \(B\) as it moves from its initial position to \(C\).

6. \begin{figure}[h] \end{figure} The region \(R\) is bounded by part of the curve with equation \(y = 4 - x ^ { 2 }\), the positive \(x\)-axis and the positive \(y\)-axis, as shown in Figure 3. The unit of length on both axes is one metre. A uniform solid \(S\) is formed by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis.

1. Show that the centre of mass of \(S\) is \(\frac { 5 } { 8 } \mathrm {~m}\) from \(O\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8374fa0f-cb28-497f-8696-877d7d0762f1-09_702_584_1138_676} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} Figure 4 shows a cross section of a uniform solid \(P\) consisting of two components, a solid cylinder \(C\) and the solid \(S\). The cylinder \(C\) has radius 4 m and length \(l\) metres. One end of \(C\) coincides with the plane circular face of \(S\). The point \(A\) is on the circumference of the circular face common to \(C\) and \(S\). When the solid \(P\) is freely suspended from \(A\), the solid \(P\) hangs with its axis of symmetry horizontal.
2. Find the value of \(l\).

7. \begin{figure}[h] \end{figure} A particle is projected from the highest point \(A\) on the outer surface of a fixed smooth sphere of radius \(a\) and centre \(O\). The lowest point \(B\) of the sphere is fixed to a horizontal plane. The particle is projected horizontally from \(A\) with speed \(\frac { 1 } { 2 } \sqrt { } ( g a )\). The particle leaves the surface of the sphere at the point \(C\), where \(\angle A O C = \theta\), and strikes the plane at the point \(P\), as shown in Figure 5.

1. Show that \(\cos \theta = \frac { 3 } { 4 }\).
2. Find the angle that the velocity of the particle makes with the horizontal as it reaches \(P\).