Edexcel M3 (Mechanics 3) 2011 January

1. A particle \(P\) moves on the positive \(x\)-axis. When the distance of \(P\) from the origin \(O\) is \(x\) metres, the acceleration of \(P\) is \(( 7 - 2 x ) \mathrm { m } \mathrm { s } ^ { - 2 }\), measured in the positive \(x\)-direction. When \(t = 0 , P\) is at \(O\) and is moving in the positive \(x\)-direction with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance of \(P\) from \(O\) when \(P\) first comes to instantaneous rest.
   (6)

\begin{figure}[h] \end{figure} A toy is formed by joining a uniform solid hemisphere, of radius \(r\) and mass \(4 m\), to a uniform right circular solid cone of mass \(k m\). The cone has vertex \(A\), base radius \(r\) and height \(2 r\). The plane face of the cone coincides with the plane face of the hemisphere. The centre of the plane face of the hemisphere is \(O\) and \(O B\) is a radius of its plane face as shown in Figure 1. The centre of mass of the toy is at \(O\).

1. Find the value of \(k\). A metal stud of mass \(\lambda m\) is attached to the toy at \(A\). The toy is now suspended by a light string attached to \(B\) and hangs freely at rest. The angle between \(O B\) and the vertical is \(30 ^ { \circ }\).
2. Find the value of \(\lambda\).

3. \begin{figure}[h] \end{figure} The region \(R\) is bounded by the curve with equation \(y = \mathrm { e } ^ { x }\), the line \(x = 1\), the line \(x = 2\) and the \(x\)-axis as shown in Figure 2. A uniform solid \(S\) is formed by rotating \(R\) through \(2 \pi\) about the \(x\)-axis.

1. Show that the volume of \(S\) is \(\frac { 1 } { 2 } \pi \left( \mathrm { e } ^ { 4 } - \mathrm { e } ^ { 2 } \right)\).
2. Find, to 3 significant figures, the \(x\)-coordinate of the centre of mass of \(S\).

1. A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds its displacement, \(x\) metres, from the origin \(O\) is given by \(x = 5 \sin \left( \frac { 1 } { 3 } \pi t \right)\).
   1. Prove that \(P\) is moving with simple harmonic motion.
   2. Find the period and the amplitude of the motion.
   3. Find the maximum speed of \(P\).
   The points \(A\) and \(B\) on the positive \(x\)-axis are such that \(O A = 2 \mathrm {~m}\) and \(O B = 3 \mathrm {~m}\).
2. Find the time taken by \(P\) to travel directly from \(A\) to \(B\).

5. \begin{figure}[h] \end{figure} A small ball \(P\) of mass \(m\) is attached to the ends of two light inextensible strings of length \(l\). The other ends of the strings are attached to fixed points \(A\) and \(B\), where \(A\) is vertically above \(B\). Both strings are taut and \(A P\) is perpendicular to \(B P\) as shown in Figure 3. The system rotates about the line \(A B\) with constant angular speed \(\omega\). The ball moves in a horizontal circle.

1. Find, in terms of \(m , g , l\) and \(\omega\), the tension in \(A P\) and the tension in \(B P\).
2. Show that \(\omega ^ { 2 } > \frac { g \sqrt { } 2 } { l }\).

6. \begin{figure}[h] \end{figure} A small ball of mass \(3 m\) is attached to the ends of two light elastic strings \(A P\) and \(B P\), each of natural length \(l\) and modulus of elasticity \(k m g\). The ends \(A\) and \(B\) of the strings are attached to fixed points on the same horizontal level, with \(A B = 2 l\). The mid-point of \(A B\) is \(C\). The ball hangs in equilibrium at a distance \(\frac { 3 } { 4 } l\) vertically below \(C\) as shown in Figure 4.

1. Show that \(k = 10\) The ball is now pulled vertically downwards until it is at a distance \(\frac { 12 } { 5 } l\) below \(C\). The ball is released from rest.
2. Find the speed of the ball as it reaches \(C\).

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light rod of length \(l\). The other end of the rod is attached to a fixed point \(O\). The rod can turn freely in a vertical plane about \(O\). The particle is projected with speed \(u\) from a point \(A\), where \(O A\) makes an angle \(\alpha\) with the upward vertical through \(O\) and \(0 < \alpha < \frac { \pi } { 2 }\). When \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\) the speed of \(P\) is \(v\) as shown in Figure 5.

1. Show that \(v ^ { 2 } = u ^ { 2 } + 2 g l ( \cos \alpha - \cos \theta )\). It is given that \(\cos \alpha = \frac { 3 } { 5 }\) and that \(P\) moves in a complete vertical circle.
2. Show that \(u > 2 \sqrt { } \left( \frac { g l } { 5 } \right)\). As the rod rotates the least tension in the rod is \(T\) and the greatest tension is \(5 T\).
3. Show that \(u ^ { 2 } = \frac { 33 } { 10 } \mathrm { gl }\).