Edexcel M3 (Mechanics 3) 2013 January

1. A particle \(P\) is moving along the positive \(x\)-axis. When the displacement of \(P\) from the origin is \(x\) metres, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the acceleration of \(P\) is \(9 x \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

When \(x = 2 , v = 6\)
Show that \(v ^ { 2 } = 9 x ^ { 2 }\).
(4)

2. \begin{figure}[h] \end{figure} A uniform solid consists of a right circular cone of radius \(r\) and height \(k r\), where \(k > \sqrt { } 3\), fixed to a hemisphere of radius \(r\). The centre of the plane face of the hemisphere is \(O\) and this plane face coincides with the base of the cone, as shown in Figure 1.

1. Show that the distance of the centre of mass of the solid from \(O\) is $$\frac { \left( k ^ { 2 } - 3 \right) r } { 4 ( k + 2 ) }$$ The point \(A\) lies on the circumference of the base of the cone. The solid is suspended by a string attached at \(A\) and hangs freely in equilibrium. The angle between \(A O\) and the vertical is \(\theta\), where \(\tan \theta = \frac { 11 } { 14 }\)
2. Find the value of \(k\).

1. A particle \(P\) of mass 0.6 kg is moving along the \(x\)-axis in the positive direction. At time \(t = 0 , P\) passes through the origin \(O\) with speed \(15 \mathrm {~ms} ^ { - 1 }\). At time \(t\) seconds the distance \(O P\) is \(x\) metres, the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the resultant force acting on \(P\) has magnitude \(\frac { 12 } { ( t + 2 ) ^ { 2 } }\) newtons. The resultant force is directed towards \(O\).
   1. Show that \(v = 5 \left( \frac { 4 } { t + 2 } + 1 \right)\).
   2. Find the value of \(x\) when \(t = 5\)
      

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(2 a\) and modulus of elasticity \(6 m g\). The other end of the string is attached to a fixed point \(A\). The particle moves with constant speed \(v\) in a horizontal circle with centre \(O\), where \(O\) is vertically below \(A\) and \(O A = 2 a\), as shown in Figure 2 .

1. Show that the extension in the string is \(\frac { 2 } { 5 } a\).
2. Find \(v ^ { 2 }\) in terms of \(a\) and \(g\).

5. A particle \(P\) is moving in a straight line with simple harmonic motion on a smooth horizontal floor. The particle comes to instantaneous rest at points \(A\) and \(B\) where \(A B\) is 0.5 m . The mid-point of \(A B\) is \(O\). The mid-point of \(O A\) is \(C\). The mid-point of \(O B\) is \(D\). The particle takes 0.2 s to travel directly from \(C\) to \(D\). At time \(t = 0 , P\) is moving through \(O\) towards \(A\).

1. Show that the period of the motion is \(\frac { 6 } { 5 } \mathrm {~s}\).
2. Find the distance of \(P\) from \(B\) when \(t = 2 \mathrm {~s}\).
3. Find the maximum magnitude of the acceleration of \(P\).
4. Find the maximum speed of \(P\).

6. \begin{figure}[h] \end{figure} A smooth hollow cylinder of internal radius \(a\) is fixed with its axis horizontal. A particle \(P\) moves on the inner surface of the cylinder in a vertical circle with radius \(a\) and centre \(O\), where \(O\) lies on the axis of the cylinder. The particle is projected vertically downwards with speed \(u\) from point \(A\) on the circle, where \(O A\) is horizontal. The particle first loses contact with the cylinder at the point \(B\), where \(\angle A O B = 150 ^ { \circ }\), as shown in Figure 3. Given that air resistance can be ignored,

1. show that the speed of \(P\) at \(B\) is \(\sqrt { } \left( \frac { a g } { 2 } \right)\),
2. find \(u\) in terms of \(a\) and \(g\). After losing contact with the cylinder, \(P\) crosses the diameter through \(A\) at the point \(D\). At \(D\) the velocity of \(P\) makes an angle \(\theta ^ { \circ }\) with the horizontal.
3. Find the value of \(\theta\).

7. A particle \(P\) of mass 1.5 kg is attached to the mid-point of a light elastic string of natural length 0.30 m and modulus of elasticity \(\lambda\) newtons. The ends of the string are attached to two fixed points \(A\) and \(B\), where \(A B\) is horizontal and \(A B = 0.48 \mathrm {~m}\). Initially \(P\) is held at rest at the mid-point, \(M\), of the line \(A B\) and the tension in the string is 240 N .

1. Show that \(\lambda = 400\) The particle is now held at rest at the point \(C\), where \(C\) is 0.07 m vertically below \(M\). The particle is released from rest at \(C\).
2. Find the magnitude of the initial acceleration of \(P\).
3. Find the speed of \(P\) as it passes through \(M\).