Edexcel M3 (Mechanics 3) 2014 June

1. A particle \(P\) moves in a straight line with simple harmonic motion. The period of the motion is \(\frac { \pi } { 4 }\) seconds. At time \(t = 0 , P\) is at rest at the point \(A\) and the acceleration of \(P\) has magnitude \(20 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

Find

1. the amplitude of the motion,
2. the greatest speed of \(P\) during the motion,
3. the time \(P\) takes to travel a total distance of 1.5 m after it has first left \(A\).

2. \begin{figure}[h] \end{figure} A uniform lamina \(L\) is in the shape of an equilateral triangle of side \(2 a\). The lamina is placed in the \(x y\)-plane with one vertex at the origin \(O\) and an axis of symmetry along the \(x\)-axis, as shown in Figure 1. Use algebraic integration to find the \(x\) coordinate of the centre of mass of \(L\).

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass 3 kg is attached by two light inextensible strings to two fixed points \(A\) and \(B\) on a fixed vertical pole. Both strings are taut and \(P\) is moving in a horizontal circle with constant angular speed \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\). String \(A P\) is inclined at \(30 ^ { \circ }\) to the vertical. String \(B P\) has length 0.4 m and \(A\) is 0.4 m vertically above \(B\), as shown in Figure 2 . Find the tension in

1. \(A P\),
2. \(B P\).

1. At time \(t = 0\), a particle \(P\) of mass 0.4 kg is at the origin \(O\) moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the \(x\)-axis in the positive \(x\) direction. At time \(t\) seconds, \(t \geqslant 0\), the resultant force acting on \(P\) has magnitude \(\frac { 4 } { ( t + 5 ) ^ { 2 } } \mathrm {~N}\) and is directed away from \(O\).
   1. Show that the speed of \(P\) cannot exceed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   The particle passes through the point \(A\) when \(t = 2\) and passes through the point \(B\) when \(t = 7\)
2. Find the distance \(A B\).
3. Find the gain in kinetic energy of \(P\) as it moves from \(A\) to \(B\).

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(2 m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). Initially the particle is at the point \(A\) where \(O A = a\) and \(O A\) makes an angle \(60 ^ { \circ }\) with the downward vertical. The particle is projected downwards from \(A\) with speed \(u\) in a direction perpendicular to the string, as shown in Figure 3. The point \(B\) is vertically below \(O\) and \(O B = a\). As \(P\) passes through \(B\) it strikes and adheres to another particle \(Q\) of mass \(m\) which is at rest at \(B\).

1. Show that the speed of the combined particle immediately after the impact is $$\frac { 2 } { 3 } \sqrt { u ^ { 2 } + a g } .$$
2. Find, in terms of \(a , g , m\) and \(u\), the tension in the string immediately after the impact. The combined particle moves in a complete circle.
3. Show that \(u ^ { 2 } \geqslant \frac { 41 a g } { 4 }\).

6. A particle of mass \(m\) is attached to one end of a light elastic string, of natural length \(6 a\) and modulus of elasticity 9 mg . The other end of the string is attached to a fixed point \(A\) on a ceiling. The particle hangs in equilibrium at the point \(B\), where \(B\) is vertically below \(A\) and \(A B = ( 6 + p ) a\).

1. Show that \(p = \frac { 2 } { 3 }\) The particle is now released from rest at a point \(C\) vertically below \(B\), where \(A C < \frac { 22 } { 3 } a\).
2. Show that the particle moves with simple harmonic motion.
3. Find the period of this motion.
4. Explain briefly the significance of the condition \(A C < \frac { 22 } { 3 } a\). The point \(D\) is vertically below \(A\) and \(A D = 8 a\). The particle is now released from rest at \(D\). The particle first comes to instantaneous rest at the point \(E\).
5. Find, in terms of \(a\), the distance \(A E\).

7. \begin{figure}[h] \end{figure} Diagram not drawn to scale A uniform right circular solid cylinder has radius \(3 a\) and height \(2 a\). A right circular cone of height \(\frac { 3 a } { 2 }\) and base radius \(2 a\) is removed from the cylinder to form a solid \(S\), as shown in Figure 4. The plane face of the cone coincides with the upper plane face of the cylinder and the centre \(O\) of the plane face of the cone is also the centre of the upper plane face of the cylinder.

1. Show that the distance of the centre of mass of \(S\) from \(O\) is \(\frac { 69 a } { 64 }\). The point \(A\) is on the open face of \(S\) such that \(O A = 3 a\), as shown in Figure 4. The solid is now suspended from \(A\) and hangs freely in equilibrium.
2. Find the angle between \(O A\) and the horizontal.
   (3) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e5b08946-7311-4cf7-9c5f-5f309a1feed7-13_543_826_1653_557} \captionsetup{labelformat=empty} \caption{Figure 5}
   \end{figure} The solid is now placed on a rough inclined plane with the face through \(A\) in contact with the inclined plane, as shown in Figure 5. The solid rests in equilibrium on this plane. The coefficient of friction between the plane and \(S\) is 0.6 and the plane is inclined at an angle \(\phi ^ { \circ }\) to the horizontal. Given that \(S\) is on the point of sliding down the plane,
3. show that \(\phi = 31\) to 2 significant figures.