Edexcel M3 (Mechanics 3) 2016 June

1. A particle \(P\) of mass 0.5 kg is moving along the positive \(x\)-axis under the action of a resultant force. The force acts along the \(x\)-axis. At time \(t\) seconds, \(P\) is \(x\) metres from the origin \(O\) and is moving away from \(O\) in the positive \(x\) direction with speed \(\frac { 12 } { x + 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
   1. Find the magnitude of the force acting on \(P\) when \(x = 3\)
   Given that \(x = 4\) when \(t = 2\)
2. find the value of \(t\) when \(x = 10\)

2. \begin{figure}[h] \end{figure} Figure 1 shows a uniform triangular lamina \(A B C\) in which \(A B = 6 \mathrm {~cm} , B C = 9 \mathrm {~cm}\) and angle \(A B C = 90 ^ { \circ }\). The centre of mass of the lamina is \(G\). Use algebraic integration to find the distance of \(G\) from \(A B\).
(6)

3. One end of a light elastic string, of natural length 1.5 m and modulus of elasticity 14.7 N ,
3. One end of a light elastic string, of natural length 1.5 m and modulus of elasticity 14.7 N , is attached to a fixed point \(O\) on a ceiling. A particle \(P\) of mass 0.6 kg is attached to the free end of the string. The particle is held at \(O\) and released from rest. The particle comes to instantaneous rest for the first time at the point \(A\). Find

1. the distance \(O A\),
2. the magnitude of the instantaneous acceleration of \(P\) at \(A\).
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4c1c51ff-6ae8-402d-b303-b656d26e4230-05_620_956_118_500} \captionsetup{labelformat=empty} \caption{igure 2}
   \end{figure} A uniform solid \(S\) consists of two right circular cones of base radius \(r\). The smaller cone has height \(2 h\) and the centre of the plane face of this cone is \(O\). The larger cone has height \(k h\) where \(k > 2\). The two cones are joined so that their plane faces coincide, as shown in Figure 2.
3. Show that the distance of the centre of mass of \(S\) from \(O\) is $$\frac { h } { 4 } ( k - 2 )$$ The point \(A\) lies on the circumference of the base of one of the cones. The solid is suspended by a string attached at \(A\) and hangs freely in equilibrium. Given that \(r = 3 h\) and \(k = 6\)
4. find the size of the angle between \(A O\) and the vertical.

4. A uniform solid \(S\) consists of two right circular cones of base rate
has height \(2 h\) and the centre of the plane face of this cone is \(O\). T
\(k h\) where \(k > 2\). The two cones are joined so that their plane fac
Figure 2 .

1. Show that the distance of the centre of mass of \(S\) from \(O\) is
   \(\frac { h } { 4 } ( k - 2 )\)
   

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to the ends of two light inextensible strings. The other ends of the strings are attached to fixed points \(A\) and \(B\), where \(B\) is vertically below \(A\) and \(A B = l\). The particle is moving with constant angular speed \(\omega\) in a horizontal circle. Both strings are taut and inclined at \(30 ^ { \circ }\) to \(A B\), as shown in Figure 3.

1. 1. Show that the tension in \(A P\) is \(\frac { m \sqrt { 3 } } { 6 } \left( 2 g + l \omega ^ { 2 } \right)\)
   2. Find the tension in \(B P\).
2. Show that the time taken by \(P\) to complete one revolution is less than \(\pi \sqrt { \frac { 2 l } { g } }\)

6. One end of a light inextensible string of length \(l\) is attached to a particle \(P\) of mass \(2 m\). The other end of the string is attached to a fixed point \(A\). The particle is hanging freely at rest with the string vertical. The particle is then projected horizontally with speed \(\sqrt { \frac { 7 g l } { 2 } }\) (a) Find the speed of \(P\) at the instant when the string is horizontal.
(4) When the string is horizontal and \(P\) is moving upwards, the string comes into contact with a small smooth peg which is fixed at the point \(B\), where \(A B\) is horizontal and \(A B < l\). The particle then describes a complete semicircle with centre \(B\).
(b) Show that \(A B \geqslant \frac { 1 } { 2 } l\)

7. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic spring, of natural length 1.2 m and modulus of elasticity 15 N . The other end of the spring is attached to a fixed point \(A\) on a smooth horizontal table. The particle is placed on the table at the point \(B\) where \(A B = 1.2 \mathrm {~m}\). The particle is pulled away from \(B\) to the point \(C\), where \(A B C\) is a straight line and \(B C = 0.8 \mathrm {~m}\), and is then released from rest.

1. 1. Show that \(P\) moves with simple harmonic motion with centre \(B\).
   2. Find the period of this motion.
2. Find the speed of \(P\) when it reaches \(B\). The point \(D\) is the midpoint of \(A B\).
3. Find the time taken for \(P\) to move directly from \(C\) to \(D\). When \(P\) first comes to instantaneous rest a particle \(Q\) of mass 0.3 kg is placed at \(B\). When \(P\) reaches \(B\) again, \(P\) strikes and adheres to \(Q\) to form a single particle \(R\).
4. Show that \(R\) also moves with simple harmonic motion.
5. Find the amplitude of this motion.