Edexcel M3 (Mechanics 3) 2008 June

1. \begin{figure}[h] \end{figure} A light elastic spring, of natural length \(L\) and modulus of elasticity \(\lambda\), has a particle \(P\) of mass \(m\) attached to one end. The other end of the spring is fixed to a point \(O\) on the closed end of a fixed smooth hollow tube of length \(L\). The tube is placed horizontally and \(P\) is held inside the tube with \(O P = \frac { 1 } { 2 } L\), as shown
in Figure 1. The particle \(P\) is released and passes through the open end of the tube with speed \(\sqrt { } ( 2 g L )\).

1. Show that \(\lambda = 8 \mathrm { mg }\). The tube is now fixed vertically and \(P\) is held inside the tube with \(O P = \frac { 1 } { 2 } L\) and \(P\) above \(O\). The particle \(P\) is released and passes through the open top of the tube with speed \(u\).
2. Find \(u\).

2. A particle \(P\) moves with simple harmonic motion and comes to rest at two points \(A\) and \(B\) which are 0.24 m apart on a horizontal line. The time for \(P\) to travel from \(A\) to \(B\) is 1.5 s . The midpoint of \(A B\) is \(O\). At time \(t = 0 , P\) is moving through \(O\), towards \(A\), with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the value of \(u\).
2. Find the distance of \(P\) from \(B\) when \(t = 2 \mathrm {~s}\).
3. Find the speed of \(P\) when \(t = 2 \mathrm {~s}\).

3. \begin{figure}[h] \end{figure} Figure 2 shows a particle \(B\), of mass \(m\), attached to one end of a light elastic string. The other end of the string is attached to a fixed point \(A\), at a distance \(h\) vertically above a smooth horizontal table. The particle moves on the table in a horizontal circle with centre \(O\), where \(O\) is vertically below \(A\). The string makes a constant angle \(\theta\) with the downward vertical and \(B\) moves with constant angular speed \(\omega\) about \(O A\).

1. Show that \(\omega ^ { 2 } \leqslant \frac { g } { h }\). The elastic string has natural length \(h\) and modulus of elasticity \(2 m g\).
   Given that \(\tan \theta = \frac { 3 } { 4 }\),
2. find \(\omega\) in terms of \(g\) and \(h\).

4. \begin{figure}[h] \end{figure} A uniform solid hemisphere, of radius \(6 a\) and centre \(O\), has a solid hemisphere of radius \(2 a\), and centre \(O\), removed to form a bowl \(B\) as shown in Figure 3.

1. Show that the centre of mass of \(B\) is \(\frac { 30 } { 13 } a\) from \(O\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-07_735_614_1126_662} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} The bowl \(B\) is fixed to a plane face of a uniform solid cylinder made from the same material as \(B\). The cylinder has radius \(2 a\) and height \(6 a\) and the combined solid \(S\) has an axis of symmetry which passes through \(O\), as shown in Figure 4.
2. Show that the centre of mass of \(S\) is \(\frac { 201 } { 61 } a\) from \(O\). The plane surface of the cylindrical base of \(S\) is placed on a rough plane inclined at \(12 ^ { \circ }\) to the horizontal. The plane is sufficiently rough to prevent slipping.
3. Determine whether or not \(S\) will topple. \section*{
   \includegraphics[max width=\textwidth, alt={}]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-08_56_366_251_178}
   }

1. A particle \(P\) of mass \(m\) 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 \(O\). The particle is released from rest with the string taut and \(O P\) horizontal.
   1. Find the tension in the string when \(O P\) makes an angle of \(60 ^ { \circ }\) with the downward vertical.
   A particle \(Q\) of mass \(3 m\) is at rest at a distance \(a\) vertically below \(O\). When \(P\) strikes \(Q\) the particles join together and the combined particle of mass \(4 m\) starts to move in a vertical circle with initial speed \(u\).
2. Show that \(u = \sqrt { } \left( \frac { g a } { 8 } \right)\). The combined particle comes to instantaneous rest at \(A\).
3. Find
   1. the angle that the string makes with the downward vertical when the combined particle is at \(A\),
   2. the tension in the string when the combined particle is at \(A\).
      \section*{LU
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1. A particle \(P\) of mass 0.5 kg moves along the positive \(x\)-axis. It moves away from the origin \(O\) under the action of a single force directed away from \(O\). When \(O P = x\) metres, the magnitude of the force is \(\frac { 3 } { ( x + 1 ) ^ { 3 } } \mathrm {~N}\) and the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   Initially \(P\) is at rest at \(O\).
   1. Show that \(v ^ { 2 } = 6 \left( 1 - \frac { 1 } { ( x + 1 ) ^ { 2 } } \right)\).
   2. Show that the speed of \(P\) never reaches \(\sqrt { } 6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   3. Find \(x\) when \(P\) has been moving for 2 seconds.
      
   \section*{LL
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