Edexcel M3 (Mechanics 3)

1. A particle \(P\) moves on the positive \(x\)-axis. When the displacement of \(P\) from \(O\) is \(x\) metres, its acceleration is \(( 6 - 4 x ) \mathrm { m } \mathrm { s } ^ { - 2 }\), measured in the direction of \(x\) increasing. Initially \(P\) is at \(O\) and the velocity of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction \(O x\).

Find the distance of \(P\) from \(O\) when \(P\) is instantaneously at rest.

2. A light elastic string \(A B\) has one end \(A\) attached to a fixed point on a ceiling. A particle \(P\) of mass 0.3 kg is attached to \(B\). When \(P\) hangs in equilibrium with \(A B\) vertical, \(A B = 100 \mathrm {~cm}\). The particle \(P\) is replaced by another particle \(Q\) of mass 0.5 kg . When \(Q\) hangs in equilibrium with \(A B\) vertical, \(A B = 110 \mathrm {~cm}\). Find

1. the natural length of the string,
2. the modulus of elasticity of the string.

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(3 a\). The other end of the string is attached to a fixed point \(A\) which is a vertical distance \(a\) above a smooth horizontal table. The particle moves on the table in a circle whose centre \(O\) is vertically below \(A\), as shown in Fig. 1. The string is taut and the speed of \(P\) is \(2 \sqrt { } ( a g )\). Find

1. the tension in the string,
2. the normal reaction of the table on \(P\).

4. \begin{figure}[h] \end{figure} A small smooth bead \(B\) of mass 0.2 kg is threaded on a smooth horizontal wire. The point \(A\) is on the same horizontal level as the wire and at a perpendicular distance \(d\) from the wire. The point \(O\) is the point on the wire nearest to \(A\), as shown in Fig. 2. The bead experiences a force of magnitude \(5 ( A B )\) newtons in the direction \(B A\) towards \(A\). Initially \(B\) is at rest with \(O B = 2 \mathrm {~m}\).

1. Prove that \(B\) moves with simple harmonic motion about \(O\), with period \(\frac { 2 \pi } { 5 } \mathrm {~s}\).
2. Find the greatest speed of \(B\) in the motion.
3. Find the time when \(B\) has first moved a distance 3 m from its initial position.

5. In a "test your strength" game at an amusement park, competitors hit one end of a small lever with a hammer, causing the other end of the lever to strike a ball which then moves in a vertical tube whose total height is adjustable. The ball is attached to one end of an elastic spring of natural length 3 m and modulus of elasticity 120 N . The mass of the ball is 2 kg . The other end of the spring is attached to the top of the tube. The ball is modelled as a particle, the spring as light and the tube is assumed to be smooth. The height of the tube is first set at 3 m . A competitor gives the ball an initial speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the height to which the ball rises before coming to rest. The tube is now adjusted by reducing its height to 2.5 m . The spring and the ball remain unchanged.
2. Find the initial speed which the ball must now have if it is to rise by the same distance as in part (a).
   (5 marks)

6. (a) Show, by integration, that the centre of mass of a uniform right cone, of radius \(a\) and height \(h\), is a distance \(\frac { 3 } { 4 } h\) from the vertex of the cone. \begin{figure}[h] \end{figure} A uniform right cone \(C\), of radius \(a\) and height \(h\), has vertex \(A\). A solid \(S\) is formed by removing from \(C\) another cone, of radius \(\frac { 2 } { 3 } a\) and height \(\frac { 1 } { 2 } h\), with the same axis as \(C\). The plane faces of the two cones coincide, as shown in Fig. 3.
(b) Find the distance of the centre of mass of \(S\) from \(A\).

7. A smooth solid hemisphere is fixed with its plane face on a horizontal table and its curved surface uppermost. The plane face of the hemisphere has centre \(O\) and radius \(a\). The point \(A\) is the highest point on the hemisphere. A particle \(P\) is placed on the hemisphere at \(A\). It is then given an initial horizontal speed \(u\), where \(u ^ { 2 } = \frac { 1 } { 2 } ( a g )\). When \(O P\) makes an angle \(\theta\) with \(O A\), and while \(P\) remains on the hemisphere, the speed of \(P\) is \(v\).

1. Find an expression for \(v ^ { 2 }\).
2. Show that, when \(\theta = \arccos 0.9 , P\) is still on the hemisphere.
3. Find the value of \(\cos \theta\) when \(P\) leaves the hemisphere.
4. Find the value of \(v\) when \(P\) leaves the hemisphere. After leaving the hemisphere \(P\) strikes the table at \(B\).
5. Find the speed of \(P\) at \(B\).
6. Find the angle at which \(P\) strikes the table. \section*{Alternative Question 2:}
   1. Two light elastic strings \(A B\) and \(B C\) are joined at \(B\). The string \(A B\) has natural length 1 m and modulus of elasticity 15 N . The string \(B C\) has natural length 1.2 m and modulus of elasticity 30 N . The ends \(A\) and \(C\) are attached to fixed points 3 m apart and the strings rest in equilibrium with \(A B C\) in a straight line.
   Find the tension in the combined string \(A C\).