Edexcel M3 (Mechanics 3) 2024 January

1. A spacecraft \(S\) of mass \(m\) moves in a straight line towards the centre, \(O\), of a planet.

The planet is modelled as a fixed sphere of radius \(R\).
The spacecraft \(S\) is modelled as a particle.
The gravitational force of the planet is the only force acting on \(S\).
When \(S\) is a distance \(x ( x \geqslant R )\) from \(O\)

• the gravitational force is directed towards \(O\) and has magnitude \(\frac { m g R ^ { 2 } } { 2 x ^ { 2 } }\)
• the speed of \(S\) is \(v\)
  1. Show that

$$v ^ { 2 } = \frac { g R ^ { 2 } } { x } + C$$ where \(C\) is a constant. When \(x = 3 R , v = \sqrt { 3 g R }\)

Find, in terms of \(g\) and \(R\), the speed of \(S\) as it hits the surface of the planet.

2. \begin{figure}[h] \end{figure} A light elastic spring has natural length \(l\) and modulus of elasticity \(\lambda\) One end of the spring is attached to a point \(A\) on a smooth plane.
The plane is inclined at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\) A particle \(P\) of mass \(m\) is attached to the other end of the spring. Initially \(P\) is held at the point \(B\) on the plane, where \(A B\) is a line of greatest slope of the plane. The point \(B\) is lower than \(A\) and \(A B = 2 l\), as shown in Figure 1 .
The particle is released from rest at \(B\) and first comes to instantaneous rest at the point \(C\) on \(A B\), where \(A C = 0.7 l\)

1. Use the principle of conservation of mechanical energy to show that $$\lambda = \frac { 100 } { 91 } m g$$
2. Find the acceleration of \(P\) when it is released from rest at \(B\).

3. \begin{figure}[h] \end{figure} The shaded region in Figure 2 is bounded by the \(x\)-axis, the line with equation \(x = 2\) and the curve with equation \(y = \frac { 1 } { 4 } x ( 3 - x )\).
This region is rotated through \(2 \pi\) radians about the \(x\)-axis, to form a solid of revolution which is used to model a uniform solid \(S\). The volume of \(S\) is \(\frac { 2 } { 5 } \pi\)

1. Use the model and algebraic integration to show that the \(x\) coordinate of the centre of mass of \(S\) is \(\frac { 31 } { 24 }\) The solid \(S\) is placed with its circular face on a rough plane which is inclined at \(\alpha ^ { \circ }\) to the horizontal. The plane is sufficiently rough to prevent \(S\) from sliding. The solid \(S\) is on the point of toppling.
2. Find the value of \(\alpha\)

4. \begin{figure}[h] \end{figure} Figure 3 shows a thin hollow right circular cone fixed with its circular rim horizontal.
The centre of the circular rim is \(O\). The vertex \(V\) of the cone is vertically below \(O\).
The radius of the circular rim is \(4 a\) and \(O V = 3 a\).
A particle \(P\) of mass \(m\) moves in a horizontal circle of radius \(r ( 0 < r < 4 a )\) on the inner surface of the cone. The coefficient of friction between \(P\) and the inner surface of the cone is \(\frac { 1 } { 4 }\)
The particle moves with a constant angular speed.
Show that the maximum possible angular speed is \(\sqrt { \frac { 16 g } { 13 r } }\)

1. (a) Use algebraic integration to show that the centre of mass of a uniform semicircular disc of radius \(r\) and centre \(O\) is at a distance \(\frac { 4 r } { 3 \pi }\) from the diameter through \(O\) [You may assume, without proof, that the area of a circle of radius \(r\) is \(\pi r ^ { 2 }\) ]

A uniform lamina L is in the shape of a semicircle with centre \(B\) and diameter \(A C = 8 a\). The semicircle with diameter \(A B\) is removed from \(L\) and attached to the straight edge \(B C\) to form the template \(T\), shown shaded in Figure 4. \begin{figure}[h] \end{figure} The distance of the centre of mass of \(T\) from \(A C\) is \(d\).
(b) Show that \(d = \frac { 4 a } { \pi }\) The template \(T\) is freely suspended from \(A\) and hangs in equilibrium with \(A C\) at an angle \(\theta\) to the downward vertical.
(c) Find the exact value of \(\tan \theta\)

1. The fixed point \(A\) is vertically above the fixed point \(B\), with \(A B = 3 l\)

A light elastic string has natural length \(l\) and modulus of elasticity \(4 m g\) One end of the string is attached to \(A\) and the other end is attached to a particle \(P\) of mass \(m\) A second light elastic string also has natural length \(l\) and modulus of elasticity \(4 m g\) One end of this string is attached to \(P\) and the other end is attached to \(B\). Initially \(P\) rests in equilibrium at the point \(E\), where \(A E B\) is a vertical straight line.

1. Show that \(A E = \frac { 13 } { 8 } l\) The particle \(P\) is now held at the point that is a distance \(2 l\) vertically below \(A\) and released from rest. At time \(t\), the vertical displacement of \(P\) from \(E\) is \(x\), where \(x\) is measured vertically downwards.
2. Show that \(\ddot { x } = - \frac { 8 g } { l } x\)
3. Find, in terms of \(g\) and \(l\), the speed of \(P\) when it is \(\frac { 1 } { 8 } l\) below \(E\).
4. Find the length of time, in each complete oscillation, for which \(P\) is more than \(1.5 l\) from \(A\), giving your answer in terms of \(g\) and \(l\)

7. \begin{figure}[h] \end{figure} A thin smooth hollow spherical shell has centre \(O\) and radius \(r\). Part of the shell is removed to form a bowl with a plane circular rim. The bowl is fixed with the circular rim uppermost and horizontal. The point \(A\) is the lowest point of the bowl, as shown in Figure 5. The point \(B\) is on the rim of the bowl, with \(O B\) at an angle \(\theta\) to the upward vertical, where \(\tan \theta = \frac { 12 } { 5 }\)
A small ball is placed in the bowl at \(A\). The ball is projected from \(A\) with horizontal speed \(u\) and moves in the vertical plane \(A O B\). The ball stays in contact with the bowl until it reaches \(B\). At the instant when the ball reaches \(B\), the speed of the ball is \(v\).
By modelling the ball as a particle and ignoring air resistance,

1. use the principle of conservation of mechanical energy to show that $$v ^ { 2 } = u ^ { 2 } - \frac { 36 } { 13 } g r$$
2. show that \(u ^ { 2 } \geqslant \frac { 41 } { 13 } g r\) The point \(C\) is such that \(B C\) is a diameter of the rim of the bowl.
   Given that \(u ^ { 2 } = 4 g r\)
3. use the model to show that, after leaving the inner surface of the bowl at \(B\), the ball falls back into the bowl before reaching \(C\).