Edexcel M3 (Mechanics 3) 2023 June

1. In this question you must show all stages in your working.

Solutions relying on calculator technology are not acceptable. \begin{figure}[h] \end{figure} The finite region \(R\), shown shaded in Figure 1, is bounded by the \(x\)-axis, the line with equation \(x = 3\), the curve with equation \(y = \sqrt { ( x + 1 ) }\) and the \(y\)-axis.
Find the \(\boldsymbol { y }\) coordinate of the centre of mass of a uniform lamina in the shape of \(R\).

2. \begin{figure}[h] \end{figure} A light elastic string \(A B\) has modulus of elasticity \(2 m g\) and natural length \(k a\), where \(k\) is a constant.
The end \(A\) of the elastic string is attached to a fixed point. The other end \(B\) is attached to a particle of mass \(m\). The particle is held in equilibrium, with the elastic string taut, by a force that acts in a direction that is perpendicular to the string. The line of action of the force and the elastic string lie in the same vertical plane. The string makes an angle \(\theta\) with the downward vertical at \(A\), as shown in Figure 2. Given that the length \(A B = \frac { 21 } { 10 } a\) and \(\tan \theta = \frac { 3 } { 4 }\), find the value of \(k\).

1. A uniform solid right circular cone \(C\) has base radius \(r\), height \(H\) and vertex \(V\). A uniform solid \(S\), shown in Figure 3, is formed by removing from \(C\) a uniform solid right circular cone of height \(h ( h < H )\) that has the same base and axis of symmetry as \(C\).

\begin{figure}[h] \end{figure}

1. Show that the distance of the centre of mass of \(S\) from \(V\) is $$\frac { 1 } { 4 } ( 3 H - h )$$ The solid \(S\) is suspended by two vertical light strings. The first string is attached to \(S\) at \(V\) and the second string is attached to \(S\) at a point on the circumference of the circular base of \(S\).
   The solid \(S\) hangs freely in equilibrium with its axis of symmetry horizontal.
   The tension in the first string is \(T _ { 1 }\) and the tension in the second string is \(T _ { 2 }\)
2. Find \(\frac { T _ { 1 } } { T _ { 2 } }\), giving your answer in terms of \(H\) and \(h\), in its simplest form.

4. \begin{figure}[h] \end{figure} A car is travelling round a circular track. The track is banked at an angle \(\alpha\) to the horizontal, as shown in Figure 4. The car and driver are modelled as a particle.
The car moves round the track with constant speed in a horizontal circle of radius \(r\).
When the car is moving with speed \(\frac { 1 } { 2 } \sqrt { g r }\) round the circle, there is no sideways friction between the tyres of the car and the track.

1. Show that \(\tan \alpha = \frac { 1 } { 4 }\) The sideways friction between the tyres of the car and the track has coefficient of friction \(\mu\), where \(\mu < 4\) The maximum speed at which the car can move round the circle without slipping sideways is \(V\).
2. Find \(V\) in terms of \(\mu , r\) and \(g\).

1. The centre of the Earth is the point \(O\) and the Earth is modelled as a fixed sphere of radius \(R\).
   At time \(t = 0\), a particle \(P\) is projected vertically upwards with speed \(U\) from a point \(A\) on the surface of the Earth.

At time \(t\) seconds, where \(t \geqslant 0\)

• \(\quad P\) is a distance \(x\) from \(O\)
• \(P\) is moving with speed \(v\)
• \(P\) has acceleration of magnitude \(\frac { g R ^ { 2 } } { x ^ { 2 } }\) directed towards \(O\)

Air resistance is modelled as being negligible.

1. Show that \(v ^ { 2 } = \frac { 2 g R ^ { 2 } } { x } + U ^ { 2 } - 2 g R\) Particle \(P\) is first moving with speed \(\frac { 1 } { 2 } \sqrt { g R }\) at the point \(B\).
2. Given that \(U = \sqrt { g R }\) find, in terms of \(R\), the distance \(A B\).
3. Find, in terms of \(g\) and \(R\), the smallest value of \(U\) that would ensure that \(P\) never comes to rest, explaining your reasoning.

6. \begin{figure}[h] \end{figure} 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 \(P\) is held at rest with the string taut and horizontal and is then projected vertically downwards with speed \(u\), as shown in Figure 5. Air resistance is modelled as being negligible.
At the instant when the string has turned through an angle \(\theta\) and the string is taut, the tension in the string is \(T\).

1. Show that \(T = \frac { m u ^ { 2 } } { a } + 3 m g \sin \theta\) Given that \(u = 2 \sqrt { \frac { 3 a g } { 5 } }\)
2. find, in terms of \(a\) and \(g\), the speed of \(P\) at the instant when the string goes slack.
3. Hence find, in terms of \(a\), the maximum height of \(P\) above \(O\) in the subsequent motion.

1. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(l\). The other end of the string is attached to a fixed point on a ceiling. The particle \(P\) hangs in equilibrium at a distance \(D\) below the ceiling.

The particle \(P\) is now pulled vertically downwards until it is a distance \(3 l\) below the ceiling and released from rest. Given that \(P\) comes to instantaneous rest just before it reaches the ceiling,

1. show that \(D = \frac { 5 l } { 3 }\)
2. Show that, while the elastic string is stretched, \(P\) moves with simple harmonic motion, with period \(2 \pi \sqrt { \frac { 2 l } { 3 g } }\)
3. Find, in terms of \(g\) and \(l\), the exact time from the instant when \(P\) is released to the instant when the elastic string first goes slack.