Edexcel FM2 AS (Further Mechanics 2 AS) 2022 June

1. \begin{figure}[h] \end{figure} A uniform plane lamina is in the shape of an isosceles trapezium \(A B C D E F\), as shown shaded in Figure 1.

• \(B C E F\) is a square
• \(A B = C D = a\)
• \(B C = 3 a\)
  1. Show that the distance of the centre of mass of the lamina from \(A D\) is \(\frac { 11 a } { 8 }\)

The mass of the lamina is \(M\)
The lamina is suspended by two light vertical strings, one attached to the lamina at \(A\) and the other attached to the lamina at \(F\) The lamina hangs freely in equilibrium, with \(B F\) horizontal.

Find, in terms of \(M\) and \(g\), the tension in the string attached at \(A\)

2. \begin{figure}[h] \end{figure} Uniform wire is used to form the framework shown in Figure 2.
In the framework

• \(A B C D\) is a rectangle with \(A D = 2 a\) and \(D C = a\)
• \(B E C\) is a semicircular arc of radius \(a\) and centre \(O\), where \(O\) lies on \(B C\)

The diameter of the semicircle is \(B C\) and the point \(E\) is such that \(O E\) is perpendicular to \(B C\). The points \(A , B , C , D\) and \(E\) all lie in the same plane.

1. Show that the distance of the centre of mass of the framework from \(B C\) is $$\frac { a } { 6 + \pi }$$ The framework is freely suspended from \(A\) and hangs in equilibrium with \(A E\) at an angle \(\theta ^ { \circ }\) to the downward vertical.
2. Find the value of \(\theta\). The mass of the framework is \(M\).
   A particle of mass \(k M\) is attached to the framework at \(B\).
   The centre of mass of the loaded framework lies on \(O A\).
3. Find the value of \(k\).

1. A cyclist is travelling around a circular track which is banked at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\)

The cyclist moves with constant speed in a horizontal circle of radius \(r\).
In an initial model,

• the cyclist and her cycle are modelled as a particle
• the track is modelled as being rough so that there is sideways friction between the tyres of the cycle and the track, with coefficient of friction \(\mu\), where \(\mu < \frac { 4 } { 3 }\)
  Using this model, the maximum speed that the cyclist can travel around the track in a horizontal circle of radius \(r\), without slipping sideways, is \(V\).
  1. Show that \(V = \sqrt { \frac { ( 3 + 4 \mu ) r g } { 4 - 3 \mu } }\)

In a new simplified model,

• the cyclist and her cycle are modelled as a particle
• the motion is now modelled so that there is no sideways friction between the tyres of the cycle and the track

Using this new model, the speed that the cyclist can travel around the track in a horizontal circle of radius \(r\), without slipping sideways, is \(U\).

Find \(U\) in terms of \(r\) and \(g\).

Show that \(U < V\).

1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\) in the direction of \(x\) increasing, where

$$v = \frac { 1 } { 2 } \left( 3 \mathrm { e } ^ { 2 t } - 1 \right) \quad t \geqslant 0$$ The acceleration of \(P\) at time \(t\) seconds is \(a \mathrm {~ms} ^ { - 2 }\)

1. Show that \(a = 2 v + 1\)
2. Find the acceleration of \(P\) when \(t = 0\)
3. Find the exact distance travelled by \(P\) in accelerating from a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)