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 \(U < V\).