Banked track – no friction (find speed or radius)

1. \begin{figure}[h] \end{figure} A car moves round a bend in a road which is banked at an angle \(\alpha\) to the horizontal, as shown in Fig. 1. The car is modelled as a particle moving in a horizontal circle of radius 100 m . When the car moves at a constant speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), there is no sideways frictional force on the car. Find, in degrees to one decimal place, the value of \(\alpha\).

7 It is required to model the motion of a car of mass \(m \mathrm {~kg}\) travelling at a constant speed \(v \mathrm {~ms} ^ { - 1 }\) around a circular portion of banked track. The track is banked at \(30 ^ { \circ }\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{0501e5a4-2137-4e7d-98ff-2ee81941cbf3-5_414_624_356_242} In a model, the following modelling assumptions are made.

• The track is smooth.
• The car is a particle.
• The car follows a horizontal circular path with radius \(r \mathrm {~m}\).
  1. Show that, according to the model, \(\sqrt { 3 } \mathrm { v } ^ { 2 } = \mathrm { gr }\).

For a particular portion of banked track, \(r = 24\).

Find the value of \(v\) as predicted by the model. A car is being driven on this portion of the track at the constant speed calculated in part (b). The driver finds that in fact he can drive a little slower or a little faster than this while still moving in the same horizontal circle.

Explain

1. A car moves round a bend which is banked at a constant angle of \(\theta ^ { \circ }\) to the horizontal.

When the car is travelling at a constant speed of \(80 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) there is no sideways frictional force on the car. The car is modelled as a particle moving in a horizontal circle of radius 500 m .

1. Find the value of \(\theta\).
2. Identify one limitation of this model. The speed of the car is increased so that it is now travelling at a constant speed of \(90 \mathrm { kmh } ^ { - 1 }\) The car is still modelled as a particle moving in a horizontal circle of radius 500 m .
3. Describe the extra force that will now be acting on the car, stating the direction of this force.

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A vehicle of mass 1200 kg is moving with a constant speed of \(40\text{ ms}^{-1}\) around a horizontal circular path which is on a test track banked at an angle of 60° to the horizontal. There is no tendency to sideslip at this speed. The vehicle is modelled as a particle.

1. Calculate the normal reaction of the track on the vehicle. [3]
2. Determine
   1. the radius of the circular path,
   2. the angular speed of the vehicle and clearly state its units. [6]
3. What further assumption have you made in your solution to (b)? Briefly explain what effect this assumption has on the radius of the circular path. [2]