7. A cyclist is travelling round a circular bend of radius 25 m on a track which is banked at an angle of \(35 ^ { \circ }\) to the horizontal.
In a model of the situation, the cyclist and her bicycle are represented by a particle of mass 60 kg and air resistance and friction are ignored.
Using this model and assuming that the cyclist is not slipping,
- find, correct to 3 significant figures, the speed at which she is travelling.
In tests it is found that the cyclist must travel at a minimum speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to prevent the bicycle from slipping down the slope. A more refined model is now used with a coefficient of friction between the bicycle and the track of \(\mu\).
Using this model,
- show that \(\mu = 0.227\), correct to 3 significant figures,
- find, correct to 2 significant figures, the maximum speed at which the cyclist can travel without slipping up the slope.
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