Question

A cyclist speeding at 6m/s in a circle of 18 m radius makes an angle theta with the vertical. Calculate theta. Also determine the minimum possible value of coefficient of friction between the tyres and the ground.

Answer 1

This is not a particularly difficult problem once you understand the forces that are present on the cyclist in the cyclist's own frame of reference. Look at the diagram below:



The centrifugal force is pseudoforce as the result of the centripetal acceleration that the cyclist maintains at all times to stay on the circle. This is equal to [math]\displaystyle\frac{mv^2}{r}[/math], where [math]v[/math] is the speed of the cyclist along the circle. Then there is a second force, that of gravity, which is of course [math]mg[/math][1]. The resultant of both these forces on individual elements of the cyclist may be considered to pass through the cyclist's center of mass. So that the cyclist does not bend and fall down, the angular acceleration about the ground should be zero, that is to say the sum of all torques on the cyclist about the ground should be zero. The axis of rotation for both the centrifugal force and the force of gravity is a line parallel to the line passing through the center of the two wheels and therefore the magnitude of the torque induced by each will be equal to the other. It can be seen that the torque induced by the centrifugal force is [math]\displaystyle\frac{mv^2x}{r}\sin\theta[/math], where [math]x[/math] is the distance between the center of mass of the cyclist and the bottom of the bicycle (that is, the arm of the torque) and [math]\theta[/math] is the cyclist's angle of inclination to the ground. The torque of the force of gravity is [math]mgx\cos\theta[/math]. To balance each other out, the two must be equal and hence, we conclude that [math]\displaystyle\tan\theta = \frac{gr}{v^2}[/math].

[1] There are additionally forces, actually in the form of the force of friction between the tires of the bicycle and the ground. But since these are applied at the axis of rotation, they do not create any torque and hence can be omitted in the calculations