Question
A cyclist tends to negotiate a curved track. Obtain an expression for the angle which he will have to make with vertical.
Solution
Consider a cyclist of negotiating a curve of radius r with velocity v.
Weight of the cyclist, W = mg
In order to provide the necessary centripetal force, the cyclist leans through angle
in inward direction as shown in figure above.
The various forces acting on the cyclist are:
(i) Weight Mg acting vertically downward at the centre of gravity of cycle and the cyclist.
(ii) The reaction R of the ground onto the cyclist, acting along a line, making angle
with the vertical.
The cyclist while taking the turn is in equilibrium, therefore the vertical component Rcosθ of the normal reaction R will balance the weight of the cyclist.
The horizontal component R sinθ will provide the necessary centripetal force to the cyclist.
i.e.,
...(1)
and
...(2)
Dividing (2) by (1), we have
Therefore, the cyclist should lean inwards at angle
given by 
Weight of the cyclist, W = mg
In order to provide the necessary centripetal force, the cyclist leans through angle
in inward direction as shown in figure above. The various forces acting on the cyclist are:
(i) Weight Mg acting vertically downward at the centre of gravity of cycle and the cyclist.
(ii) The reaction R of the ground onto the cyclist, acting along a line, making angle
with the vertical. The cyclist while taking the turn is in equilibrium, therefore the vertical component Rcosθ of the normal reaction R will balance the weight of the cyclist.
The horizontal component R sinθ will provide the necessary centripetal force to the cyclist.
i.e.,
...(1) and
...(2) Dividing (2) by (1), we have
Therefore, the cyclist should lean inwards at angle
given by 
