Explain why a cyclist bends while negotiating a curve road?

Explanation

(i) Consider a cyclist negotiating a circular level road (not banked) of radius r with a speed v about the center O as shown in figure.

(ii) Let m be the mass of the system, which includes cycle and cyclist and C be the center of gravity of this system.
(iii) Let us consider horizontal is along x-axis and vertical is along Z-axis.
(iv) In this rotating frame, the centrifugal force `(mv^(2))/(r)` acts away from center O and passing through the center of gravity C as shown below.

(v) As the system is in rotational equilibrium, the net torque must be zero. Thus,
`vec(tau)_("net") = 0`
`-mg AB + (mv^(2))/(r) BC = 0`
(vi) Here, the clock wise moment (mg AB) is taken as negative and the anti clockwise moment `((mv^(2))/(r) BC)` is taken as positive.
`mg AB = (mv^(2))/(r) BC`
(vii) But from `Delta` ABC, AB = AC sin `theta` & BC = AC cos `theta`. Therefore, the above equation can be written as,
`mg AC sin theta = (mv^(2))/(r)AC cos theta`
`tan theta = (v^(2))/(rg)`
`theta = tan^(-1) ((v^(2))/(rg))`
(vii) It shows that while negotiating a circular road of radius r at velocity v, a cyclist has to bend an angle `theta` from vertical, to avoid a fall.