Discuss the bending of a cyclist in curves.

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, `vectau_(" net ") =0`
`-mgAB+(mv^2)/(r )BC=0`

(vi) Here, the clock wise moment (mg AB) is taken as negative and the anti clodkwise moment
`((mv^2)/(r )BC)` is taken as positive.
`mgAB=(mv^2)/(r )BC`
(vii) But from `triangleABC, 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))`
(viii) 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.