Explain why a cyclist bends while negotiating a curve road?

Explanation

Let us consider a cyclist negotating a circular level road ( not banked of radius r with a speed v. The cycle and the system is C and it goes in a circle of radius r with center at O / Let us choose the line OC as X-axis and the vertical line through O as Z- axis as shown in figure
The system as a frame is rotating about Z - axis the system is at rest in this rotating frame .TO solve problems in rotating frame or reference , we have to apply a centrifugal force ( pseudo force ) on the system which will be `( mv^2 )/(r ) ` this force will act through the centrer of gravity .The forces acting on the system are ,(i ) gravitational force (mg ) , (ii ) normal force (N ) , (iii ) frictional force (f ) and (iv ) Centrifugal force `((mv^2)/(r ))` . As the system is in equilibrium in the rotational frame of reference , the n e t external force and n e t external torque must be zero . Let us consider all torques about the point A in figure .
For rotational equilibrium ,
` vec(tau)_(n e t) =0`
The torque due to the gravitional force about about point A is ( mg AB ) which causes a clockwise turn that is taken as negative . The torque due to the centripetal force is
`((mv^2)/(r) B C)` which causes an anticlockwise turn that is taken as positive .
`- mg AB + (mv^2)/(r) B C =0`
`mg AB = (mv^2 )/(r ) BC `
from ` Delta ABC ,`
`AB = AC sin theta and BC = AC cos theta `
` Mg AC sin theta = ( mv^2 )/(r ) AC cos theta `
` tan theta = (v^2)/(rg ) `
` theta = tan^(-1) ((v^2)/(rg ))`