JEE Main & Advanced Physics Two Dimensional Motion Bending of A Cyclist

A cyclist provides himself the necessary centripetal force by leaning inward on a horizontal track, while going round a curve. Consider a cyclist of weight mg taking a turn of radius r with velocity \[\upsilon \]. In order to provide the necessary centripetal force, the cyclist leans through angle  \[\theta \] inwards as shown in figure.

The cyclist is under the action of the following forces :

The weight mg acting vertically downward at the centre of gravity of cycle and the cyclist.

The reaction R of the ground on cyclist. It will act along a line-making angle \[\theta \] with the vertical.

The vertical component R cos \[\theta \] of the normal reaction R will balance the weight of the cyclist, while the horizontal component R sin \[\theta \] will provide the necessary centripetal force to the cyclist.

\[R\sin \theta =\frac{m{{v}^{2}}}{r}\]                ...(i)  

and  R cos q  = mg                    ...(ii)

Dividing equation (i) by (ii), we have           

\[\frac{R\sin \theta }{R\cos \theta }=\frac{m{{{v}^{2}}}/{r}\;}{mg}\]            

or \[\tan \theta =\frac{{{v}^{2}}}{rg}\]                               ...(iii)                        

Therefore, the cyclist should bend through an angle \[\theta ={{\tan }^{-1}}\left( \frac{{{v}^{2}}}{rg} \right)\]

It follows that the angle through which cyclist should bend will be greater, if

(i) The radius of the curve is small i.e. the curve is sharper

(ii) The velocity of the cyclist is large.

Note :

• For the same reasons, an ice skater or an aeroplane has to bend inwards, while taking a turn.