Question

A cyclist rides along the circumference of a circular horizontal plane of radius R, the frictional coefficient being dependent only on distance `r' from the centre of the plane as $\mu ={\mu }_0[1-\frac{r}{R}]$ , where ${\mu }_0$ is a constant. Find the radius of the circle with the centre at the point along which the cyclist can ride with the maximum velocity.

• A. $R$
• B. $\frac{R}{2}$
• C. $\frac{R}{4}$
• D. $\frac{R}{8}$

Answer 1

The correct option is B $\frac{R}{2}$

Here centripetal force is provided by the frictional force OR $\mu mg=\frac{m{v}^2}{r}$
${\mu }_0(1-\frac{r}{R})g=\frac{v^2}{r}or2v\frac{dv}{dr}={\mu }_0(1-\frac{2r}{R})gif\frac{dv}{dr}=0$ $\therefore r=\frac{R}{2}$