Question

At what angle to the vertical must a motorcyclist lean so as to negotiate a bend with a radius of 280m at a speed of 180km/h.

Answer 1

Let's apply Newton's Second Law of Motion in projections on axis $x$- and $y$:

$$Ncos\theta=mg,$$ $$Nsin\theta=\dfrac{mv^2}{r}.$$

Dividing the second equation by the first one, we get:

$$tan\theta=\dfrac{v^2}{rg},$$ $$\theta=tan^{-1}(\dfrac{v^2}{rg}),$$ $$\theta=tan^{-1}(\dfrac{(50\ \dfrac{m}{s})^2}{280\ m\cdot9.8\ \dfrac{m}{s^2}})=42^{\circ}.$$