Question

A section of a high speed test track is circular with a radius of curvature R = 1860 m.

At what angle of θ should the track be inclined so that a car traveling at 61.0 m/s (136 mph) would keep moving in a circle if there is oil on that section of the track, i.e., it would not slip sideways even with zero friction on that section. (Hint: The car's vertical acceleration is zero.)

Answer 1

Using Newton's second law, the car is experiencing a net force parallel to the banked curve such that

`mg \sin(\theta) = \frac{mv^2}r \implies \sin(\theta) = (v^2)/(rg)`

where
`v` is the tangential speed of the car and
`r` is the radius of the curve. Solve for
`\theta` :

`\sin(\theta) = (\left(61.0(\rm m)/(\rm s)\right)^2)/((1860\,\mathrm m)g) \approx 0.204 \implies \theta = \sin^(-1)(0.204) \approx \boxed{11.8^\circ}`