A 600-kg car is going around a banked curve with a radius of 110 m at a steady speed of 24.5 m/s. What is the appropriate banking angle so that the car stays on its path without the assistance of
Friction?
A) $29.1 ^ { \circ }$
B) $13.5 ^ { \circ }$
C) $33.8 ^ { \circ }$
D) $60.9 ^ { \circ }$
E) $56.2 ^ { \circ }$

Question

Quizplus · Accepted Answer

The appropriate banking angle can be found using the formula for the centripetal force required to keep the car moving in a circle of radius $r$ at speed $v$, which is given by $F_c = \frac{mv^2}{r}$, where $m$ is the mass of the car, $v$ is its velocity, and $r$ is the radius of the curve. For a banked curve without friction, the centripetal force is provided by the component of the gravitational force acting down the slope of the bank, which depends on the banking angle $\theta$. The correct relationship that equates the gravitational force component to the necessary centripetal force, and allows solving for $\theta$, is $\tan(\theta) = \frac{v^2}{rg}$, where $g$ is the acceleration due to gravity (approximately $9.8 m/s^2$). Substituting the given values ($v = 24.5 m/s$, $r = 110 m$, $g = 9.8 m/s^2$) into this formula and solving for $\theta$ yields an angle of approximately $29.1^\circ$.

A 600-Kg Car Is Going Around a Banked Curve with a Radius