Question

A vehicle moves on a banked curve of radius \( r \) with banking angle \( \theta \). What is the speed \( v \) of the vehicle to avoid slipping without friction?

• A. \( \sqrt{r g \tan \theta} \)
• B. \( \sqrt{r g \cot \theta} \)
• C. \( \sqrt{\dfrac{r g}{\tan \theta}} \)
• D. \( \sqrt{r g \sin \theta} \)

Hint:

On a frictionless banked road, the formula for safe speed is derived from balancing vertical forces and using centripetal force. Remember: \( v = \sqrt{r g \tan \theta} \).

Answer 1

The Correct Option is A

Step 1: Analyze forces on a frictionless banked road.
The horizontal component of the normal force supplies the centripetal force:
\[N \sin \theta = \frac{mv^2}{r}\]Step 2: Apply vertical equilibrium to express \( N \):
\[N \cos \theta = mg \quad \Rightarrow \quad N = \frac{mg}{\cos \theta}\]Step 3: Substitute \( N \) into the centripetal force equation:\[\frac{mg}{\cos \theta} \cdot \sin \theta = \frac{mv^2}{r}\Rightarrow mg \tan \theta = \frac{mv^2}{r}\]Step 4: Isolate \( v \):\[v^2 = r g \tan \theta \quad \Rightarrow \quad v = \sqrt{r g \tan \theta}\]