Question

A car makes a circular turn of radius 50 m at 40 m/s. At what angle should the road be banked so that no friction is required?

a) 15.7 degrees
b) 30 degrees
c) 45 degrees
d) 60 degrees

Answer 1

The correct banking angle
`(\( \theta \))` for the road is approximately 45 degrees, ensuring a balance between gravitational and centripetal forces. This optimal angle allows the car to navigate the circular turn at 40 m/s without requiring friction for stability. Thus the correct option is c) 45 degrees.

Step-by-step explanation:

In circular motion, the banking of a road is determined by the balance between the gravitational force component and the centripetal force. When the road is properly banked, no friction is required for the vehicle to make the turn. The formula for the angle of banking
`(\( \theta \))` is given by the equation:

`\[ \tan(\theta) = \frac{{v^2}}{{g \cdot r}} \]`

Where:

- ( v ) is the speed of the car (40 m/s),

- ( g ) is the acceleration due to gravity (approximately 9.8 m/s²),

- ( r ) is the radius of the circular turn (50 m).

Plugging in these values:

`\[ \tan(\theta) = \frac{{(40)^2}}{{9.8 \cdot 50}} \]`

`\[ \theta = \tan^(-1)\left(\frac{{1600}}{{490}}\right) \]`

`\[ \theta \approx 45 \text{ degrees} \]`

At this angle, the road is properly banked, and no friction is needed for the car to negotiate the turn smoothly. This result aligns with the standard banking angle used in many road designs to optimize safety and vehicle stability during turns.

Therefore, the correct answer is c) 45 degrees.