Question

What will be the maximum speed of a car on a circular road of radius $12 \text{ m}$ if the coefficient of friction between the tyres and the road is $0.3$? $g = 10 \text{ms}^{-2}$}

• A. $3.6 \text{ ms}^{-1}$
• B. $36 \text{ ms}^{-1}$
• C. $60 \text{ ms}^{-1}$
• D. $10 \text{ ms}^{-1}$
• E. $6 \text{ ms}^{-1}$

Hint:

For flat circular roads, the maximum safe speed is independent of the mass of the vehicle. It only depends on the friction, radius, and gravity.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
For a car turning on a flat circular road, the necessary centripetal force is provided entirely by the static friction between the tires and the road.
Step 2: Key Formula or Approach:
The maximum safe speed \(v_{\text{max}}\) before slipping occurs when the required centripetal force equals the maximum static friction force:
\[ \frac{mv_{\text{max}}^2}{r} = \mu mg \] Solving for \(v_{\text{max}}\) gives: \(v_{\text{max}} = \sqrt{\mu rg}\).
Step 3: Detailed Explanation:
Given values:
Radius \(r = 12\) m
Coefficient of friction \(\mu = 0.3\)
Acceleration due to gravity \(g = 10 \text{ m/s}^2\)
Substitute these values into the formula:
\[ v_{\text{max}} = \sqrt{0.3 \times 12 \times 10} \] \[ v_{\text{max}} = \sqrt{3 \times 12} \] \[ v_{\text{max}} = \sqrt{36} = 6 \text{ m/s} \] Step 4: Final Answer:
The maximum speed is 6 ms\(^{-1}\).