The co-effiicent of friction between the road surface and the tyre of an automobile is μ. At what speed v, the automobile can travel safely without skidding around a cur ve of radiu s r. Take m as mass of automobile and g as acceleration due to gravity.
Understanding Safe Speed on a Curved Road
When an automobile travels around a curved road, it requires a centripetal force directed towards the center of the curve to change its direction. This centripetal force is provided by the force of static friction between the tyres of the automobile and the road surface.
Centripetal Force Requirement
The centripetal force (\(F_c\)) required for the automobile to move in a circle of radius \(r\) at a speed \(v\) is given by the formula:
\(F_c = \frac{mv^2}{r}\)
where:
- \(m\) is the mass of the automobile.
- \(v\) is the speed of the automobile.
- \(r\) is the radius of the curve.
Friction Force Providing Centripetal Force
On a level road, the normal force (\(N\)) acting on the automobile is equal to its weight (\(mg\)), where \(g\) is the acceleration due to gravity. The maximum possible force of static friction (\(F_{f,max}\)) between the tyres and the road surface is proportional to the normal force and is given by:
\(F_{f,max} = \mu N\)
Since \(N = mg\) on a level road, the maximum static friction force is:
\(F_{f,max} = \mu mg\)
where \(\mu\) is the coefficient of friction between the road surface and the tyres.
Calculating Maximum Safe Speed Without Skidding
For the automobile to travel safely without skidding around the curve, the required centripetal force must be less than or equal to the maximum static friction force available. The maximum safe speed (\(v_{max}\)) is reached when the required centripetal force is exactly equal to the maximum static friction force:
\(F_c = F_{f,max}\)
\(\frac{mv^2}{r} = \mu mg\)
We can cancel the mass \(m\) from both sides of the equation:
\(\frac{v^2}{r} = \mu g\)
Now, we can solve for \(v^2\):
\(v^2 = \mu g r\)
Taking the square root of both sides to find the maximum safe speed \(v\):
\(v = \sqrt{\mu g r}\)
This equation shows that the maximum safe speed without skidding on a curved road depends on the coefficient of friction (\(\mu\)), the acceleration due to gravity (\(g\)), and the radius of the curve (\(r\)). It does not depend on the mass of the automobile.
Comparing this result with the given options, we find that it matches option 2.
Important Questions from Circular motion
A particle is moving in a circle with uniform speed. Which of the following quantities is constant?
A particle is moving in a circle with uniform speed. It has constant
The vehicle moving on a level circular path will exert pressure such that _____.
Which of the following statement is incorrect about a body undergoing a uniform circular motion?
If the speed and radius of a body moving in a circular path are doubled, then the magnitude of centripetal acceleration will be
