A curve in a level road has a radius 75 m. The maximum speed of a car turning this curved road can be 30 m/s without skidding. If radius of curved road is changed to 48 m and the coefficient of friction between the tyres and the road remains same, then maximum allowed speed would be ______ m/s.Correct answer is '24'. Can you explain this answer?

Question

Answer

Given Information

Radius of the original curved road = 75 m

Maximum speed without skidding on the original curved road = 30 m/s

Radius of the new curved road = 48 m

Coefficient of friction between the tires and the road remains the same

Explanation

The maximum speed at which a car can turn a curved road without skidding depends on the radius of the curve and the coefficient of friction between the tires and the road. To understand why the maximum speed decreases when the radius of the curve is reduced, we need to consider the forces acting on the car as it turns the curve.

When a car turns a curve, there are two main forces acting on it: the gravitational force (mg) and the frictional force (f) between the tires and the road. The frictional force provides the necessary centripetal force to keep the car moving in a curved path.

Centripetal Force

The centripetal force required to keep an object moving in a curved path is given by the equation:

F = (mv^2) / r

Where:
F = Centripetal force
m = Mass of the car
v = Velocity of the car
r = Radius of the curve

Maximum Speed Calculation

To find the maximum speed at which a car can turn a curved road without skidding, we need to calculate the maximum centripetal force that can be provided by the frictional force between the tires and the road. The maximum frictional force is given by the equation:

f_max = μN

Where:
f_max = Maximum frictional force
μ = Coefficient of friction
N = Normal force (equal to the weight of the car, mg)

We can substitute the value of f_max into the centripetal force equation to find the maximum speed:

f_max = (mv_max^2) / r

Simplifying the equation, we get:

v_max = √(f_max * r / m)

Comparison of Maximum Speeds

Let's calculate the maximum speed for both the original and new curved roads using the given information:

For the original curved road:
Radius (r) = 75 m
Maximum speed (v_max) = 30 m/s

For the new curved road:
Radius (r) = 48 m
Maximum speed (v_max) = ?

We know that the coefficient of friction remains the same. Therefore, the maximum speed on the new curved road can be calculated using the same formula as before:

v_max = √(f_max * r / m)

Substituting the values for the new curved road, we have:

v_max = √(f_max * 48 / m)

Since the coefficient of friction and the mass of the car are the same for both cases, the only difference is the radius of the curve. As the radius decreases, the maximum speed also decreases. Therefore, the maximum allowed speed on the new curved road is less than 30 m/s.

The correct answer is 24 m/s.

Answer

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