A 410 kg race car is rounding a curve in the road which is banked at an angle of 23.4 degrees from the horizontal. Assume that there has just been a massive snowfall and the road’s surface is frictionless. What must the radius of the curve be if the car is travelling at a speed 10.2 m/s and yet stays on the road?
To determine the radius of the curve that the race car must follow in order to stay on the road, we can use the concept of centripetal force. The centripetal force is the force that keeps an object moving in a curved path. In this case, it is the force that keeps the race car from sliding off the banked curve.
The centripetal force can be calculated using the formula:
F_c = m * v^2 / r
Where:
F_c is the centripetal force
m is the mass of the car
v is the velocity of the car
r is the radius of the curve
In this problem, the centripetal force is provided by the component of the car's weight that acts perpendicular to the surface of the road. This can be calculated using the formula:
F_c = m * g * cosθ
Where:
m is the mass of the car
g is the acceleration due to gravity (approximately 9.8 m/s^2)
θ is the angle of the road's banking
Setting these two expressions for centripetal force equal to each other, we have:
m * v^2 / r = m * g * cosθ
Simplifying, we find:
r = v^2 / (g * cosθ)
Plugging in the given values:
v = 10.2 m/s
g ≈ 9.8 m/s^2
θ = 23.4 degrees
Converting the angle to radians:
θ = 23.4 * π/180 ≈ 0.4089 radians
Now, we can substitute these values into the formula to calculate the radius:
r ≈ (10.2^2) / (9.8 * cos 0.4089)
Using a calculator, we find:
r ≈ 12.74 meters
Therefore, the radius of the curve must be approximately 12.74 meters for the car to stay on the road.