Determine the banking angle needed to make a high-speed turn off a highway exit ramp with a radius of 720 m if the vehicle is travelling at 80 km/h. Assume there is no friction.
To determine the banking angle needed to make a high-speed turn off a highway exit ramp, we need to use the concept of centripetal force. The centripetal force is the force that acts towards the center of the circular path and keeps the object moving in a curved path.
In this scenario, the centripetal force required to keep the vehicle moving in a circular path is provided by the horizontal component of the normal force exerted by the road surface on the vehicle. The normal force is the force exerted by a surface to support the weight of an object resting on it.
To find the banking angle, we first need to calculate the required centripetal force using the following formula:
F = (mass of the vehicle) × (centripetal acceleration)
The centripetal acceleration can be calculated using the formula:
a = (velocity^2) / radius
where
F = centripetal force
m = mass of the vehicle
v = velocity
r = radius of the curve
a = centripetal acceleration
Given:
Velocity (v) = 80 km/h = 80,000 m / 3,600 s ≈ 22.22 m/s
Radius (r) = 720 m
First, let's calculate the centripetal acceleration:
a = (v^2) / r
a = (22.22 m/s)^2 / 720 m
a ≈ 0.686 m/s^2
Next, we can calculate the required centripetal force. Since there is no friction, the centripetal force is provided solely by the normal force. The horizontal component of the normal force is:
F = m × a
To find the banking angle, we can use the following formula:
tan(banking angle) = a / g
where
g = acceleration due to gravity (approximately 9.8 m/s^2)
Let's substitute the values into the formula:
tan(banking angle) = 0.686 m/s^2 / 9.8 m/s^2
tan(banking angle) ≈ 0.07
To find the banking angle, we need to take the inverse tangent (arctan) of both sides:
banking angle ≈ arctan(0.07)
banking angle ≈ 4 degrees
Therefore, the banking angle needed to make a high-speed turn off the highway exit ramp is approximately 4 degrees.