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, you 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, the centripetal force is provided by the horizontal component of the normal force acting on the vehicle.
To find the banking angle, we'll need to consider the forces acting on the vehicle in the horizontal direction. In the absence of friction, there are only two forces:
1. The vertical component of the vehicle's weight acting downwards.
2. The horizontal component of the normal force acting towards the center of the circular path.
The vertical component of the weight is simply mg, where m is the mass of the vehicle and g is the acceleration due to gravity (approximately 9.8 m/s²). The horizontal component of the normal force is Nsin(θ), where N is the magnitude of the normal force and θ is the banking angle.
These two forces need to balance each other for the vehicle to move in a curve without slipping. Therefore, we can equate them:
Nsin(θ) = mg
Now, we need to express the normal force N in terms of the known quantities, such as the vehicle's mass and the radius of the curve.
To do that, we'll consider the vertical equilibrium of forces. The vertical component of the normal force, Ncos(θ), has to balance the vehicle's weight mg:
Ncos(θ) = mg
Solving this equation for N gives:
N = mg / cos(θ)
We can substitute this expression for N in our original equation:
(mg / cos(θ)) * sin(θ) = mg
Simplifying, we get:
tan(θ) = v² / (g * r)
where v is the velocity of the vehicle and r is the radius of the curve.
In this case, the velocity v is given as 80 km/h, which needs to be converted to m/s:
v = 80 km/h * (1000 m/1 km) * (1 h/3600 s) = 22.22 m/s
The radius of the curve, r, is given as 720 m.
Plugging these values into the equation, we have:
tan(θ) = (22.22 m/s)² / (9.8 m/s² * 720 m)
Solving for θ, we get:
θ = arctan((22.22 m/s)² / (9.8 m/s² * 720 m))
Using a calculator, the approximate value of θ is 35.7 degrees.
Therefore, the banking angle needed to make a high-speed turn off a highway exit ramp with a radius of 720 m and a velocity of 80 km/h is approximately 35.7 degrees.