The speed limit around a curve of radius 85 meters is 30 miles per hour. What angle should the curve be banked at if a car is to be able to go around the curve at that speed with no friction?
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To determine the angle at which the curve should be banked, we need to consider the forces acting on the car as it goes around the curve. In the absence of friction, the car would be subject to two forces: the gravitational force pulling it downward and the normal force exerted by the road perpendicular to the car's motion.
The gravitational force can be resolved into two components: one perpendicular to the curve, which contributes to the normal force, and one parallel to the curve, which contributes to the car's acceleration.
The normal force, in turn, can be resolved into two components: one perpendicular to the road surface, which balances the gravitational force, and one parallel to the road surface, which provides the necessary centripetal force to keep the car moving in a curved path.
We can use these forces and the known values to find the angle of banking using the following steps:
Step 1: Convert the speed limit from miles per hour to meters per second.
Since the radius is given in meters, it is more convenient to work with a speed unit in meters per second. To convert 30 miles per hour to meters per second, we multiply by the conversion factor: 1 mile = 1609 meters, and 1 hour = 3600 seconds.
30 miles/hour * (1609 meters/mile) * (1 hour/3600 seconds) = 13.41 meters/second
So the speed limit of 30 miles per hour is approximately 13.41 meters per second.
Step 2: Calculate the required centripetal acceleration using the speed and radius.
The centripetal acceleration is given by the equation: a = v^2 / r, where v is the velocity and r is the radius of the curve.
Substituting the given values, we get:
a = (13.41 m/s)^2 / 85 m = 2.105 m/s^2
Step 3: Determine the angle of banking using the acceleration and gravitational force.
The angle of banking can be calculated using the equation: tan(theta) = a / g, where a is the centripetal acceleration and g is the acceleration due to gravity.
Substituting the values, we get:
tan(theta) = 2.105 m/s^2 / 9.8 m/s^2 (approximate value of g)
Taking the inverse tangent (arctan) of both sides, we can find the angle theta:
theta = arctan(2.105 / 9.8) ≈ 12.7 degrees
So, the curve should be banked at approximately 12.7 degrees for a car to be able to go around the curve at 30 miles per hour with no friction.