A car goes around a curve on a road that is banked at an angle of 30 degrees. Even though the road is slick the car will stay on the road without any friction between its tires and the road when its speed is 24.0 m/s. WHAT IS THE RADIUS OF THE CURVE?
http://www.batesville.k12.in.us/physics/Phynet/Mechanics/Circular%20Motion/banked_no_friction.htm
To find the radius of the curve, we can use the concept of centripetal force. When a car goes around a curve, there must be a force acting towards the center of the circle to keep the car moving in a curved path. In this case, the force is provided by the horizontal component of the car's weight, since there is no friction between the tires and the road.
The formula for centripetal force is:
F = (mv^2) / r
Where:
F = Centripetal force
m = Mass of the car
v = Velocity of the car
r = Radius of the curve
In this case, since we're dealing with no friction, the only force acting towards the center is the horizontal component of the car's weight. This force is given by:
F = mg * sinθ
Where:
m = Mass of the car
g = Acceleration due to gravity (9.8 m/s^2)
θ = Angle of the road (bank angle)
Equating the two formulas for force, we get:
mg * sinθ = (mv^2) / r
Simplifying the equation, we can solve for the radius (r):
r = (v^2) / (g * sinθ)
Now we can plug in the values given in the question:
v = 24.0 m/s
θ = 30 degrees = 0.52 radians
g = 9.8 m/s^2
r = (24.0^2) / (9.8 * sin(0.52))
Calculating the value, we find:
r ≈ 118.61 meters
Therefore, the radius of the curve is approximately 118.61 meters.