highway curve in northern Minnesota has a radius of 230 m. The curve is banked so that a car traveling at 21 m/s and will not skid sideways, even if the curve is coated with a frictionless glaze of ice. At what angle to the horizontal is the curve banked?
63
To find the angle at which the curve is banked, we can use the concept of centripetal force. On a banked curve, the force of gravity and the normal force together provide the centripetal force required for the car to move in a circular path.
Let's break down the forces acting on the car:
1. Gravity: The weight of the car acts vertically downward. Its vertical component is mg, where m is the mass of the car and g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. Normal force: The normal force acts perpendicular to the surface of the road. It has both vertical and horizontal components.
3. Centripetal force: This force acts radially inward and is responsible for maintaining the circular motion of the car. It is equal to mv^2 / r, where m is the mass of the car, v is its velocity, and r is the radius of the curve.
Considering the vertical components of the forces, we can equate them:
mg = Normal force * cos(theta) -- Equation 1
Where theta is the angle of the banked curve.
Now, considering the horizontal components of the forces, we equate them to the centripetal force:
Normal force * sin(theta) = (mv^2) / r -- Equation 2
To find theta, we can rewrite Equation 1 as:
Normal force = mg / cos(theta)
Substituting this into Equation 2, we get:
(mg / cos(theta)) * sin(theta) = (mv^2) / r
Now we can simplify and solve for theta:
tan(theta) = (v^2) / (r * g)
tan(theta) = (21^2) / (230 * 9.8)
Using a calculator, we find that tan(theta) is approximately 0.20.
Finally, we can take the inverse tangent (arctan) of both sides to find theta:
theta = arctan(0.20)
theta is approximately 11.31 degrees.
Therefore, the curve is banked at an angle of approximately 11.31 degrees to the horizontal.