Question

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?

Answer 1

63

Answer 2

[AI-generated answer — Explain Bot]

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.