A car goes around a curve on a road that is banked at an angle of 32.5 . 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 19.0 .

What is the radius of the curve?

5.7 To determine the radius of the curve, we can use the concept of centripetal force.

First, let's identify the forces acting on the car when it goes around the curve without any friction. There are two forces involved: the gravitational force (mg) that pulls the car downward, and the normal force (N) that acts perpendicular to the surface of the road.

Since the road is banked at an angle, the normal force can be resolved into two components: one component acting perpendicular to the road surface (N⊥) and the other component acting parallel to the road surface (N∥).

The gravitational force can also be resolved into two components: one component acting perpendicular to the road surface (mg⊥) and the other component acting parallel to the road surface (mg∥).

In the absence of friction, the inward component of the normal force (N⊥) provides the centripetal force required to keep the car moving in a circle.

Using trigonometry, we can relate the forces acting on the car to the angle of the curve and the radius of the curve.

First, let's find the components of the gravitational force:

mg⊥ = mg * cos(32.5°)
mg∥ = mg * sin(32.5°)

Since the car is moving without any friction, the centripetal force is provided solely by the inward component of the normal force:

N⊥ = Fc

Substituting the expressions for N⊥, mg⊥, and Fc:

mg * cos(32.5°) = mg * sin(32.5°)

Simplifying the equation:

cos(32.5°) = sin(32.5°)

At this point, you can see that both sides of the equation are equal. This happens because the angle of the curve (32.5°) is specifically chosen such that the car can maintain a constant speed without any friction.

However, to find the radius of the curve, we need to use the equation for centripetal force:

Fc = (mv^2) / r

where m is the mass of the car, v is the velocity, and r is the radius of the curve.

Rearranging the formula to solve for r:

r = (mv^2) / Fc

Using the values given in the problem (speed = 19.0 m/s), you need the mass of the car and the centripetal force in order to calculate the radius. If these values are not provided, you may not be able to determine the radius.