A car goes around a curve on a road that is banked at an angle of 30.5 . 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?

force up the hill=force down the hill

mv^2/r cos30.5=mg Sin30.5

solve for r.

plugging everything in do I get 84.15? That seems a little high...

nvm. i got it. 84 was wrong... thanks for your help!

i get 62.536 when i do it. what is the answer

Smack-dab what I was lokonig for—ty!

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

The centripetal force acting on the car is provided by the horizontal component of the normal force exerted by the road. This force is directed towards the center of the curve.

Let's break down the forces acting on the car in the horizontal direction:

1. Weight (mg): This force is acting vertically downward and does not contribute to the centripetal force.

2. Normal force (N): This force is acting perpendicular to the road surface and can be resolved into vertical and horizontal components.

3. Friction force (f): Since the problem states that there is no friction between the tires and the road, this force can be ignored.

Considering the equilibrium of forces in the horizontal direction, we have:

centripetal force = horizontal component of the normal force

The centripetal force is given by:

Fc = m * (v^2 / r)

Where
- Fc is the centripetal force
- m is the mass of the car
- v is the velocity of the car
- r is the radius of the curve

The horizontal component of the normal force is given by:

Nh = N * cos(theta)

Where
- Nh is the horizontal component of the normal force
- N is the normal force
- theta is the angle of banking

Since the car is on the verge of slipping, the centripetal force is equal to the horizontal component of the normal force:

Fc = Nh

Substituting the equations, we get:

m * (v^2 / r) = N * cos(theta)

Given that the angle of banking (theta) is 30.5 degrees, the speed (v) is 19.0 m/s, and there is no friction (f), we can solve for the radius (r):

r = (m * v^2) / (N * cos(theta))

To find the radius of the curve, we need to know the mass of the car and the normal force exerted on the car by the road.