A car is travelling around a banked curve at a constant speed of 84.0 km/h. If the radius of curvature of the path is 67.0 m, and there is no friction between the car and the surface, at what angle (in degrees) should the bank be set at, relative to the horizontal, in order for the car not to slip in either direction?

Question

A car is travelling around a banked curve at a constant speed of 84.0 km/h. If the radius of curvature of the path is 67.0 m, and there is no friction between the car and the surface, at what angle (in degrees) should the bank be set at, relative to the horizontal, in order for the car not to slip in either direction?

Answer 1

v = 84.0 * 1000m/km/3600 s/h = 84/3.6 m/s

Ac = v^2/R
tan A = (v^2/R)/g

Answer 2

To determine the angle at which the banked curve should be set, we need to consider the forces acting on the car. Since there is no friction, the car will rely on the normal force and the gravitational force to keep it moving in a curved path.

1. Draw a free-body diagram of the car on the banked curve. Identify the forces acting on the car:

- The gravitational force (mg) acting vertically downwards, where m represents the mass of the car and g represents the acceleration due to gravity.
- The normal force (N) acting perpendicular to the surface of the banked curve.
- The centripetal force (Fc) acting towards the center of the circular path.

2. Decompose the gravitational force into its components. The vertical component (mgcosθ) contributes to the normal force, and the horizontal component (mgsinθ) contributes to the centripetal force.

3. Equate the centripetal force to the horizontal component of the gravitational force:

Fc = mgsinθ

4. The centripetal force can also be expressed as:

Fc = (mv^2) / r

where v is the velocity of the car and r is the radius of curvature.

5. Set the two expressions for centripetal force equal to each other:

mgsinθ = (mv^2) / r

6. Cancel out the mass (m) from both sides of the equation:

gsinθ = v^2 / r

7. Rearrange the equation to solve for the angle (θ):

θ = arcsin(v^2 / (rg))

8. Plug in the given values: v = 84.0 km/h = 23.3 m/s, r = 67.0 m, and g = 9.8 m/s^2:

θ = arcsin((23.3^2) / (67.0 * 9.8))

9. Calculate the angle using a scientific calculator, or use the arcsin function on a calculator. The angle should be approximately 26.7 degrees.

Therefore, the banked curve should be set at an angle of approximately 26.7 degrees relative to the horizontal to prevent the car from slipping in either direction.