A. A car is traveling at a speed of 55 mi/hr along a banked highway having a radius of curvature of 500 ft. At what angle should the road be banked in order that a zero friction force is needed for the car to go around this curve?
To find the angle at which the road should be banked, we need to consider that a zero friction force is needed for the car to go around the curve. This means that the net force acting on the car in the radial direction (towards the center of the curve) should be zero.
Let's break down the forces acting on the car:
1. The gravitational force (mg) acting vertically downward.
2. The normal force (N) exerted by the road, perpendicular to the surface.
3. The friction force (f) acting horizontally towards the center of the curve.
Since we want the friction force to be zero, the net radial force acting on the car should be zero. This can be achieved by equating the gravitational force component in the radial direction (mg*sinθ) to the centripetal force (mv^2 / r), where:
- m is the mass of the car,
- v is the velocity of the car,
- r is the radius of curvature of the road, and
- θ is the angle at which the road is banked.
The equation would be: mg*sinθ = mv^2 / r
Now, solve for θ:
1. Convert the speed from miles per hour to feet per second:
- 55 mi/hr = (55 * 5280 ft) / (1 hr * 3600 s) = 80.67 ft/s
2. Plug in the given values into the equation:
- mg*sinθ = mv^2 / r
- (m * 32.2 ft/s^2) * sinθ = (m * 80.67^2 ft^2/s^2) / 500 ft
3. Simplify the equation by canceling the mass:
- 32.2 ft/s^2 * sinθ = (80.67^2 ft^2/s^2) / 500 ft
4. Solve for sinθ:
- sinθ = (80.67^2 ft^2/s^2) / (500 ft * 32.2 ft/s^2)
- sinθ = 0.400
5. Take the inverse sine (sin^(-1)) of both sides to find θ:
- θ = sin^(-1)(0.400)
- θ ≈ 23.58 degrees
Therefore, the angle at which the road should be banked is approximately 23.58 degrees.