On a banked race track, the smallest circular path on which cars can move has a radius r1 = 103 m, while the largest has a radius r2 = 157 m, as the drawing illustrates. The height of the outer wall is 18 m.
To find the height of the inner wall, you need to determine the difference in height between the inner and outer walls of the banked race track.
First, let's define some variables:
- Height of the outer wall: h1 = 18 m
- Radius of the inner circular path: r1 = 103 m
- Radius of the outer circular path: r2 = 157 m
- Height of the inner wall: h2 (what we're trying to find)
Now, let's analyze the problem:
1. Identify the relevant concepts:
- Banked race track: The track is built at an incline, allowing vehicles to safely navigate curves at high speeds.
- Centripetal force: The force required to keep an object moving in a curved path.
- Friction: The force that opposes the motion of an object on a surface.
2. Understand the forces involved:
- The centripetal force required to keep the cars moving in a circular path is provided by a combination of vertical forces:
- The component of the car's weight acting perpendicular to the track surface (mg * cosθ), where θ is the angle of the incline.
- The normal force (N) exerted by the track surface upward on the cars.
- The frictional force (f) between the tires of the cars and the track surface prevents the cars from sliding down the incline.
3. Apply the principles of equilibrium:
- When the cars are moving at a constant speed along the banked track, the net force acting on them in the vertical direction is zero (since there is no vertical acceleration).
- This means that the sum of the vertical components of the forces (mg * cosθ and N) must equal zero.
4. Calculate the angle of the incline (θ):
- Since the cars are moving at a constant speed, the vertical component of the car's weight must be equal to the normal force (N): mg * cosθ = N.
- Rearranging the equation, we can solve for θ: θ = cos⁻¹(N / mg)
- To find N, we need to consider the forces on the car at the outer wall. The only additional force acting on the car is the height of the outer wall (h1).
- The vertical component of the car's weight is given by mg * cosθ = (mg * h1) / r2 (considering the right-angled triangle formed by the weight, height, and radius).
- Rearranging the equation, we find N = mg * r2 / h1.
- Plugging this value of N into the equation for θ, we get θ = cos⁻¹((mg * r2) / (h1 * mg)) = cos⁻¹(r2 / h1).
5. Find the height of the inner wall (h2):
- Since the inner wall has a smaller radius (r1) than the outer wall (r2), the angle of the incline (θ) will be larger at the inner wall.
- We can use the same equation for θ, but with the radius r1: θ = cos⁻¹(r1 / h2).
- Rearranging the equation, we can solve for h2: h2 = r1 / cos(θ).
- Plugging in the value of θ from step 4, we can calculate h2.