On a banked race track, the smallest circular path on which cars can move has a radius of 112 m, while the largest has a radius of 165 m, as the drawing illustrates. The height of the outer wall is 18 m. Find (a) the smallest and (b) the largest speed at which cars can move on this track without relying on friction.
To find the smallest and largest speeds at which cars can move on the banked race track without relying on friction, we can use the concept of centripetal force.
Let's start by analyzing the forces acting on a car moving on the banked track:
1. Gravitational force (mg): The weight of the car acts vertically downward.
2. Normal force (N): The perpendicular force exerted by the track on the car, which prevents it from sinking into the track or flying off. It can be split into two components:
- Vertical component (Nv): Acts in the upward direction, balancing the gravitational force (Nv = mg).
- Horizontal component (Nh): Acts inward towards the center of the circular path.
3. Centripetal force (Fc): The force acting towards the center of the circular path, responsible for keeping the car moving in a circle.
Now, let's calculate the required variables. The height of the outer wall (h) is given as 18 m.
(a) Smallest speed:
For the smallest circular path with a radius of 112 m, we assume the car is moving at the smallest speed without relying on friction. In this case, the normal force (N) acts at an angle towards the center of the circle, making the car move in a circular path.
Using trigonometry:
sinθ = h / r - where θ is the angle of inclination with the horizontal plane.
Substituting the given values:
sinθ = 18 / 112
θ ≈ 0.1604 rad
The horizontal component of the normal force (Nh) can be calculated:
Nh = N * cosθ ≈ mg * cosθ ≈ mg * cos(0.1604 rad)
Since the horizontal component of the normal force (Nh) provides the centripetal force (Fc), we can equate these two forces:
Nh = Fc
mg * cos(0.1604 rad) = (mv^2) / r - where m is the mass of the car and v is the speed.
Simplifying, we can solve for v:
v = √(g * r * cos(0.1604 rad))
Substituting the known values:
v = √(9.8 m/s^2 * 112 m * cos(0.1604 rad))
Calculating this expression will give us the smallest speed at which cars can move on the track without relying on friction.
(b) Largest speed:
For the largest circular path with a radius of 165 m, we follow the same steps as above to calculate the largest speed.
v = √(g * r * cos(θ))
Substituting the known values:
v = √(9.8 m/s^2 * 165 m * cos(θ))
Calculate this expression to find the largest speed at which cars can move on the track without relying on friction.
Note: It is crucial to check the calculated values and units to ensure accuracy. Additionally, assume the absence of any other external forces such as wind resistance or air resistance, which might affect the motion of the cars.