On a banked race track, the smallest circular path on which cars can move has a radius of 119 m, while the largest has a radius of 159 m, as the drawing illustrates. The height of the outer wall is 16.8 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 speed at which cars can move on the banked race track without relying on friction, we can use the concept of centripetal force.
(a) Smallest Speed:
For the smallest circular path, with a radius of 119 m, we need to determine the speed at which the centripetal force is provided solely by the component of gravitational force acting towards the center of the circle. This is given by the equation:
mg = (mv²) / r
Where:
m = mass of the car (which cancels out in the equation)
g = acceleration due to gravity (9.8 m/s²)
v = speed of the car
r = radius of the smallest circular path (119 m)
Rearranging the equation, we get:
v = √(rg)
Plugging in the values, we have:
v = √(9.8 m/s² * 119 m)
v = √(1166.2 m²/s²)
v ≈ 34.16 m/s
Therefore, the smallest speed at which cars can move on the track without relying on friction is approximately 34.16 m/s.
(b) Largest Speed:
For the largest circular path, with a radius of 159 m, we need to determine the speed at which the centripetal force is provided solely by the component of gravitational force acting towards the center of the circle. Using the same equation as above, we have:
v = √(rg)
Plugging in the values, we have:
v = √(9.8 m/s² * 159 m)
v = √(1558.2 m²/s²)
v ≈ 39.47 m/s
Therefore, the largest speed at which cars can move on the track without relying on friction is approximately 39.47 m/s.