On a banked race track, the smallest circular path on which cars can move has a radius of 101 m, while the largest has a radius of 163 m, as the drawing illustrates. The height of the outer wall is 19.0 m. Find (a) the smallest and (b) the largest speed at which cars can move on this track without relying on friction.
without relying on friction, I assume you mean that the cars scrape along the wall, since they cannot grip the road without 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:
To find the smallest speed, we need to consider the car moving on the smallest circular path with a radius of 101 m.
The diagram shows that the height of the outer wall is 19.0 m. This height, along with the radius, forms a right-angled triangle.
Using trigonometry, we can find the angle between the horizontal and the inclined plane of the track. Let's call this angle θ.
sin(θ) = height / radius
sin(θ) = 19.0 / 101
θ ≈ 0.1883 radians
Now, the net force acting on the car is the horizontal component of the car's weight. The vertical component of the car's weight is balanced by the normal force from the road surface.
The horizontal component of the car's weight provides the centripetal force to keep the car moving in a circular path.
F_c (centripetal force) = m (mass) * v^2 (velocity) / r (radius)
Since there is no friction to provide any additional force, the centripetal force is only the horizontal component of the car's weight.
F_c = m * g * sin(θ)
Equating the two expressions for centripetal force, we have:
m * g * sin(θ) = m * v^2 / r
Simplifying, we find:
v^2 = g * r * sin(θ)
Plugging in the values:
v^2 = 9.8 m/s^2 * 101 m * sin(0.1883 radians)
v^2 ≈ 187.26
v ≈ √187.26 ≈ 13.68 m/s
Therefore, the smallest speed at which cars can move on this track without relying on friction is approximately 13.68 m/s.
(b) Largest Speed:
To find the largest speed, we need to consider the car moving on the largest circular path with a radius of 163 m.
Using the same calculations as above, we find:
v^2 = g * r * sin(θ)
Plugging in the values:
v^2 = 9.8 m/s^2 * 163 m * sin(0.1883 radians)
v^2 ≈ 302.90
v ≈ √302.90 ≈ 17.39 m/s
Therefore, the largest speed at which cars can move on this track without relying on friction is approximately 17.39 m/s.