On a banked race track, the smallest circular path on which cars can move has a radius of 114 m, while the largest has a radius of 170 m, as the drawing illustrates. The height of the outer wall is 17.5 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.
(a) Smallest speed:
To find the smallest speed, we need to consider the car moving on the smallest circular path with a radius of 114 m. At this small radius, the gravitational force acting downwards can contribute partially to the necessary centripetal force.
Let's calculate the minimum speed at which the car can move without relying on friction:
Step 1: Determine the gravitational force component acting towards the center of the circular path.
- The gravitational force component acting towards the center can be found by multiplying the weight of the car (mg) by the sine of the banking angle (θ).
- The banking angle (θ) can be found using the given height of the outer wall (17.5 m) and the radius of the circular path (114 m).
- Using the trigonometric relationship: sine (θ) = opposite / hypotenuse, we have:
sine (θ) = 17.5 m / 114 m
θ = arcsin (17.5 / 114)
Step 2: Calculate the gravitational force component:
- The gravitational force component (Fg) towards the center is given by:
Fg = mg * sin(θ)
(where m is the mass of the car and g is the acceleration due to gravity)
Step 3: Calculate the centripetal force required:
- The centripetal force (Fc) required for the car to move on the circular path is given by:
Fc = m * v^2 / r
(where v is the speed of the car and r is the radius of the circular path)
Since the car is not relying on friction to move, the centripetal force is provided by the combination of gravitational force and the normal force, N (which acts perpendicular to the surface of the track).
Step 4: Set up the equation equating the centripetal force and the gravitational force components:
- Fc = Fg + N
- Since N is equal to the weight of the car (mg) and the car is not relying on friction, we have:
Fc = Fg + mg
Step 5: Substitute the expressions for Fc and Fg:
- m * v^2 / r = m * g * sin(θ) + m * g
- Cancel out the mass (m) on both sides:
v^2 / r = g * sin(θ) + g
v^2 = r * (g * sin(θ) + g)
v = √(r * (g * sin(θ) + g))
Now we can substitute the given values to find the minimum speed:
- r = 114 m
- g = 9.8 m/s^2 (approximate acceleration due to gravity)
Substituting these values into the equation above, we can find the minimum speed (v) at which the car can move on the smallest circular path without relying on friction.
(b) Largest speed:
To find the largest speed, we need to consider the car moving on the largest circular path with a radius of 170 m. At this larger radius, the gravitational force acting downwards is perpendicular to the circular path, and there is no contribution to the necessary centripetal force.
The largest speed can be found using the same equation as in step 5, but this time the gravitational force component will not be considered since it does not contribute to the centripetal force.
Substituting the given values into the equation, we can find the largest speed (v) at which the car can move on the largest circular path without relying on friction.
Note: Both the smallest and largest speeds calculated in this manner assume idealized conditions without considering any other forces or factors that may affect the actual speed of the cars on the banked race track.
v=[r*g*tan(theta)]^.5
theta=height/(max radius - min radius)
To solve for smallest speed, substitute min radius for "r" in velocity equation.
To solve for largest speed, substitute max radius for "r" in velocity equation.
Good luck! Greetings from UCF!