Suppose that Lance Armstrong1 is using a 180 mm chain ring and pedaling at a rate of 80 revolutions per minute. The wheels on his road bike have a diameter of 622mm. What diameter sprocket on the rear wheel does he need to maintain a speed of 30 kilometers per hour? In your solution, show your work, quote formulas you use, and explain your reasoning.
To find the diameter of the sprocket on the rear wheel that Lance Armstrong needs to maintain a speed of 30 kilometers per hour, we can use the formula for the gear ratio.
The gear ratio is defined as the ratio of the number of teeth on the front chainring to the number of teeth on the rear sprocket.
Given that Lance Armstrong is using a 180 mm chainring, which is equivalent to a circumference of:
C = 2 * π * r
C = 2 * π * (180 mm / 2)
C = 360 * π mm
To convert the speed from kilometers per hour to meters per minute, we divide by 60:
Speed = 30 km/h
Speed = (30,000 m / 60) m/min
Speed = 500 m/min
The distance traveled by the bike in one minute is equal to the chainring circumference multiplied by the number of revolutions per minute. Since the distance traveled in one minute is equal to the speed of the bike (when there is no slippage), we can set up the following equation:
Distance traveled in one minute = Speed = Chainring circumference * Revolutions per minute
Rear Sprocket circumference = Chainring circumference * (Number of teeth on the front chainring / Number of teeth on the rear sprocket)
Let's assume the number of teeth on the rear sprocket is 'x'. Therefore, the rear sprocket's diameter is equal to the circumference of the sprocket:
Rear Sprocket diameter = π * x
By substituting the values and rearranging the equation, we can solve for the number of teeth on the rear sprocket (x):
Rear Sprocket circumference = Chainring circumference * (Number of teeth on the front chainring / Number of teeth on the rear sprocket)
Rear Sprocket circumference = 360 * π * (Number of teeth on the front chainring / Number of teeth on the rear sprocket)
π * x = 360 * π * (180 mm / 2) / (80 * 500)
Simplifying the equation:
x = (360 * 180 mm) / (80 * 500)
Converting the answer from millimeters to meters:
x = (360 * 0.18 m) / (80 * 500)
Finally, calculating the value of x:
x = 0.009 m = 9 mm
Therefore, Lance Armstrong would need a rear sprocket with a diameter of 9 mm to maintain a speed of 30 kilometers per hour.
To determine the diameter of the sprocket on the rear wheel that Lance Armstrong needs to maintain a speed of 30 kilometers per hour, we need to consider the gear ratios involved.
First, let's convert the given speed from kilometers per hour to meters per second. Since there are 1000 meters in a kilometer and 3600 seconds in an hour, the conversion factor is 1000/3600 = 0.27778.
So, the speed in meters per second is 30 km/h * 0.27778 = 8.33333 m/s.
Next, we can calculate the gear ratio by dividing the linear speed of the bike by the linear speed of the pedal stroke. The linear speed of the bike can be calculated by multiplying the diameter of the rear wheel by π, and the linear speed of the pedal stroke can be calculated by multiplying the diameter of the chain ring by π.
Given:
Chain ring diameter (D1): 180 mm
Pedal revolutions per minute (RPM): 80
Rear wheel diameter (D2): 622 mm
Speed (v): 8.33333 m/s
First, let's convert all the diameters from millimeters to meters:
D1 = 180 mm = 0.18 m
D2 = 622 mm = 0.622 m
Now, let's calculate the linear speed of the bike:
Linear speed of the bike = D2 * π
Next, let's calculate the linear speed of the pedal stroke:
Linear speed of the pedal stroke = D1 * π
The gear ratio is given by the linear speed of the bike divided by the linear speed of the pedal stroke:
Gear ratio = (D2 * π) / (D1 * π) = D2 / D1
Now we can solve for the diameter of the sprocket on the rear wheel (D3) that will maintain a speed of 8.33333 m/s (or 30 km/h):
D3 / D1 = D2 / D1
D3 = (D2 * D1) / D1
D3 = D2
So, the diameter of the sprocket on the rear wheel that Lance Armstrong needs to maintain a speed of 30 kilometers per hour is equal to the diameter of the rear wheel.
In this case, the diameter of the sprocket on the rear wheel is 622 mm or 0.622 m.
Note: It is important to assume that Lance Armstrong is pedaling at a constant rate of 80 revolutions per minute throughout the calculation.