Angular and Linear Speeds bicycle example

[madlog]

An image of a 15 speed bicycle is shown below. Since the tension on the chain is the same at both ends, when the chain is on a front gear and back gear, the linear speed of the front gear is equal to the linear speed of the back gear.

The gear ratios are shown in the table. The tire has a 26 inch outer diameter.

Front gears (pedals)​	Back gears (wheel)​
smallest 30 teeth (2.5 inch radius)​	smallest 11 teeth (0.9 inch radius)​
largest 52 teeth (4.3 inch radius)​	largest 25 teeth (2.1 inch radius)​

Your task is to find the speed of the bicycle given the speed of the front gear. Question 1 will walk you through the process and question 2 will have you try the process more independently. The steps for the entire process as follows:

1. 1. Find the linear speed of the front gear given its angular speed
   2. Find the linear speed of the back gear (the tension in the chain makes that equal to the linear speed of the front gear)
   3. Find the angular speed of the back gear based on the linear speed of the back gear
   4. Find the angular speed of the back wheel based on the angular speed of the back gear
   5. Find the linear speed of the back wheel based on the angular speed of the back wheel

1a. If the pedals are rotating the front gear at a rate of 60 rpm, what is the linear speed of the outside of the small front gear? (This is step 1) Show your work.(5 points)

1b. Using the linear speed from part a, if the chain is on the largest back gear, what is the angular speed of the rear wheel? Remember that the linear speed of the front gear is equal to the linear speed of the back gear (Steps 2 and 3) Show your work. (5 points)

1c. Using the angular speed from part b, what is the linear speed (in miles per hour) of the outside of the rear tire? (Steps 4 and 5) Show your work. (5 points)

2. Find the linear speed of the rear tire (in miles per hour) if the chain is on the largest front gear and the smallest back gear, and the pedals are rotating at 60 rpm. Show your work. (10 points)

 

[Deleted member 4993]

An image of a 15 speed bicycle is shown below. Since the tension on the chain is the same at both ends, when the chain is on a front gear and back gear, the linear speed of the front gear is equal to the linear speed of the back gear.

The gear ratios are shown in the table. The tire has a 26 inch outer diameter.

Front gears (pedals)​	Back gears (wheel)​
smallest 30 teeth (2.5 inch radius)​	smallest 11 teeth (0.9 inch radius)​
largest 52 teeth (4.3 inch radius)​	largest 25 teeth (2.1 inch radius)​

Your task is to find the speed of the bicycle given the speed of the front gear. Question 1 will walk you through the process and question 2 will have you try the process more independently. The steps for the entire process as follows:

1. 1. Find the linear speed of the front gear given its angular speed
   2. Find the linear speed of the back gear (the tension in the chain makes that equal to the linear speed of the front gear)
   3. Find the angular speed of the back gear based on the linear speed of the back gear
   4. Find the angular speed of the back wheel based on the angular speed of the back gear
   5. Find the linear speed of the back wheel based on the angular speed of the back wheel

1a. If the pedals are rotating the front gear at a rate of 60 rpm, what is the linear speed of the outside of the small front gear? (This is step 1) Show your work.(5 points)

1b. Using the linear speed from part a, if the chain is on the largest back gear, what is the angular speed of the rear wheel? Remember that the linear speed of the front gear is equal to the linear speed of the back gear (Steps 2 and 3) Show your work. (5 points)

1c. Using the angular speed from part b, what is the linear speed (in miles per hour) of the outside of the rear tire? (Steps 4 and 5) Show your work. (5 points)

2. Find the linear speed of the rear tire (in miles per hour) if the chain is on the largest front gear and the smallest back gear, and the pedals are rotating at 60 rpm. Show your work. (10 points)

Where are you stuck?

 

[Cubist]

Since the tension on the chain is the same at both ends, when the chain is on a front gear and back gear, the linear speed of the front gear is equal to the linear speed of the back gear.

This explanation isn't correct. It's easy enough to apply exactly the same force to two wheels of different inertias, both initially at rest, and I guarantee that one wheel will soon start to spin faster than the other.

The reason that the linear speeds of the two gears are the same is that a chain is not extensible.