Applications of Circular Motion: Bicycle Chain Drive Problem

Applications of Circular Motion

Bicycle Chain Drive Problem

This problem explores the relationship between rotational motion and linear displacement using the example of a bicycle's chain drive mechanism. Understanding this application is important for connecting concepts of angles, arc length, and the geometry of circles to real-world mechanical systems.

• Rotational Motion: When the pedals of a bicycle are rotated, they turn the front gear (chainring), which in turn moves the chain and rotates the rear wheel.

• Given Data:

  • Radius of pedal gear (chainring): 4.03 in

  • Radius of rear gear (sprocket): 1.35 in

  • Radius of bicycle wheel: 13.9 in

  • Pedals rotated through: 180° (which is $\pi$ radians)

Key Concepts and Formulas

• Arc Length: The distance a point on a circle travels when the circle is rotated through an angle $\theta$ (in radians) is given by: $s = r\theta$ where $s$ is the arc length, $r$ is the radius, and $\theta$ is the angle in radians.

• Gear Ratio: The ratio of the number of teeth (or radii) of the front and rear gears determines how many times the rear wheel turns for each pedal rotation: $\text{Gear Ratio} = \dfrac{\text{Radius of Chainring}}{\text{Radius of Sprocket}}$ For this problem: $\text{Gear Ratio} = \dfrac{4.03}{1.35}$

• Wheel Circumference: The distance the bicycle moves in one full rotation of the wheel is the circumference: $C = 2\pi r$ where $r$ is the radius of the wheel.

Step-by-Step Solution

1. Find the arc length the chain travels as the pedals rotate 180°: Angle in radians: $180^\circ = \pi$ radians Arc length on chainring: $s = r\theta = 4.03 \times \pi$ $s \approx 12.66$ inches

2. Determine how many times the rear sprocket turns: The same length of chain passes over the rear sprocket, so the angle turned by the rear sprocket is: $\theta_{\text{rear}} = \dfrac{s}{r_{\text{rear}}} = \dfrac{12.66}{1.35} \approx 9.38$ radians

3. Find how many wheel revolutions this causes: Each full revolution is $2\pi$ radians, so number of revolutions: $\text{Revolutions} = \dfrac{\theta_{\text{rear}}}{2\pi} \approx \dfrac{9.38}{6.283} \approx 1.49$ Additional info: In a real bicycle, the rear sprocket is attached to the wheel, so the wheel turns the same number of times as the rear sprocket.

4. Calculate the distance the bicycle moves: Distance = (Number of wheel revolutions) × (Wheel circumference) Wheel circumference: $C = 2\pi \times 13.9 \approx 87.3$ inches Distance: $1.49 \times 87.3 \approx 130$ inches

Summary Table: Key Quantities

Quantity	Value	Formula
Arc length (chainring)	12.66 in	$4.03 \times \pi$
Angle turned (rear sprocket)	9.38 radians	$\dfrac{12.66}{1.35}$
Wheel circumference	87.3 in	$2\pi \times 13.9$
Distance moved	130 in	$1.49 \times 87.3$

Example Application

Example: If the pedals are rotated through a different angle, such as 360°, simply double the arc length and repeat the steps above to find the new distance traveled.

Additional info: This problem demonstrates the use of radian measure, arc length, and proportional reasoning in a mechanical context, which are key Precalculus skills.