Lecture Notes: Spinning Bicycle Wheel

Key Concepts

Angular Speed (ω): Measured in RPM (revolutions per minute) and needs to be converted to radians per second for calculations.
Angular Acceleration (α): The change in angular speed over time.
Angular Displacement (θ): The total angle through which a point or line has been rotated in a specified sense about a specified axis.

First Part: Calculate the average angular acceleration when a cyclist starts cycling from rest to a final angular speed.
Second Part: Calculate the average angular acceleration, angular displacement, and number of revolutions when the cyclist applies brakes to stop the wheel.

Initial Conditions:
- Initial Angular Speed (ω_i) = 0 rad/s
- Final Angular Speed (ω_f) = 200 RPM or 20.9 rad/s (conversion: 200 * π / 30)
- Time (Δt) = 4 seconds
Average Angular Acceleration (α):
- Formula: α = (ω_f - ω_i) / Δt
- Calculation: α = (20.9 - 0) / 4 = 5.22 rad/s²
- Interpretation: A positive value indicates increasing speed.*

Initial Conditions:
- Initial Angular Speed (ω_i) = 20.9 rad/s
- Final Angular Speed (ω_f) = 0 rad/s
- Time (Δt) = 20 seconds
Average Angular Acceleration (α):
- Formula: α = (ω_f - ω_i) / Δt
- Calculation: α = (0 - 20.9) / 20 = -1.05 rad/s²
- Interpretation: A negative value indicates decreasing speed.
Angular Displacement (θ):
- Using the formula: ω_f² = ω_i² + 2 * α * θ
- Calculation for θ: 0 = 20.9² + 2 * (-1.05) * θ
- Solve for θ: θ = 208 radians
Number of Revolutions:
- 1 revolution = 2π radians
- Calculation: θ / 2π = 208 / 2π ≈ 33 revolutions

Calculations involved transforming linear motion equations to rotational motion equivalents.
Understanding the sign of angular acceleration helps interpret whether the speed is increasing or decreasing.

Successfully calculated average angular acceleration, angular displacement, and number of revolutions for both scenarios: starting motion and braking.
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