1. Understand the Concepts
* Angular Velocity (ω): The rate at which an object rotates, measured in radians per second (rad/s).
* Angular Acceleration (α): The rate at which angular velocity changes, measured in radians per second squared (rad/s²).
* Revolutions: One complete rotation of a circle.
2. Convert Units
* Initial Angular Velocity (ω₀): 100 rad/s (already in the correct unit)
* Final Angular Velocity (ω): We need to convert 50 revolutions to radians/second:
* 1 revolution = 2π radians
* 50 revolutions = 50 * 2π = 100π radians
* Since the wheel *decelerates* to 50 revolutions, its final angular velocity is 0 rad/s.
3. Apply the Angular Kinematic Equation
We'll use the following equation to relate initial angular velocity, final angular velocity, angular acceleration, and the number of revolutions (which we'll convert to radians):
ω² = ω₀² + 2αθ
Where:
* ω = final angular velocity (0 rad/s)
* ω₀ = initial angular velocity (100 rad/s)
* α = angular acceleration (what we want to find)
* θ = angular displacement (in radians)
4. Calculate Angular Displacement (θ)
* The wheel rotates 50 revolutions, so θ = 50 revolutions * 2π radians/revolution = 100π radians
5. Solve for Angular Acceleration (α)
Plug the values into the equation:
0² = (100 rad/s)² + 2α (100π radians)
Simplify and solve for α:
-10000 rad²/s² = 200πα
α = -10000 rad²/s² / (200π radians)
α ≈ -15.92 rad/s²
Answer:
The angular acceleration of the wheel is approximately -15.92 rad/s². The negative sign indicates that the wheel is decelerating (slowing down).
