10.1 Angular Acceleration - College Physics

Key Concepts

• Angular Velocity (ω): Time rate of change of angle. It can be related to linear velocity (v) by the equation ( v = rω ), where ( r ) is the radius.
• Angular Acceleration (α): Rate of change of angular velocity over time. Formula: ( α = \frac{Δω}{Δt} ).
  • Units: radians per second squared (rad/s²).
  • Positive α indicates an increase in angular velocity, while negative α indicates a decrease.

Applications and Examples

Example 10.1: Bike Wheel

Scenario: A bicycle wheel spins from rest to 250 rpm in 5 seconds.

• a) Angular Acceleration (α):
  • Convert rpm to rad/s: ( 250 \text{ rpm} = 26.2 \text{ rad/s} )
  • Calculate ( α = \frac{26.2 \text{ rad/s}}{5 \text{ s}} = 5.24 \text{ rad/s}^2 )
• b) Deceleration when brakes are applied:
  • Given ( α = -87.3 \text{ rad/s}^2 )
  • Time to stop: ( t = \frac{-26.2 \text{ rad/s}}{-87.3 \text{ rad/s}^2} = 0.300 \text{ s} )

Example 10.2: Motorcycle Wheel

Scenario: Motorcycle accelerates from 0 to 30 m/s in 4.2 s.

• Find Angular Acceleration:
  • Linear acceleration ( a_t = \frac{30 \text{ m/s}}{4.2 \text{ s}} = 7.14 \text{ m/s}^2 )
  • Wheel radius ( r = 0.320 \text{ m} )
  • Angular acceleration ( α = \frac{a_t}{r} = \frac{7.14 \text{ m/s}^2}{0.320 \text{ m}} = 22.3 \text{ rad/s}^2 )

Relationship Between Angular and Linear Quantities

• Linear acceleration ( a_t ) is proportional to angular acceleration ( α ):
  • Formula: ( a_t = rα )
• Greater angular acceleration results in greater linear acceleration.
• Table 10.1 provides a comparison of rotational and translational quantities:
  • ( θ ) (angular position) corresponds to ( x ) (linear position)
  • ( ω ) corresponds to ( v )
  • ( α ) corresponds to ( a )

Check Your Understanding

• Angular acceleration is a vector with magnitude and direction.
  • Direction is denoted by + or - sign.
  • Example: A gymnast's angular momentum during a flip.

Additional Concepts

• Tangential Acceleration (a_t): Changes in velocity magnitude, tangent to the circular path.
• Centripetal Acceleration (a_c): Changes direction of velocity, towards the center.

Take-Home Experiment

• Experiment: Rotate on a chair and sketch out angle, angular velocity, and angular acceleration over time.

Ladybug Revolution Activity

• Interactive exploration of rotational motion, including concepts like angular velocity and acceleration.

Note: Angular acceleration relationships mirror those in linear motion, important for understanding both types of motion in real-world applications.