Lecture 6b: Accelerated Motion and Uniform Circular Motion

Key Concepts:

• Tangential (Linear) and Angular Acceleration:
  • Previously discussed their definitions and relationships.
  • Focus on motion accelerated both tangentially and angularly.

Tangential vs Angular Quantities:

• Tangential (Linear/Translational) Quantities:

  • Velocity, acceleration, displacement (arc length 's').
  • Example equations:
    • ( x = v_0 t + \frac{1}{2} a t^2 )
    • ( v = v_0 + \frac{v}{2} t )

• Angular (Rotational) Quantities:

  • Angular velocity (\omega), angular acceleration (\alpha), angular displacement (\theta).
  • Related through: ( s = r\theta ), ( v = r\omega ), ( a = r\alpha )

Acceleration in Motion:

• Both motion types (rolling without slipping) relate the same linear and angular equations due to the arc length (s) being equal to displacement.

Example Problems:

1. CD Angular Acceleration:

   • CD spins from 480 rpm to 210 rpm over 74 minutes.
   • Calculate average angular acceleration (\alpha) using:
     • ( \alpha = \frac{\omega - \omega_0}{t} )
   • Convert units to radians/s².

2. Car Tire Rotation:

   • Car accelerates from 20 m/s with 1.5 m/s² over 0.48 seconds.
   • Calculate angle through which tire rotates using ( \theta = \frac{s}{r} ), with s calculated using linear motion equations.

Uniform Circular Motion:

• Period of Rotation (T): Time in seconds per revolution.
• Tangential Speed (v_t): (v_t = \frac{2\pi r}{T})
  • Uniform implies constant speed but changing velocity direction.
  • Demonstrated change in velocity direction causes acceleration.

Centripetal and Centrifugal Forces:

• Centripetal Acceleration:
  • Directed toward the circle's center.
  • Magnitude: ( a_c = \frac{v_t^2}{r} = r\omega^2 = \frac{4\pi^2r}{T^2})
  • Uniform circular motion involves continuous change in direction, not speed magnitude.

Combining Accelerations:

• Tangential acceleration changes speed magnitude.
• Centripetal acceleration changes speed direction.
• Combined using vector addition of (a_t) and (a_c).

Example Problem: Fan Blade Acceleration

• Blades increase speed from 1.5 rad/s with 2 rad/s² angular acceleration.
• Calculate total acceleration and angle between total and centripetal accelerations after 0.5s.

Summary:

• Tangential and angular accelerations are related; centripetal is separate but can be combined with tangential for total acceleration.
• Importance of understanding terms, units, and equation applications for different motion scenarios.