Lecture Notes: Rotational Dynamics

Introduction to Rotational Dynamics

• Transition from kinematics to rotational dynamics.
• Previously studied:
  • 1D and 2D kinematics (displacement, velocity, acceleration).
  • Projectiles and independent perpendicular vectors.
• New focus: How objects move in circles or rotate.

Problem Example

• Four objects on an inclined ramp with same mass and radius:
  • Green: Spherical shell (basketball)
  • Red: Solid sphere (pool ball)
  • Blue: Ring (cylindrical shell)
  • Gold: Solid cylinder (can of tuna)
• Despite same mass and radius, objects finish race at different times:
  • Solid sphere wins, cylinder second, ring last.
• Importance of mass distribution in rotational behavior.

Key Concepts in Rotational Dynamics

Rotational Angle (Theta - θ)

• Describes rotational motion, analogous to linear distance.
• Defined as the arc length (s) divided by radius (r).
• Measured in radians (rad).
• 2π radians = 1 revolution = 360 degrees.

Angular Velocity (Omega - ω)

• Rate of change of rotational angle, analogous to linear velocity.
• Measured in radians per second (rad/s).
• Formula: ω = Δθ / Δt and ω = v / r (where v is linear velocity).

Angular Acceleration (Alpha - α)

• Rate of change of angular velocity, analogous to linear acceleration.
• Measured in radians per second squared (rad/s²).
• Formula: α = Δω / Δt

Examples and Applications

Example 1: Car Tire

• Calculate angular velocity of a car tire with a 0.3m radius at 54 km/h.
• Steps:
  1. Convert speed to m/s.
  2. Use ω = v / r to find ω.

Example 2: Bicycles and Brakes

• Student spins bicycle wheel to 250 RPM in 5s.
• Calculating angular acceleration:
  • Convert RPM to rad/s.
  • Use α = Δω / Δt.
• Braking example:
  • Use α = Δω / Δt with negative acceleration.

Relationship Between Circular and Linear Motion

• Linear acceleration (aₜ) is tangent to circular path.
• Centripetal acceleration (aₙ) is directed towards center.
• aₜ changes speed; aₙ keeps object in circular path.
• Formula: aₜ = αr, showing relationship between linear and angular acceleration.

Rotational Kinematics Formulas

• Analogous to linear kinematic equations:
  • ω = ω₀ + αt analogous to v = v₀ + at.
  • θ = ω₀t + 0.5αt² analogous to d = v₀t + 0.5at².

Practical Problem Solving

Deep-Sea Fisherman Example

• Fishing reel with radius of 4.5 cm.
• Given angular acceleration and time, calculate:
  • Final angular velocity: Use ω = ω₀ + αt.
  • Linear speed of line: v = ωr.
  • Revolutions made: θ in revolutions (θ / 2π).
  • Linear distance of line: s = θr.

Conclusion

• This lecture introduced basic principles of rotational dynamics.
• Established the parallels between rotational and linear motion.
• Provided foundational knowledge for more advanced topics in rotational dynamics.