Lecture Notes: Circular and Rotational Motion

Introduction

• Transition from linear (translational) motion to circular motion
• Circular motion with non-constant angular speed

Key Concepts

Angular Displacement (Theta)

• Previously referred to as rotation angle
• Definition: Ratio of arc length (s or x) to radius of curvature
• Units: Radians

Angular Velocity

• Rate of change of angle
• Related to linear speed: linear speed over radius
• Units: Radians per second

Angular Acceleration (Alpha)

• Rate of change of angular velocity
• Similar to translational acceleration, but angular
• Units: Radians per second squared
• Formula: Translational acceleration (a) = Radius x Angular acceleration (alpha)

Vector Direction

• Direction determined using the right-hand rule
• Similar patterns between angular and linear velocity/acceleration
  • Speeding up: Vectors have same direction
  • Slowing down: Vectors have opposite directions

Rotational vs. Translational Concepts

• Same equations apply in rotational dynamics as in translational
• Translational position (X, Y, R, Z) vs Angular position (Theta)
• Translational velocity vs Angular velocity
• Translational acceleration vs Angular acceleration

Kinematic Equations in Rotation

• Similar to translational kinematic equations
• Do not mix rotational and linear variables in the same equation

Example Problem: Motorcycle Wheels

• Calculate angular acceleration of wheels
• Given: Linear acceleration, time, radius
• Method: Convert linear variables to angular variables or vice versa

Moment of Inertia

• Rotational equivalent of mass
• Depends on mass distribution relative to rotation axis
• Formula: Sum of mass times position squared
• Units: Kilogram meter squared

Example Problem: Moment of Inertia

• Calculating for two masses on a rod
• Rotating around middle vs rotating around one end

Solid Objects and Moment of Inertia

• Use integrals for continuous mass distribution
• Reference table for common shapes and their moment of inertia
• Shapes include hoops, cylinders, discs, rods, spheres, etc.

Conclusion

• Next class will continue with rotational dynamics
• Reminder to complete checkpoints and prepare questions for class discussion