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In-Depth Notes on Angular Kinematics

Objectives of Angular Kinematics

  • Understand how to describe angular motion.

  • Measure properties of angular motion like distance, displacement, velocity, and acceleration.

  • Explore the relationship between angular motion and linear motion.

  • Recognize practical applications of angular kinematics.

Definitions of Kinematics

  • Kinematics: The study of description of motion, focusing on the geometry of motion. It answers "what" happened in motion.

  • Angular Motion: Involves the motion of a body or body part around a fixed axis, measured in degrees or radians.

Relative versus Absolute Angles

  • Relative Angle: Angle formed between two adjacent body segments; zero when in anatomical reference position.

  • Absolute Angle: The angle of a single body segment with respect to a fixed line of reference, either horizontal or vertical.

Angular Distance and Displacement

  • Angular Distance (Φ): The total angle (in degrees or radians) that a body has rotated through.

  • Angular Displacement (θ): The difference between the initial and final angles, can be positive (counterclockwise) or negative (clockwise).

  • Right-hand Rule: Method to determine direction for angular measurement.

Measurement of Angular Distance

  1. Revolutions: 1 revolution = 360°.

  2. Degrees: One way to represent angular distance; 360° = 1 full revolution.

  3. Radians: Defined such that the angle subtended by an arc of a circle is equal to the radius of the circle.

    • 1 revolution = 2π radians ≈ 6.28 radians.

    • Conversion formulas:

      • Degrees to Radians: multiply by π/180.

      • Radians to Degrees: multiply by 180/π.

Relationships Between Linear and Angular Quantities

  • Linear Displacement (L): For a point on a rotating body, it is the product of the radius (r) and the angular displacement (θ in radians).

    • Formula: L = r × θ.

  • Example: If wheels of different diameters rotate for one complete revolution, their linear distances will differ based on their radius.

Angular Velocity and Speed

  • Angular Speed (σ): Change in angular distance over time. Units: degrees/s, radians/s, revolutions/s.

  • Angular Velocity (ω): Angular displacement over time. Computed as:

    • ω = θ/Δt.

Tangential Velocity

  • The linear path taken by an object after release from a circular motion.

  • Tangential Velocity (vT): Related to angular velocity by the formula:

    • vT = ω × r.

Example Calculation of Angular Velocity

  • Consider a golf club moving through a 170° angle in 0.4 seconds:

    • Calculation: ω = θ/t = 170°/0.4 s.

Angular Acceleration

  • Angular Acceleration (α): Rate of change of angular velocity.

    • Formula: α = (ωf - ωi)/t.

  • Describes changes in rotation speed or direction.

Centripetal Acceleration

  • Maintained in circular motion with constant angular velocity.

  • Direction of linear acceleration is always towards the center of rotation, caused by centripetal force, with an opposite force being centrifugal force.