Lecture Notes: Relating Angular Kinematics to Linear Motion

Key Concepts

• Angular Displacement: Measures rotation in radians. Example: angle of a baseball bat swing.
• Linear Displacement: Distance covered by the tip of the bat.

Relationship Between Angular and Linear Quantities

• Displacement: Angle in radians, arc length (linear distance), radius.
  • Equation: ( \Theta = \frac{S}{R} )
• Velocity:
  • Linear velocity is proportional to angular velocity and radius.
  • Equation: ( v_{linear} = r , \omega )
• Acceleration:
  • Tangential (along the path) and radial (toward the center).
  • Tangential: ( a_{t} = r , \alpha )
  • Radial: ( a_{r} = \frac{v^2}{r} ) or ( a_{r} = r , \omega^2 )

Application in Sports

• Objective: Maximize linear speed (e.g., volleyball spike, baseball throw).
• Radius & Arc Length: Increasing radius increases arc length and potential linear velocity.

Practical Examples

• Baseball Swing: Contact point on bat affects distance due to radius/arc length relationship.
• Throwing/Striking Motions: Full joint extension increases radius, thus increasing linear speed.
• Javelin Throw: Emphasize quick rotation and full extension for maximum release velocity.

Acceleration Types

• Tangential Acceleration: Changes in angular velocity affect tangential speed.
• Radial Acceleration: Maintains circular path; equivalent to centripetal force.

Resultant Acceleration

• Combination of tangential and radial accelerations.
• Equation: ( a_{r}^2 = a_{t}^2 + a_{radial}^2 )

Example Problems

• Conversions: Degrees to radians (e.g., 235 degrees = 4.1 radians).
• Angular Velocity: Calculate change in angle over time.
• Segment Angles: Calculate angles using endpoints or known segments.

Qualitative vs. Quantitative Analysis

• Qualitative: Subjective assessments (e.g., feel, smoothness).
• Quantitative: Numerical measurements for performance analysis.

Practice and Application

• Work through practice problems to understand concepts.
• Attend lectures for further clarification and examples.