Linear and Angular Kinematics

Linear and Angular Kinematics I

Course Information

• Instructor: Chris Kuenze, PhD, ATC

Knowledge Check #1 - Cyclist Example

• Scenario: A professional cyclist rides from Charlottesville to Harrisonburg.

  • Displacement: 61 km

  • Distance: 110 km (country roads)

  • Time: 5 hours and 10 minutes

• Task: Calculate the cyclist's average speed.

Required Data

• Total Distance Traveled: 110 km, which converts to:

  • $110 ext{ km} = 110,000 ext{ meters}$

• Total Time Taken: 5 hours and 10 minutes, converted to seconds:

  • $5 ext{ hr} = 5 imes 3600 + 10 imes 60 = 18,600 ext{ seconds}$

Calculation of Speed

• Formula for Speed:

  • Average speed is calculated using the formula:

  • $ext{Average speed} = rac{ ext{Total Distance}}{ ext{Total Time}}$

• Substituting Values:

  • $ext{Average speed} = rac{110,000 ext{ m}}{18,600 ext{ s}} ext{ m/s}$

• Result:

  • $ext{Average speed} ext{ (approx)} = 5.91 ext{ m/s}$

Linear Acceleration

• Formula Explanation:

  • Angular acceleration denoted as: $ext{α}$

  • Acceleration is defined as:

  • $ext{Acceleration} = rac{ ext{Change in Velocity}}{ ext{Change in Time}}$

• Sample Calculation:

  • Given:

  • Starting Velocity: $0 ext{ m/s}$

  • End Velocity: $10.44 ext{ m/s}$

  • Start Time: $0 ext{ s}$

  • End Time: $9.58 ext{ s}$

• Calculation of Changes:

  • Change in Velocity:

  • $ext{Change in Velocity} = 10.44 ext{ m/s} - 0 ext{ m/s} = 10.44 ext{ m/s}$

  • Change in Time:

  • $ext{Change in Time} = 9.58 ext{ s} - 0 ext{ s} = 9.58 ext{ s}$

Resulting Acceleration

• Final Calculation:

  • $ext{Acceleration} = rac{10.44 ext{ m/s}}{9.58 ext{ s}} = 1.09 ext{ m/s}^2$

Linear and Angular Kinematics II

• Key Takeaways:

  • Relationship between axis of rotation and plane of motion.

  • Basic metrics for describing angular kinematics including:

  • Angular velocity (ω)

  • Angular acceleration (α)

  • Directional systems for motion.

Axes of Rotation

• Concept: As motion occurs, joints turn around an axis related to the motion plane.

• Types of Axes:

  • Mediolateral Axis of Rotation: Runs side to side

  • Anteroposterior Axis of Rotation: Runs front to back

  • Longitudinal Axis of Rotation: Runs top to bottom

Rotary (Angular) Motion

• Definition: Movement in which each point on a segment moves through the same angle at the same time and at a constant distance from the center of rotation.

Degrees of Freedom

• Definition: Indicates the movement options available for a segment.

• Six Degrees of Freedom: An unconstrained segment can rotate around three axes, giving:

  • $ext{Degrees of Freedom} = 3 ext{ axes} imes 2 ext{ directions} = 6$

Conversion Between Units

• Linear vs Rotary Motion Units:

  • Linear motion is measured in meters.

  • Rotational motion is measured in radians:

  • $1.0 ext{ radian} ext{ (approx)} = 57.3 ext{ degrees}$

Example Conversion from Degrees to Radians

• 72 Degrees Conversion:**

  • $ext{Radians} = 72° imes rac{ ext{π}}{180} = 1.25664 ext{ radians}$

  • To convert radians back to degrees:

  • $ext{Degrees} = ext{Radians} imes rac{180}{ ext{π}}$

  • Example: $1.4 ext{ rad} imes rac{180}{ ext{π}} ext{ (gives approximately } 80.21°$

Direction of Rotation

• Signs assigned to rotation direction:

  • Clockwise: (-)

  • Counterclockwise: (+)

Angular Kinematics Example

• Starting Position: $- rac{ ext{π}}{6} ext{ rad}$

• End Position: $rac{ ext{π}}{6} ext{ rad}$

Calculating Angular Displacement

• Formula:

  • $ext{Angular Displacement} = ext{End Position} - ext{Starting Position}$

  • Calculation:

  • ext{Displacement} = rac{ ext{π}}{6} - ig(- rac{ ext{π}}{6}ig) = rac{2 ext{π}}{6} = rac{ ext{π}}{3} ext{ rad}

Angular Velocity Calculation

• Given Time: Change in Time is 3 seconds.

• Angular Velocity Calculation:

  • $ext{Angular Velocity} = rac{ ext{Angular Displacement}}{ ext{Change in Time}} = rac{ rac{ ext{π}}{3}}{3 ext{ s}} = rac{ ext{π}}{9} ext{ rad/s}$

Angular Acceleration

• Definition: Rate of change of angular velocity with respect to time.

• Formula for Angular Acceleration:

  • $ext{Angular Acceleration} ( ext{α}) = rac{ ext{Δω (rad/s)}}{ ext{Δt (s)}}$

• Example Calculation:

  • Starting Velocity: $0 ext{ rad/s}$

  • End Velocity: $- rac{ ext{π}}{2} ext{ rad/s}$

  • Change in Time: 4 seconds.

  • Calculation:

  • Step 1: $- rac{ ext{π}}{2} - 0 = - rac{ ext{π}}{2}$

  • Step 2: $ext{Angular Acceleration} = rac{- rac{ ext{π}}{2}}{4 ext{ s}} = - rac{ ext{π}}{8} ext{ rad/s}^2$

Summary of Coordinate System

• Axes Defined:

  • X-axis: Medial-Lateral

  • Right = Positive

  • Left = Negative

  • Y-axis: Anterior-Posterior

  • Anterior = Positive

  • Posterior = Negative

  • Z-axis: Longitudinal

  • Superior = Positive

  • Inferior = Negative

Upcoming Topics

• Future Lectures include:

  • Biomechanics and Motor Control

  • Linear and Angular Kinematics

  • Forces in Human Movement

  • Muscle Mechanics