Week 3: Module 1 Exploring the Relationship Between Linear and Angular Quantities Connecting Linear and Angular Quantities in Human Movement by ADDU BOUT INSING ME SOSAR Learning Competency: Demonstrate the relationship between linear and angular quantities using real-life applications such as human movement and ergonomic design Lesson Overview In this lesson, students will: 0 Review the basic definitions of linear and angular quantities. Apply these concepts to body movement (e.g., swinging an arm). Understand how this relationship is used in ergonomic design (e.g. tool handles, chair lovers). Perform a simple activity that connects linear speed and angular motion. Concept Map with Definitions Linear Quantity Displacement (x) Velocity (v) Acceleration Radius (r) Definition The change in position of an object in a straight line; measured in meters (m). The rate of change of displacement over time; describes how fast and in what direction an object moves; measured in meters per second (m/s). The rate of change of velocity over time; tells how quickly an object is speeding up or slowing down; measured in m/s². Angular Quantity Angular Displacement (0) Angular Velocity (0) The distance from the axis of rotation to the point of interest (e.g., a hand, a tool handle); measured in meters (m). Angular Acceleration (a) Definition The angle through which an object rotates about a fixed axis, measured in radians (rad) The rate of change of angular displacement, how fast something rotates; measured in radians per second (rad/s). The rate of change of angular velocity, tells how quickly an object is spinning faster or slower, measured in rad/s². Used as a factor linking linear and angular quantities in formulas like: x=r0, v=rm, a =ral
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1) Core ideas (plain words)Linear = straight-line. Displacement (x), Velocity (v), Acceleration (a).Angular = rotation. Angle (θ, in radians), Angular velocity (ω), Angular acceleration (α), Radius (r).Link formulas:Displacement along an arc: x = r θLinear speed from rotation: v = r ωTangential accel (speeding up/slowing down spin): a_t = r αCentripetal accel (pull to center while turning): a_c = v² / r = r ω²2) Human movement examplesSwinging your arm: Shoulder = axis, upper arm = radius, hand moves in an arc.Longer arm (bigger r) → bigger arc x and faster v for the same ω.Kicking a ball: Hip/knee rotate (ω); shoe tip speed v = r ω determines kick power.Running: Foot rotates about ankle before toe-off; larger r or ω → higher shoe tip speed.3) Ergonomic design usesTool handles (hammer, wrench): Slightly longer r gives higher v at the head for the same hand motion; but too long increases effort and control issues.Door handles/levers: Longer lever arm (r) reduces needed force/torque.Chair levers/desk setup: Pivot positions set r so small angles (θ) give comfortable linear adjustments (x = r θ), preventing overreaching.4) Mini-calculation (show the link)Arm length r = 0.70 m, angular speed ω = 3 rad/s.Hand speed v = r ω = 0.70 × 3 = 2.1 m/s.If the arm sweeps θ = 1 rad:Arc distance x = r θ = 0.70 × 1 = 0.70 m.If it speeds up at α = 2 rad/s²:Tangential accel a_t = r α = 0.70 × 2 = 1.4 m/s².Centripetal (while spinning) a_c = r ω² = 0.70 × 9 = 6.3 m/s².5) Simple class activityTie a string of known length (r) to a small weight; spin it at steady speed.Measure 10 full turns in time T to get ω = (2π × 10)/T.Compute v = r ω and compare with how “fast” it looks.Shorten/lengthen r and observe: for same ω, v changes in proportion to r.Key takeawaysRadians make x = r θ and v = r ω work cleanly.Bigger r or ω → bigger v; good for power but may raise required control and force.Ergonomics balances r and ω to maximize efficiency and safety.
