Compare angular rates with radius-dependent point motion
A planar rigid body rotating about one fixed axis shares angular velocity and angular acceleration across its points. That does not mean every point has the same linear speed. At perpendicular radius r, speed magnitude is r|ω|, tangential acceleration is rα and inward acceleration magnitude is rω².
Learning outcome: Hold radius fixed while comparing ω=−2,0,+2, then double radius at one rate. Retain inward and tangential components separately and explain which direction changes with ω and which does not.
Starting recap
Declare the axis, angular units, recipient and valid contact or material branch. Distinguish supplied shape factors from independently measured inertia.
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For r=0.5 m,ω=2 rad/s andα=2 rad/s², velocity is(−0.5,0.866025) m/s. Acceleration is about(−2.23205,−0.133975) m/s², with magnitude√5. Doubling r doubles every linear component.
Open the experimentPredict
Reversing angular velocity while holding angular acceleration fixed changes:
Experiment
Hold radius fixed while comparing ω=−2,0,+2, then double radius at one rate. Retain inward and tangential components separately and explain which direction changes with ω and which does not.
Resolve the motion of one point on a rigid rotation
Worked teaching model · synthetic values · Instantaneous planar rigid rotation at angle 30 degrees counterclockwise from east, angular acceleration +2 rad/s². All points share omega and alpha; no centrifugal interaction is added in the inertial frame. No physical rotating machinery.
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Velocity components at the selected instant
Spatial axes use the same pixels per physical unit, preserving geometry. 1: velocity vector (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Velocity components at the selected instant
velocity vector
| east velocity (m/s) | north velocity (m/s) |
|---|---|
| 0 | 0 |
| -0.5 | 0.866025 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
Acceleration components at the selected instant
Spatial axes use the same pixels per physical unit, preserving geometry. 1: total acceleration vector (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Acceleration components at the selected instant
total acceleration vector
| east acceleration (m/s²) | north acceleration (m/s²) |
|---|---|
| 0 | 0 |
| -2.23205 | -0.133975 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Point speed magnitude | 1 | m/s |
| Tangential acceleration component | 1 | m/s² |
| Inward acceleration magnitude | 2 | m/s² |
| East velocity | -0.5 | m/s |
| North velocity | 0.866025 | m/s |
| East acceleration | -2.23205 | m/s² |
| North acceleration | -0.133975 | m/s² |
| Acceleration magnitude | 2.23607 | m/s² |
Shared angular rates do not make all rigid-body points share the same linear speed.
Use v=r omega tangentially, a_t=r alpha and a_in=r omega². The inward direction is toward the axis even when omega changes sign; tangential direction follows signed alpha. At omega=0 the instantaneous inward term vanishes but tangential acceleration can remain. These are kinematic components of acceleration, not extra physical forces.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Shared angular rates do not make all rigid-body points share the same linear speed.
Use v=r omega tangentially, a_t=r alpha and a_in=r omega². The inward direction is toward the axis even when omega changes sign; tangential direction follows signed alpha. At omega=0 the instantaneous inward term vanishes but tangential acceleration can remain. These are kinematic components of acceleration, not extra physical forces.
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Why does zero instantaneous velocity not establish zero acceleration?
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