SCIENCE · INVESTIGATE AND EXPLAIN

Differentiate the radial basis of a rigid rotation

Derive inward and tangential acceleration from changing direction as well as changing speed.

Learning outcome: Compare alpha=0 and 2 at radius 1, then halve radius. Reconstruct radial/tangential and Cartesian components independently and verify squared magnitudes agree. Explain which components are kinematic requirements rather than named forces.

Starting recap

Declare the fixed axis, integrate the actual mass or load distribution, and test constraint feasibility before interpreting a numerical result.

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Try

r=1,alpha=2 gives speed 3 m/s, inward acceleration 9, tangential acceleration 2 and total sqrt 85. For alpha=0, tangential acceleration vanishes but inward acceleration remains.

Open the experiment

Predict

At constant nonzero angular speed, is the acceleration of an off-axis point zero?

Your prediction and reason

Experiment

Compare alpha=0 and 2 at radius 1, then halve radius. Reconstruct radial/tangential and Cartesian components independently and verify squared magnitudes agree. Explain which components are kinematic requirements rather than named forces.

Differentiate the rotating radial basis

Worked teaching model · synthetic values · Rigid planar rotation about a fixed axis, omega0=1 rad/s,theta0=0. Evaluate at 1 s. Compare inward and tangential acceleration, not an extra centrifugal interaction in the inertial frame.

Selected radius: 1 m Constant angular acceleration: 2 rad/s²

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Rigid radius versus speed

Axes rescale independently. Curves join sampled coordinates; a joining segment is not additional measured evidence. Read physical units rather than screen angles or apparent slopes.

Rigid radius versus speed: numerical coordinates

radius (m)	speed (m/s)
0	0
0.025	0.075
0.05	0.15
0.075	0.225
0.1	0.3
0.125	0.375
0.15	0.45
0.175	0.525
0.2	0.6
0.225	0.675
0.25	0.75
0.275	0.825
0.3	0.9
0.325	0.975
0.35	1.05
0.375	1.125
0.4	1.2
0.425	1.275
0.45	1.35
0.475	1.425
0.5	1.5
0.525	1.575
0.55	1.65
0.575	1.725
0.6	1.8
0.625	1.875
0.65	1.95
0.675	2.025
0.7	2.1
0.725	2.175
0.75	2.25
0.775	2.325
0.8	2.4
0.825	2.475
0.85	2.55
0.875	2.625
0.9	2.7
0.925	2.775
0.95	2.85
0.975	2.925
1	3

Calculated values under the declared model
Quantity	Value	Unit
Angular position at 1 s	2	rad
Angular velocity at 1 s	3	rad/s
Tangential speed	3	m/s
Inward acceleration magnitude	9	m/s²
Tangential acceleration component	2	m/s²
Acceleration x	1.92673	m/s²
Acceleration y	-9.01597	m/s²
Total acceleration magnitude	9.21954	m/s²

The basis-vector derivative supplies inward acceleration even at constant angular speed.

Differentiate er to omega e_theta and e_theta to −omega er. All radii share omega and alpha in a rigid body, but linear speed and both acceleration magnitudes scale with radius. These components describe acceleration, not two extra forces.

Derive before checking. Retain contrasting captures and explain one assumption that could fail; calculated evidence is not actual laboratory work.

The basis-vector derivative supplies inward acceleration even at constant angular speed.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Separate rigid angular motion from point motion

Every point of a planar rigid body shares angular position change, omega and alpha about its fixed axis. Its distance r from the axis determines linear arc length r theta and tangential speed r omega. Points on the axis have zero linear speed even when the body rotates. Equal angular velocity is not equal linear velocity.

Differentiate a moving basis

Write position r er with fixed radius. The unit vectors depend on angle: d(er)/dt=omega e_theta and d(e_theta)/dt=−omega er. Therefore velocity is r omega e_theta and acceleration is r alpha e_theta−r omega² er. The inward term comes from changing direction; it remains at constant angular speed.

Resolve components in a fixed frame

At angle theta, er=(cos theta,sin theta) and e_theta=(−sin theta,cos theta). Thus ax=−r omega² cos theta−r alpha sin theta and ay=−r omega² sin theta+r alpha cos theta. These are Cartesian acceleration components, not extra interactions to add to a force diagram.

Evaluate the original packet

At t=1 with omega 0=1,alpha=2 and r=1, theta=2 rad and omega=3 rad/s. Inward acceleration magnitude 9 and tangential component 2 give total sqrt 85≈9.21954 m/s². A point at radius.5 has half the speed and half both acceleration components, while retaining the same angular rates.

Avoid a rotating-frame force shortcut

In the stipulated inertial frame, an inward net force supplies the required inward acceleration. There is no extra outward force on the point merely because the trajectory curves. If analysis is moved to a rotating frame, additional apparent-force terms need a separately declared framework. The present derivation does not certify the material stresses that maintain rigidity.

Worked original case

r=1,alpha=2 gives speed 3 m/s, inward acceleration 9, tangential acceleration 2 and total sqrt 85. For alpha=0, tangential acceleration vanishes but inward acceleration remains.

Investigation

Public numerical transfer practice

Try a new practice case

A point at radius 2 m rotates at omega 3 rad/s with alpha 0. What is inward acceleration magnitude in m/s²?

Numerical result in the requested units

This public, unsaved check is not a secure assessment. Explain the derivative, integral or boundary in the retained course evidence workspace.

Sources and evidence boundary

Original teaching and numerical cases checked 30 October 2026 against the official AP Physics C Mechanics framework and OpenStax fixed-axis rotation. Supplied/generated evidence is not physical measurement, a load rating or laboratory participation. No rotating machinery, hanging weights or mechanical construction is requested. Public feedback and retained self-reports do not certify independent learner mastery.

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Public practice checks a specific claim. It does not establish independent mastery.

Which assumption fails if different points no longer share the same angular velocity?

Your explanation and limitations

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