Differentiate an unwrapped angular record

Declare an axis and an angular coordinate

Choose a fixed axis and use the right-hand rule to name positive rotation. Angular position is an unwrapped coordinate in radians: several revolutions are not reduced to a single angle between zero and 2pi when tracking accumulated motion. A change of angle, angular velocity and angular acceleration have distinct meanings and units.

Integrate a nonconstant angular acceleration

The assigned alpha(t)=ct gives omega(t)=omega 0+ct²/2 and theta(t)=omega 0t+ct³/6 after imposing theta(0)=0. Both constants matter. Substituting alpha at the endpoint into a constant-alpha formula would replace an integral with a different assumed history. Differentiate the derived expressions to check the original rate law.

Distinguish signed displacement from total swept angle

A zero of omega can mark reversal. If omega 0 and c have opposite signs, a positive candidate time is sqrt(−2omega 0/c); check whether it lies inside the observation interval. Total swept angle is the integral of absolute omega, requiring a split at an interior reversal. Signed angle can cancel across that split.

Compare a concrete reversal

For omega 0=−1 and c=1 over 0–3 s, reversal occurs at sqrt 2. Signed theta(3)=1.5 rad. The minimum angle is−2sqrt 2/3, so total swept angle is 1.5+4sqrt 2/3≈3.38562 rad, not 1.5. The object initially rotates negatively and later positively; a positive final angular velocity does not erase the early motion.

Recover the constant-acceleration special case

For an independent constant alpha, integrating gives omega=omega 0+alpha t and delta-theta=omega 0t+alpha t²/2. Eliminating time yields omega²−omega 0²=2alpha delta-theta; the equation does not by itself choose the sign of omega. Uniform circular motion has constant omega but the linear velocity direction of a rotating point still changes. Multi-axis precession is outside this fixed-axis lesson.

Choose whether internal rotation matters

A rigid wheel keeps its shape, but opposite rim points have different linear velocities. A single center coordinate therefore cannot describe its spin. For an orbital question that neglects the wheel’s internal spin, its center trajectory may instead be treated as one particle. Predict which representation is needed to compare two rim-point velocities: the extended-body model is necessary.

For the reversal packet above, sketch theta, omega and alpha against time before revealing numerical values. The theta graph has zero slope at t=sqrt 2, where omega crosses zero; the slope of omega there is sqrt 2, not zero. Signed area under omega gives the 1.5 rad displacement, while signed area under alpha gives the change of angular velocity. These graph operations are the derivatives and integrals already derived, not new constant-rate approximations.

Worked original case

omega 0=1,c=.5 gives theta(3)=5.25 rad, omega(3)=3.25 rad/s and alpha(3)=1.5 rad/s².

Investigation

Compare omega 0=−1,c=1 with omega 0=1,c=1. Locate any reversal, integrate absolute angular speed by intervals and retain both signed and total angle. Check the derivative at a chosen intermediate time.

Public numerical transfer practice

Try a new practice case

For omega 0=2 rad/s and alpha(t)=2t rad/s², what is omega at t=2 s in rad/s?

Numerical result in the requested units Check this practice result

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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.