Integrate a rotational torque law with its initial condition

Begin with the core rotational equation

For a rigid body about a fixed axis with constant I, net torque is I alpha=I dω/dt. A torque changes angular velocity; a nonzero angular velocity does not by itself require nonzero net torque. Inertia depends on the mass distribution and axis, not on the value of angular speed in this model.

Declare optional differential-equation enrichment

The following exact viscous-resistance solution is optional calculus enrichment of the rotational force law. The workbench stipulates I=2 kg m², drive torque 6 N m and resistance−bω. The simple law is not claimed to model every bearing, static friction threshold or actual apparatus.

Solve for angular velocity

The ODE is dω/dt=(6−bω)/2. Define steady angular velocity ωinfinity=6/b and response time tau=2/b. Separation or direct substitution gives ω=ωinfinity+(ω0−ωinfinity)exp(−t/tau). The initial condition determines approach from below, approach from above or exact steady rotation. A terminal value is not a universal upper bound on the initial speed.

Integrate again for angle

With theta 0=0, theta=ωinfinity t+(ω0−ωinfinity)tau(1−exp(−t/tau)). The angle has a long-time linear term even though angular acceleration decays to zero. For b=1,ω0=0,t=2, omega=6(1−e⁻¹)≈3.79272 rad/s and theta=12e⁻¹≈4.41455 rad. Differentiate theta to verify the velocity and its initial value.

Test an unchanged equilibrium state

For b=1 andω0=6, drive and resistance balance at every instant. Angular velocity remains 6, net torque and alpha are zero, and theta at 2 s is 12 rad. This is rotational equilibrium with motion, not static equilibrium. A changed I, axis or friction law would require a different model, not reuse of the same exponential by habit.

Worked original case

b=1,omega 0=0 gives omega(2)≈3.79272 and theta(2)≈4.41455. Starting at omega 0=8 instead approaches 6 from above.

Investigation

At b=1 compare initial omega 0=0,6,8. Derive the rate and angle, retain net torque and explain the different approach directions. Then double b and predict the steady value and response time before evaluating.

Public numerical transfer practice

Try a new practice case

For drive torque 8 N m and viscous coefficient 2 N m s/rad, what is steady angular velocity 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.