FirstPrinciplesa Physoly project
ROT-10rotationtrap · Dimensional inconsistency

The parallel-axis theorem adds Md²

  1. 01

    The tempting model

    Adding Md or shifting moment of inertia by a linear distance.

  2. 02

    Why it feels right

    The axis moved by distance d, so a linear correction feels natural.

  3. 03

    Where it breaks · how to catch it

    Your corrected moment of inertia has units kg·m instead of kg·m².

  4. 04

    The correct model

    For a parallel axis a distance d from the center-of-mass axis, I = I_cm + Md².

  5. 05

    Repair · see it, then test it

    worked example A disk has I_cm = ½MR². What is I about a tangent axis parallel to the central axis?

    Tempting move

    I = I_cm + MR.

    Correct move

    I = ½MR² + MR² = 3MR²/2.

    Test the repair ROT-10 · transfer
    A uniform rod has I_cm = ML²/12. What is its moment of inertia about a parallel axis through one end?

    Make a fresh prediction before referring to the worked example.

    Then practise it on the ladder

    R33Shift the axisadds: Moment of inertia depends on the axis; use parallel-axis theorem.