Simple and physical pendulums
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalA pendulum is one of the cleanest examples of simple harmonic motion. For a small-angle simple pendulum the period is T = 2π√(L/g): only the string length and gravity matter, not the amplitude and not the bob mass. For small-angle swings of a rigid body about a pivot, the physical pendulum, the period becomes T = 2π√(I/(mgd)), where I is the moment of inertia about the pivot and d is the distance from the pivot to the center of mass.
It all comes down to what does and does not set the period. For the simple pendulum at small angles, mass and amplitude both drop out — only length and gravity remain. For the physical pendulum, the shape of the body enters through its moment of inertia about the pivot, and the distance d must run from the pivot to the center of mass, not to the far end of the body.
Simple and physical pendulums
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Lesson
Simple and physical pendulums
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How the period of a pendulum depends on its geometry: the simple-pendulum formula T = 2π√(L/g) and why mass and, at small angles, amplitude do not appear, the physical-pendulum formula T = 2π√(I/(mgd)) and how a distributed body differs from a point bob, and the role of the parallel-axis theorem in finding I about the pivot. Closes with a ten-scenario skill check.
Diagnostic
10-item topic check
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Ten items across the three Topic 7.5 mistakes: misreading the simple-pendulum period as depending on mass or amplitude or scaling linearly with length, collapsing the physical pendulum to the simple form and dropping the moment of inertia, and using the wrong distance or wrong axis in the physical-pendulum formula. Take it cold to see what still trips you up, or after the lesson to confirm it does not.
Targeted Practice
Drill a single misconception
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Pick one mistake you keep making and drill it on its own. Two correct in a row clears it and you move on.