Approximate Motion of Compound Pendulum
Theorem
Let $\PP$ be a compound pendulum free to swing about a pivot $A$.
Let $\PP$ be pulled to one side by a small angle $\alpha$ (less than about $10 \degrees$ or $15 \degrees$) from the vertical and then released.
Let $T$ be the period of $\PP$, that is, the time through which $\PP$ takes to travel from one end of its path to the other, and back again.
Then:
| \(\ds T\) | \(\approx\) | \(\ds 2 \pi \sqrt {\dfrac {k^2 + h^2} {g h} }\) | ||||||||||||
| \(\ds \) | \(\approx\) | \(\ds 2 \pi \sqrt {\dfrac I {M g h} }\) |
where:
- $g$ is the Acceleration Due to Gravity
- $h$ is the distance from the center of mass to $A$
- $k$ is the radius of gyration of $\PP$
- $I$ is the moment of inertia of $\PP$ about $A$
- $M$ is the mass of $\PP$.
Proof
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