Physical Pendulum
A ruler, a disc or a ring swinging on a nail is not a weight on a string: its mass is spread out. Find how long one swing takes, where to put the pivot for a given period, or what g is.
What is a Physical Pendulum?
Any rigid object that swings about a pivot is a physical pendulum. Its period depends on how its mass is spread out, through the moment of inertia I about the pivot. A metre stick swinging from one end beats faster than a 1 m string pendulum, because part of its mass sits close to the pivot.
Moving the pivot towards the centre first makes the swing faster, then — very close to the centre — much slower. The fastest swing comes when the pivot is one radius of gyration k from the centre.
Period and Pivot Position
periodSwinging from the Edge
Earth, small swings| Object | Pivot | Same as a string of | Period |
|---|---|---|---|
| Rod, 1 m | one end | ⅔ L = 0.667 m | 1.64 s |
| Disc, R = 20 cm | the rim | 1.5 R = 0.30 m | 1.10 s |
| Ring, R = 20 cm | the rim | 2 R = 0.40 m | 1.27 s |
| Simple pendulum, 1 m | — | 1 m | 2.01 s |
Worked Techniques
The moment of inertia about the pivot is the one about the centre plus m d². Dividing by m gives k² + d², so the mass cancels from the period.
A physical pendulum swings like a simple pendulum of length Leq. That point below the pivot is the centre of oscillation.
For a given period there are usually two pivot positions, one on each side of d = k. Their product is k².
Sources & Verification
- D. Halliday, R. Resnick & J. Walker, Fundamentals of Physics, 10th ed., Wiley, 2014 — Ch. 10 (Rotation) and Ch. 15 (Oscillations).