Physical pendulum and interchangeability

A compound (physical) pendulum is any rigid body free to oscillate in a vertical plane about a fixed horizontal axis that does not pass through its centre of gravity.

      O  <- point of suspension
      |\
      | \  l
      |  \
      |   G  (centre of gravity)  mg
      |

Time period

Let the mass be m, O the pivot, G the centre of gravity at distance l from O, and k the radius of gyration about G. Displace the body by a small angle θ.

1. Restoring torque = −mg l sinθ ≈ −mg l θ.
2. Moment of inertia about O (parallel-axis theorem): I = mk² + ml² = m(k² + l²).
3. I d²θ/dt² = −mgl θ gives d²θ/dt² + [gl/(k² + l²)] θ = 0, which is SHM with ω² = gl/(k² + l²).

T = 2π √((k² + l²)/(g l)) = 2π √(L/g) ,   L = (k² + l²)/l = l + k²/l

L is the length of the equivalent simple pendulum.

Interchangeability

4. The centre of oscillation O′ lies on the line OG extended, at distance L from O. Its distance from G is

l′ = L − l = k²/l

5. Now suspend the body at O′. Its distance from G is l′, so the new period is

T′ = 2π √((k² + l′²)/(g l′))

6. Substitute l′ = k²/l: k² + l′² = k² + k⁴/l² = k²(l² + k²)/l². Dividing by l′ = k²/l gives (l² + k²)/l.

T′ = 2π √((k² + l²)/(g l)) = T

7. So the period about O′ equals the period about O: the centre of suspension and the centre of oscillation are interchangeable. Also, the new equivalent length is L′ = l′ + k²/l′ = k²/l + l = L, the same as before.