A "bob" or "simple" pendulum is a pendulum consisting of a single spherical (or point) mass attached to a wire of negligible weight. A "physical" pendulum has extended size and is a generalization of the bob pendulum. An example would be a bar rotating around a fixed axle. A simple pendulum can be treated as a special case of a physical pendulum with moment of inertia

(1)

where m is the bob mass and l is the wire length.

The equation of motion of a physical pendulum can be found from the torque on it,

(2)

where g is the gravitational acceleration, I is the moment of inertia, is the angular acceleration, and is the angle of the wire measured from the downward vertical . Then

(3)

Now define the resonant frequency as

(4)

so (3) can be written in the simple form

(5)

Writing the equation of motion as two coupled first-order ordinary differential equations

(6)

gives the phase portrait illustrated above.

If the pendulum oscillates but does not rotate (i.e., does not have enough energy to trace a complete circle by crossing the separatrix at ), then there is an angle at which , so all the energy is potential. Since energy is conserved, at an arbitrary angle ,

(7)

so

(8)

(9)

(10)

Coupled Pendula, Double Pendulum, Melnikov-Arnold Integral , Pendulum Gravity Determination, Pendulum Small Oscillations