Pendulum - Period of Oscillation

Period of Oscillation

The period of a pendulum gets longer as the amplitude θ₀ (width of swing) increases. The true period is shown as a solid pendulum, the simple period equation is shown as a empty pendulum

The period of swing of a simple gravity pendulum depends on its length, the local strength of gravity, and to a small extent on the maximum angle that the pendulum swings away from vertical, θ₀, called the amplitude. It is independent of the mass of the bob. If the amplitude is limited to small swings, the period T of a simple pendulum, the time taken for a complete cycle, is:

where L is the length of the pendulum and g is the local acceleration of gravity.

For small swings the period of swing is approximately the same for different size swings: that is, the period is independent of amplitude. This property, called isochronism, is the reason pendulums are so useful for timekeeping. Successive swings of the pendulum, even if changing in amplitude, take the same amount of time.

For larger amplitudes, the period increases gradually with amplitude so it is longer than given by equation (1). For example, at an amplitude of θ₀ = 23° it is 1% larger than given by (1). The period increases asymptotically (to infinity) as θ₀ approaches 180°, because the value θ₀ = 180° is an unstable equilibrium point for the pendulum. The true period of an ideal simple gravity pendulum can be written in several different forms (see Pendulum (mathematics) ), one example being the infinite series:

The difference between this true period and the period for small swings (1) above is called the circular error. In the case of a longcase clock whose pendulum is about one metre in length and whose amplitude is ±0.1 radians, the θ2 term adds a correction to equation (1) that is equivalent to 54 seconds per day and the θ4 term a correction equivalent to a further 0.03 seconds per day.

For small swings the pendulum approximates a harmonic oscillator, and its motion as a function of time, t, is approximately simple harmonic motion:

For real pendulums, corrections to the period may be needed to take into account the presence of air, the mass of the string, the size and shape of the bob and how it is attached to the string, flexibility and stretching of the string, motion of the support, and local gravitational gradients.