Pendulum (mathematics) - Small-angle Approximation

Small-angle Approximation

The differential equation given above is not easily solved, and there is no solution that can be written in terms of elementary functions. However adding a restriction to the size of the oscillation's amplitude gives a form whose solution can be easily obtained. If it is assumed that the angle is much less than 1 radian, or

,

then substituting for sin θ into Eq. 1 using the small-angle approximation,

,

yields the equation for a harmonic oscillator,

The error due to the approximation is of order θ 3 (from the Maclaurin series for sin θ).

Given the initial conditions θ(0) = θ₀ and dθ/dt(0) = 0, the solution becomes,

The motion is simple harmonic motion where θ₀ is the semi-amplitude of the oscillation (that is, the maximum angle between the rod of the pendulum and the vertical). The period of the motion, the time for a complete oscillation (outward and return) is

which is known as Christiaan Huygens's law for the period. Note that under the small-angle approximation, the period is independent of the amplitude θ₀; this is the property of isochronism that Galileo discovered.