# Nonlinear Pendulum

## Solved Problems

### Example 1.

Estimate the error in the calculation of the oscillation period of a simple pendulum at different amplitude \({\alpha_0}\) when using the standard formula \[{T_0} = 2\pi \sqrt {\frac{L}{g}}.\]

Solution.

We use the solution of the nonlinear equation of the pendulum, in which the expression for the period \(T\) is represented as a series. Taking into account the term \(n = 1,\) the formula \(T\left( {{\alpha _0}} \right)\) looks like

where \({T_0}\) is the period of oscillation, calculated by the standard formula

Thus, the term \({{\frac{1}{4}} {{\sin }^2}{\frac{{{\alpha _0}}}{2}}}\) immediately shows the deviation from the standard formula (expressed as a decimal), depending on the angle \({\alpha_0}.\)

Similarly, we take into account the contribution of the following terms of the series for \(n = 2\) and \(n = 3.\) The corresponding formulas have the form:

The graphs presented in Figure \(3\) show the value of the expression in the square brackets for the functions \({T_1}\) and \({T_2}\) (in percentage). In fact, they give the error in determining the period of oscillation when using the standard formula \({T_0}\) compared with more accurate approximations.

It is seen that the power series converges well, and in the range of angles up to \({\alpha_0} = 20^{\circ}\) it is possible to restrict the series expansion by the first term \(n = 1\) to ensure the accuracy rate of about \(1\%.\)