$$\rightleftharpoonup{xx}$$
$$\longleftharp{xx}$$,
$$\longrightharp{xx}$$,
The motion of a simple pendulum is governed by Newton’s Second Law in its rotational form, which relates the net torque on the bob to its angular acceleration. This physical law gives rise to a second-order differential equation in which the angular acceleration is proportional to the sine of the displacement angle.
Because of the sin(𝜃) term, the governing equation is a nonlinear differential equation, which is difficult to solve analytically. To simplify the mathematical model, the sine function is expanded using a Maclaurin series, which is a specific type of power series centered at zero:
\begin{equation*}\sin(\theta) = \Sigma_{n=0}^{\infty} (-1)^n \frac{\theta^{2n+1}}{(2n + 1)!} = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \dots\end{equation*}
When the pendulum's swing is small, the higher-order terms, such as the cube or fifth power of the angle, become negligible in magnitude.
This allows for the small-angle approximation, where
\begin{equation*}\sin(\theta)\approx\theta\end{equation*}
This mathematical manipulation transforms the complex nonlinear equation into a solvable linear second-order differential equation:
\begin{equation*}\jfrac{d^2\theta}{dt^2}+\jfrac{g}{L}\,\theta=0\end{equation*}
This simplification is critical for the precision of mechanical clocks. For small angles, the period of the pendulum depends primarily on its length (L) and the acceleration due to gravity (g), rather than the amplitude of the swing.
In physics and engineering, the study of angular displacement determines how a system handles energy and time. In precision devices like mechanical clocks, angular displacement is intentionally kept small. This ensures that the period of each swing remains constant, a principle known as isochronism. If the angle stays within a narrow range, the clock remains accurate regardless of slight changes in the swing's width. In seismic and safety devices, the study of the angle is used to measure force. In a seismometer, the maximum angle reached by the pendulum indicates the intensity of ground vibrations. Similarly, in industrial machinery, monitoring the angular displacement is essential for safety limits. Engineers calculate the maximum allowable angle to ensure the pendulum does not strike the protective casing or exceed the structural strength of the suspension cable.
