In this study, we will be analyzing the motion of a pendulum which is composed of a massive bob attached to a massless string suspended from a ceiling (see Fig. (0)). First, we will apply the dimensional analysis concept (i.e the units) to obtain estimates on the relations among physical quantities without carrying out rigorous calculations. Secondly, we will show the difficulties in obtaining an analytical solution to this, yet simple-looking, problem. However, certain approximations will help us obtain a solution in an analytical form after linearizing the equations to be solved. Then, we will solve the problem using numerical methods and contrast the results with those obtained from the linearized equation. In the final part, we will propose an analytical form as the solution to the problem which would perfectly match the numerical solution.
In most first year engineering and science classes at universities, this problem is often presented in a simplified form; the governing equation is linearized by assuming small angle oscillations and then is solved analytically, providing the solution which is in the form of a simple harmonic oscillator. However, the exact solution is usually left as a mystery, or the limitations of the small angle approximation is not discussed in detail. It is quite instructive to work out the problem without imposing the small angle approximation and obtain the solution to the most general equation using the numerical methods. This problem sets a very good example which exhibits the importance of being able to carry out numerical calculations.
In this study, we will cover
- the dimensional analysis (i.e units analysis) which works as a great tool to make predictions on the relations among physical quantities of systems being studied,
- how to get analytical solutions by linearizing equations,
- the importance of numerical methods to get a solution to the equations which can not be solved analytically,
- how to improve the approximate results with the help of numerical calculations, and to propose a more accurate analytical form as the solution to the pendulum motion problem.