Imagine a classical pendulum with string length L and mass M. For this problem,
ID: 1892025 • Letter: I
Question
Imagine a classical pendulum with string length L and mass M. For this problem, there is no friction.
Write down a mathematical model for how the pendulum will act based on the initial angle of displacement.
That is, find a mathematical model in differential equation form that will describe how the pendulum depends on the initial angle displacement theta of the system. You should first develop a theory for how the period should depend on the angle displacement so that your final model can be tested experimentally and plotted against the results.
Explanation / Answer
The differential equation that represents the motion of a simple pendulum is: d2X / dt2 + (g/l) * Sin X = 0 X = angle of rotation for small X, the equation can be written as d2X / dt2 + (g/l) * X = 0 The solution for the above equation is: X = X0 * Cos (sqrt(g/l) * t) X0 = initial displacement omega = 2 * pie / T = sqrt(g/l) => T = 2 * pie * sqrt(l/g) l = length g = gravitational pull