Question

The simple pendulum is often given as an example of simple harmonic motion. In this problem will will see how accurate this is.

(a) Imagine a vertical pendulum of length l and mass m. Using the forces on the pendulum and applying Newton’s second law, obtain a differential equation in terms of θ (the angle with respect to the vertical axis) and its time derivatives. Please work in polar coordinates.

(b) Show that in the limit where θ is small that you obtain a second order linear differential equation with constant coefficients. What is the general solution to this equation?

(c) Consider m=1 kg, and l=1 m. If the pendulum is released from rest at t = 0 from θ=0.05 radians, what is the specific solution for θ(t) in your expression from part b? What is the period of this solution? If instead the initial condition is that it is released from rest at θ=1.25 radians at t = 0, what happens to the period of the motion?

(d) Use Mathematica (or something equivalent) to solve the exact equation that you found in part a (without making the approximation that θ is small) for the case where the pendulum is released from rest at t = 0 from θ=0.05 radians. Make sure that you figure out how to apply initial conditions. Make a plot of the solution from 0 to 6 seconds.

(e) Determine where the solution from part d is zero. 1 (You should have some idea of where to look for the roots from the plot that you made.) Use the locations of the roots to determine the period. How does this compare the to the period found above in part c?

(f) Use Mathematica (or something equivalent) to determine a numerical solution for the case where the pendulum is released from rest at t = 0 from θ=1.25 radians. Plot the solution. Once again, find the locations of the roots to determine the period. How does this compare to the case where it was released from θ = 0.05 in part e?

(g) Based on these results, what can you say about how well simple harmonic motion approximates the behavior of a pendulum?

Answer 1

u