Try adding a calculation of the energy to the example "falling ball" script described in lecture. (The link to the example program is here) Make a graph of the energy versus time to see whether it is conserved in our numerical solution.
Write a script to solve the equation of motion of a pendulum using the Euler Method. The equation is:
where g is the gravitational acceleration, l is the length of the pendulum,
and theta is the angle of the pendulum from vertical.
The exact solution for the case where the pendulum is released at some initial
angle theta0 at t=0
(and for small values of the angle theta) is:
We will write a MATLAB script to solve this problem using the iterative method
described in class and display the results. The first step is to
rewrite the second order equation as two first order DEQ's:
Compute the motion of a 4m pendulum with initial conditions omega=0 and
theta = 0.01 (a small angle!!) for t from 0 to 12 seconds. (g is 9.8 m/s/s)
Use a time step that is small enough to give good agreement with the analytical
solution. Plot your numerical solution along with the analytical
solution for comparison and submit your script and plot.
Compute the energy of the pendulum for each time step in your solution and
plot the result. (assume mass is one kg.) Is the energy conserved? Do
you expect it to be?
Extra Credit