The oscillations of a swinging pendulum can be simulated with the following nonlinear model:

where ? = the angle of displacement, g = the gravitational constant, and I = pendulum length. For small angular displacements, the sin ? is approximately equal to ? and the model can be linearized as

Use ode45 to solve for ? as a function of time for both the linear and nonlinear models, where I = 0.6 m and g = 9.81 m/s ² . First, solve for the case where the initial condition is for a small displacement (? = ?/8 and d? = 0). Then repeat the calculation for a large displacement (? = ?/2). For each case, plot the linear and nonlinear simulations on the same plot.

d0 8 1 2 + 1/ s dt sin 0 = 0 dt + $0=0