Exploring the Ratio of Successive Thetas in an Undamped and Undriven Pendulum

[fuligni]

hello friends,

I have a question on the simple undamped and undriven pendulum. I see that according to the website:

http://www.gmi.edu/~drussell/Demos/Pendulum/Pendula.html

the angle of rotation, Theta(t), can be expressed in terms of the starting angle using the small angle approximation sin(theta) = theta.

my question is, what is the ratio of successive thetas if we measure theta in increments of half periods ? the website shows:

theta(t) = theta(t=0)*cos(wt+phi)

i am interested in the ratio: theta(t=T)/theta(t=0) = cos(wt+phi)

if we substitute w = sqrt(g/L) and t=T = 2*pi*sqrt(L/g) into the right side of this ratio we get:

ratio = cos(2*pi + phi)

but cos(2*pi) = 1.0.

I don't see where i am going wrong since shouldn't the angle decrease over time until it is zero and the pendulum is stopped.

I am curious to see if the ratio of successive theta's is constant.

thankyou,
chris

 

[outandbeyond2004]

The pendulum is UNdamped. No air resistance, no loss of energy. Real pendulums do indeed act like you think -- they stop eventually -- but this is only a theoretical pendulum.

 

[fuligni]

sorry,,
i see that without friction, the pendulum swings forever and the ratio is 1.0 as the equations show.