Homework Help: Pendulums, total energy?, and Mathematica

1. Sep 12, 2010

anonindiv

1. The problem statement, all variables and given/known data
A)Show that for a non-frictional, simple linear pendulum (Sin(theta) ~ theta) the total energy of the pendulum (K + U) or kinetic plus potential is given by

E = (1/2) m l^2 (d(theta)/dt)^2 + (1/2) mgl (theta)^2

and therefor E = (1/2) mgl(theta0)^2

theta0 = theta(t=0)

2. Relevant equations
F=ma , delta K = 1/2mv^2 , delta U = mgh

3. The attempt at a solution

Alright, so I'm essentially lost in this problem, and my last calculus class was approximately 2 years ago.
I understand that the total energy should be the sum of the potential and kinetic energies of the pendulum, so it seems that E = 1/2mv^2 +mgh. But it seems that i am stuck here. I observe that the change between the kinetic energy portion of the equation is different in that v^2 is now l^2 (d(theta)/dt)^2, and the potential mgh now appears as 1/2mgl (theta)^2, but I cannot think of how to determine how to get to that point. And therefor I am unable to get to the main portion of the problem.

{a} One more problem. use mathematica to solve (d(theta)/dt)^2 +g/lsin(theta)=0 . And, show a graph of period vs. (theta 0).

Before this course I have not used mathematica, and now I am facing difficulties. I tried using the dsolve and manipulate functions many times over the past week in an attempt to graph the problem but I have been unsuccessful.

Please note that I am NOT asking for any answers just for guidance.

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2. Sep 12, 2010

anonindiv

I managed to figure out, after drawing a simple diagram (of course!), that h in the potential equation has got to be equal to L-Lcos(theta). Also, the velocity is going to be equal to L(dtheta/dt), the length multiplied by the rate of change of the center angle. Knowing this,
k= 1/2 m (L(dtheta/dt))^2, and
U= mgh= mg (L-Lcos(theta))
so
E= 1/2 m L^2 (dtheta/dt)^2 + mg(L-Lcos(theta)), which appears to be very similar to the original equation that is given, but I am again stuck. I cant seem to find why the first given equation is equal to the second.

3. Sep 12, 2010

anonindiv

So... anybody understand how to graph in mathematica or have any words of wisdom on the above problems? Anything would be useful... :D

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