Pendulums, total energy?, and Mathematica

[anonindiv]

Homework Statement

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)

Homework Equations

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

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.

 

[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 can't seem to find why the first given equation is equal to the second.

 

[anonindiv]

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