1. Using pencil and paper, start with the energy expression...

1. Using pencil and paper, start with the energy expression for the simple pendulum given in the small angle approximation $E_i = \frac{1}{2}mL\theta_i^2 + \frac{1}{2}mgL\theta_i^2$, and answer the following questions. (a) Using the Euler method, calculate $E_{i+1}$ and show that this method fails for the simple pendulum. In particular show that in the the energy will increase with time. With what power of the time step does the energy increase? (b) Using the Euler-Cromer method, calculate $E_{i+1}$ and argue that the method has terms that average to zero over one cycle.

Transcript

00:01 Hello, let's have a look at the question.

00:02 So the question state, consider the non -linear pendulum equation.

00:08 So we have d square theta by dt square is equals to minus g upon l sin theta.

00:20 In class we obtained the conservation of energy equation as, so energy equation is given as l by 2, then we have d theta by dt whole square plus g multiplied with 1 minus cos theta is equals to e.

00:41 Now here we have to find what is the value of total energy e when the pendulum is hanging vertically downwards and is at rest, then b, what is the value of total energy e when the pendulum is balanced vertically upwards and is at rest and c, for what range of total energy values e does the pendulum loop, explain and d, show that the spectritic curve that separates the different kind of behavior for the non -linear pendulum will be given by the given equation.

01:12 So now here as we have a pendulum, its initial position is this one and when swinging it goes till here, now this angle is theta.

01:26 So now as it is given that vertically downwards, so here theta will be equals to 0, that means d theta by dt for vertically downwards will be equals to 0, that is it will be at rest.

01:48 Now therefore the equation for energy e will be equals to g multiplied with 1 minus cos 0.

01:56 So here we will have g multiplied with 1 minus 1, so this will be equals to 0 since the value of cos 0 is equals to 1.

02:05 Now this is the solution for the a part.

02:09 Now for the b part we have vertically upward.

02:16 Now for vertically upward we have that theta is equals to pi.

02:22 So vertically upward means that the pendulum will be here, this is the initial position b and vertically upwards means it will be here at some point a and this angle is a complete angle that is pi.

02:42 Therefore we have taken the value of theta as pi.

02:46 So therefore here we can say d theta by dt will be equals to 0.

02:52 Now the value of e will be equals to g multiplied with 1 minus cos pi.

02:59 So this will be g multiplied with 1 minus minus 1.

03:04 So therefore we will have e is equals to 2g...

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