Recent content by Summer95

oh cool, thank you!

First I looked at a pendulum and wrote down $$-gsin\theta=l\ddot{\theta}$$ $$l\ddot{\theta}+\frac{g}{l}sin\theta=0$$ Chain Rule $$\frac{d\dot{\theta}}{d\theta}\frac{d\theta}{dt}+\frac{g}{l}sin\theta=0$$ Separate and Integrate with bounds ##0## to ##\dot{\theta}## and ##\pi/4## to ##\theta## to...

Ok cool! I will figure out how to deal with elliptic integrals now! Thanks!

I will upload the rest of my work if you want to see it. I just integrated the equation of motion for a simple pendulum. A friend was saying this is an elliptic integral, are elliptic integrals analytically solvable? If so I'm down to try it!

But what if I want an expression with some arbitrary theta? I could do something like this but I don't know what my bounds should be if I want the Time for one period. Because at half a period the angle is pi/4 and at one period the angle is pi/4!

When you integrate the equation of motion for a simple pendulum, this is what you get. The amplitude is pi/4 so I'm saying that at t=0 the angle is pi/4. Then this integral is from the initial condition to some arbitrary angle theta at time t. My end goal is to get an expression for the period.

This is a simple pendulum with amplitude pi/4 so theta is going to be less then pi/4 but yes.

Hello All! I am trying to solve the simple pendulum without using a small angle approximation. But I end up with this integral: $$\int_{\frac{\pi}{4}}^{\theta}\frac{d\theta}{\sqrt{cos(\theta)-\frac{\sqrt{2}}{2}}}$$ Is this possible to evaluate? If so, could I get a hint about what methods to...

Homework Statement There is a collection of different force fields, for example: $$F_{x}=ln z$$ $$F_{y}=-ze^{-y}$$ $$F_{z}=e^{-y}+\frac{x}{z}$$ We are supposed to indicate whether they are conservative and find the potential energy function. Homework Equations See Above The Attempt at a...

Yes! Sorry I forgot to specify that.

Homework Statement Differential equation: ##Ay''+By'+Cy=f(t)## with ##y_{0}=y'_{0}=0## Write the solution as a convolution (##a \neq b##). Let ##f(t)= n## for ##t_{0} < t < t_{0}+\frac{1}{n}##. Find y and then let ##n \rightarrow \infty##. Then solve the differential equation with...

Problem: ##y^{\prime \prime}+2y^{\prime}+y=f(t)## with initial conditions ##y_{0}=y′_{0}=0## and ##f(t) = 1 \text{ if } 0<t<a \text{ or } 0 \text{ if } t>a ## Here is my solution, in case it might help anyone, because I was able to find so few resources on solving these: Think of the...

Here is a different example of one of the problems I can't solve: ##y^{\prime \prime}+2y^{\prime}+y=f(t)## with initial conditions ##y_{0}=y^{\prime}_{0}=0## and ##f(t)=\begin{cases} 1 & \text{if } 0<t<a \\ 0 & \text{if } t>a \end{cases}## ##\frac {d^{2}} {dt^{2}} G(t,t^{\prime})+2\frac {d}...

Ok, good! That makes sense. Unfortunately I am still not getting it, when I assume continuity at ##t^{\prime}## I seem to be able to get one of the constants pretty successfully. I get ##G(t,t^{\prime})=C*sin\omega(t-t^{\prime})## When I try to use your hint I just end up with identities that...

Suppose we have a differential equation with initial conditions ##y_{0}=y^{\prime}_{0}=0## and we need to solve it using a Green Function. Then we set up our differential equation with the right side "forcing function" as ##\delta(t^{\prime}-t)## (or with ##t^{\prime}## and ##t## switched I'm a...