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...
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.
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...
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...