Homework Statement
Consider a plane monochromatic wave incident on a flat conducting surface. The incidence angle is ##θ##. The wave is polarized perpendicular to the plane of incidence. Find the radiation pressure (time-averaged force per unit area) exerted on the surface.
Homework Equations...
Oh gosh oh gosh - I just realized that I forgot to explicitly mention that there is a spring connecting the two particles. The scenario is that there is a double well potential with one particle in each well. The particles are connected to each other by a spring. We are considering small...
Going in order of what you've written, this is my first question: are the equilibirum points the points where the potential of the wells is at a minimum? I found those to be at ##x=\pm \alpha##. But then again this does not include the spring potential...
Ok so then ##K=\begin{bmatrix}\frac{8\alpha^2\beta + k}{m} & -k/m\\-k/m & \frac{8\alpha^2\beta + k}{m}\end{bmatrix}##. I removed a negative sign from inside the matrix because it looks like you pulled it out to be in front of the ##K##.
Ok so I found the coupled equations to be
##\ddot{x_1} =(\frac{-8\alpha^2 \beta -k}{m})x_1 + (\frac{-8\alpha^3 \beta + kx_2 -2k \alpha}{m}) ##
##\ddot{x_2} =(\frac{-8\alpha^2 \beta -k}{m})x_2 +( \frac{8\alpha^3 \beta + kx_1 +2k \alpha}{m}) ##
Homework Statement
Let's say that I have a potential ##U(x) = \beta (x^2-\alpha ^2)^2## with minima at ##x=\pm \alpha##. I need to find the normal modes and vibrational frequencies. How do I do this?
Homework Equations
##U(x) = \beta (x^2-\alpha ^2)^2##
##F=-kx=-m\omega ^2 x##
##\omega =...
I expanded it out to this, but nothing cancels nicely.
##\psi (x) = \frac{A(sin(kx)sin(ka)-cos(kx)cos(ka))}{sin(ka)}+\frac{B((1/2)(sin(kx)cos(ka)+cos(kx)sin(ka)-sin(kx-ka)))}{sin(ka)}##
Should I have gone a different route?
I see 3 ways to do something with what you've suggested. Here is one attempt:
##\psi(x) = \frac{Asin(kx)sin(ka)+Bcos(kx)sin(ka)}{sin(ka)}##
##=Asin(kx) + \frac{B[(1/2)(sin(ka+kx)+sin(ka-kx))]}{sin(ka)}##
##=Asin(kx)+\frac{A}{2(e^{iKa}-cos(ka))}[sin(ka+kx)+sin(ka-kx)]##...
Homework Statement
Using the equations given, show that the wave function for a particle in the periodic delta function potential can be written in the form
##\psi (x) = C[\sin(kx) + e^{-iKa}\sin k(a-x)], \quad 0 \leq x \leq a##
Homework Equations
Given equations:
##\psi (x) =A\sin(kx) +...
What, this confuses me greatly. They slide down the plane, of course, but their potential energy comes from how far up the plane they are, no? Which would correspond to the height from the bottom (assuming there is a bottom).
Ok so then ##PE=m_1gr_{1,z}+m_2g r_{2,z}## ? I changed to ##r## because I just realized that the masses have 3 coordinates.
But I don't really understand how to transform to CoM separation coordinates.
Homework Statement
I am given the following scenario: 2 masses are connected by a spring and are on a frictionless inclined plane. They are free to rotate, oscillate, and slide down the plane.
For the potential energy of the center of mass of the system, it is ##Mgh=(m_1+m_2)gh##. But isn't...