Homework Statement
Apply saddle point approximation to the following integral:
I = ∫0∞xe-ax-b/√xdx a,b > 0
Recall that to derive Stirling formula from the Euler integral in class we required N >> 1. For the integral defined above, identify in terms of a and b appropriate parameter that...
Homework Statement
The wave function for a particle of mass m moving in the potential
V =
{ ∞ for x=0
{ 0 for 0 < x ≤ a
{ V0 for x ≥ a
is
ψ(x)
{ Asin(kx) for 0 < x ≤ a
{ Ce-Kx for x ≥ a
with
k = √[(2mE/h2)]
K =√[((2m(V0-E)/h2)]
where h is h-bar in both equations, and...
Homework Statement
Three identical small balls of mass m with uniformly distributed charge Q each hang from a string of length L. The strings are all tethered at the opposite end at the same point.
(a) Find the general equation for the angle θ of each string from vertical.
Homework...
Ok, I'm confused here. I'm continuing forward with L = mvr(ψ)sin(θ), but I don't understand why I don't need to get v, since v isn't constant? I know that when taking the time derivative of L one of the terms is r x p = 0 because they parallel
Ok, well my intuition tells me to make L=r(psi)mvsin(θ) where θ is the angle between the positive x-axis and the line from the focal point to the point in the orbit
In my notes, we proved momentum was constant by making momentum L = m(x*vy - y*vx) and converting to polar coordinates. Does this...
Homework Statement
The figure illustrates a Keplerian orbit, with Cartesian coordinates (x,y) and
plane polar coordinates (r,φ).
F = -(G*M*m)/r^2
The parametric equations for the orbit:
r(ψ) = a ( 1 − e cos ψ)
tan(φ/2) = [(1+e)/(1-e)]^(1/2)* tan(ψ/2)
t(ψ) = (T/2π) ( ψ − e sin ψ)
where ψ is the...
Homework Statement
Assume that the interior of the Earth is an incompressible fluid. The density is constant: ρ = M/V. The pressure p(r) depends on the distance r from the center of the earth. The equation for static equilibrium of a self-gravitating fluid sphere is
p(r)δA − p(r+dr)δA −...
Homework Statement
(A) A damped oscillator is described by the equation
m x′′ = −b x′− kx .
What is the condition for critical damping? Assume this condition is satisfied.
(B) For t < 0 the mass is at rest at x = 0. The mass is set in motion by a sharp impulsive force at t = 0, so...
I just realized I gave the wrong set of equations, that one doesn't match the other three I got, but it's the same concept. I see my issue now, pretty simple. Thank you for your help.
Homework Statement
Consider a 2-D elastic collision between two masses. The first mass is moving at initial speed v0 towards the second mass. The second mass is initially at rest. Mass m1 = 0.1 kg and mass m2 = 0.2 kg. The first mass recoils at 30° above the horizontal at speed v1, and the...
Homework Statement
(a) Consider a binary star system in which the two stars have masses M1 and M2 and the stars move on circular orbits separated by a distance R. Derive the formula for the period of revolution.
(b) Suppose M1= 1.22M and M2= 0.64M (where M = mass of the sun) and R= 0.63...
I realized where I went wrong, I had solved for the time t when the entire chain had fallen over the side, so I went back and solved for when x(t) = L/2. Thank you for your help.
Homework Statement
A flexible chain of mass M and length L lies on a frictionless table, with a very short portion of its length L0 hanging through a hole. Initially the chain is at rest. Find a general equation for y(t), the length of chain through the hole, as a function of time. (Hint: Use...
For the first response, I rechecked my work on part (a) and realized my signs were off, using A = b the ∫Ae^(-bt) from 0 to ∞ gives me a value of +1.
For the second response, the only solution I can think of would be the Expected value E(x) which would give E(t) = ƩtP(t) = Ʃt(Ae^(-bt)) from t...