Homework Statement
A point source of light (wavelength \lambda = 600 \, \text{nm} ) is located a distance x = 10\,\text{m} away from an opaque screen with a small circular hole of radius b. A very small photodiode is moved on an axis from very far away toward the screen. The first...
Problem
This is a conceptual problem from my self-study. I'm trying to learn the basics of group theory but this business of representations is a problem. I want to know how to interpret representations of a group in different dimensions.
Relevant Example
Take SO(3) for example; it's the...
Homework Statement
The picture of the circuit is attached; I want to find |V_{A}/V_{J}|. This seems really easy but I haven't done circuit analysis in forever.
Homework Equations
Complex impedances, Z_{C} = 1/i\omega C, Z_{R} = R.
The Attempt at a Solution
First R_{A} and C_{A} are in...
Homework Statement
An infinite wire of linear charge density \lambda lies on the z axis. An insulating cylindrical shell of radius R is concentric with the wire and can rotate freely about the z axis. The charge per unit area on the cylinder is \sigma = -\lambda/2\pi R while the mass per unit...
There's a specific problem I'm doing, but this is more of a general question. The setup is a cylinder of mass m and radius R rolling without slipping down a wedge inclined at angle \alpha of mass M, where the wedge rests on a frictionless surface. I've made the Cartesian axis centred at the...
Homework Statement
If I have a Hamiltonian matrix, \mathcal{H}, that only depends on a kinetic energy operator, do the energy eigenvalues have to be non-negative? I have an \mathcal{H} like this, and some of its eigenvalues are negative, so I was wondering if they have any physical...
I have a set of data points \{\{x_1, y_1\}, \{x_2, y_2\} ... \} each with an uncertainty \{\{dx_1, dy_1,\}, \{dx_2, dy_2\} ...\}. Is there any way of fitting a nonlinear model to the data that incorporates the uncertainties on both x and y? I know that you can use the Weights command to...
Homework Statement
Not homework, but something I'm interested in finding out. The setup is a flexible wire with left endpoint fixed at x=0 and right endpoint at x=L. You push the right endpoint with some horizontal force directed towards the left endpoint which will move the right endpoint...
Homework Statement
I want to estimate f(x)=\ln (\frac{1}{1+x}) on the interval (1/10,1) with the error on the approximation being no more than 0.1.
Homework Equations
http://en.wikipedia.org/wiki/Taylor's_theorem#Example
The Attempt at a Solution
Following the example from Wikipedia, I...
I'm trying to model the flight of a small spherical object (such as a ping-pong ball) through air at smallish velocities (\approx 5{_{m/s}}) with linear drag so that F_{d}=-bv. The problem is, I can't find a table of linear drag coefficients (b) anywhere; it's always just the normal drag...
Homework Statement
We did a lab analyzing this inverting, negative feedback circuit for a 741 op-amp:
We measured the closed-loop gain and phase shift of the signal for several values of the input frequency with R_2/R1=1000,R_2/R1=100 and R_2/R_1=10. The gain curves all looked like...
Homework Statement
Check out matt grime's post in this thread (it's the last one):
https://www.physicsforums.com/showthread.php?p=470773#post470773"
How exactly did he know that the sum could be represented as that double integral? Also, is there a method of converting sums like that...
Homework Statement
I'm teaching myself the Green's function method for ODEs, because it looks relevant to my interests. This is a (slightly contrived) problem I just came up with arbitrarily:
y''+5y'+6y=sin(x) \; \; \; ; \; \; \; y(0)=y'(0)=0
Homework Equations
i) When considered as a...
Homework Statement
I (came up with)/(heard about) a way of using Laplace transforms that I didn't think about before. The problem is that it doesn't work for some reason.
Look at following integral:
I = \int_{0}^{\infty }sin(t)dt
Say that you had no idea how to integrate something...
Homework Statement
Determine a series solution to the following ODE about x0 = 0:
xy'' + y' + xy = 0
The Attempt at a Solution
I'll try to keep this concise.
I first divided through by x and made the usual guesses for the form of the series. Subbing those in gave...
The Exposition
I'm an Astrophysics major going into my second year of University this September. I have room for one other course in fall and one course in the winter (so I have two spaces to fill.) I'm already required to take every physics course offered (6 total), and the two requisite...
Not a homework question per se, but I'm having some issues with moments of inertia. Say I wanted to calculate the I for a ring. What I would do is:
I = \int r^2dm
m = \lambda L
dm = \lambda dL
I_{ring} = \int_{0}^{L}\lambda r^2dL
And that would give the requiside mr2. My question...
Homework Statement
Say that
\sum_{k=1}^{\infty }a_k
converges and has positive terms. Does the following necessarily converge?
\sum_{k=1}^{\infty }{a_k}^{5/4}
Homework Equations
If it necessarily converges, a proof is required, if not, a counter-example is required.
The...
Homework Statement
Say that f(x) is some function whose second derivative exists and say u(x, t)=f(x + ct) for c > 0. Determine
\frac{\partial u}{\partial x}
In terms of f and its derivatives.
Homework Equations
PD Chain rule.
The Attempt at a Solution
Say that x and y are...
Homework Statement
Find an expression for the electric potential a distance z away from the center of a thin uniformly charged rod of length L on a line that bisects the rod.
Homework Equations
V=k\frac{Q}{r}
The Attempt at a Solution
Determine the electric potential due to one...
Homework Statement
Determine an expression for the electric potential at a point P a distance z away from the center of a thin uniformly charged rod on the line that bisects the rod.
Homework Equations
V=k\frac{q}{r}\hat{r}
The Attempt at a Solution
I've defined r to be the...
Homework Statement
Let
V=span(sinx,cosx)
be the subspace of Maps(R,R) generated by the functions sin(x) and cos(x), and let
D:V \to V
be the differential operator defined by
D(y)=y''+y'+y for y E V.
Show that Im(D) = V and conclude that for every f E V, the differential...
Homework Statement
If f is some function defined at points "s" and "t," is there any way to simplify the following expression?
f((f(t)-f(s))
Homework Equations
None that I know of.
The Attempt at a Solution
I've been tinkering with this for a while and so far, I've determined...
Homework Statement
Not really a problem per se; more of an issue with some aspects of linear transformations. We've learned that a linear combination of linear transformations is defined as follows:
(c_1T_1+c_2T_2)(\vec{x})=c_1T_1(\vec{x})+c_2T_2(\vec{x})\,\,\,\,\,\, \vec{x}\varepsilon...
Homework Statement
Well gentlemen, another year, another integral eh? Anyways,
\int \frac{1}{x^4+4}\,dx
I really want to do this without looking at Wolfram/Google.
Homework Equations
U-substitutions, parts, partial fractions
The Attempt at a Solution
Basically I tried to...
Homework Statement
Let \vec{u}\neq 0 be a vector in \mathbb{R}^2 and let
T:\mathbb{R}^2 \to \mathbb{R}^2 be described by
T:\vec{v} \to proj_{\vec{u}}(\vec{v})
If \vec{u}=[1,-1]
Find the standard matrix for T
Homework Equations
proj_{\vec{u}}(\vec{v})= \frac{\vec{v}\cdot\vec{u}...
Homework Statement
Let V\subset\,R^n be a subpace with dim(V) = D. Prove that any S vectors in V are linearly dependent if S > D.
Homework Equations
Rank-nullity theorem?
The Attempt at a Solution
dim(V) = D implies that there are D vectors in a basis for V. If S > D then there...
Homework Statement
Let A be a skew-symmetric n x n matrix with entries in R.
a) Prove that
u^{T}Au=0 for every u E R^{n}
b) Prove that
I_{n} + A is an invertible matrix.
Homework Equations
A^{T} = -A
The Attempt at a Solution
a)
u^{T}Au=0
Transpose both sides:
(u^{T}Au)^{T} = 0^{T}...
Homework Statement
Suppose that f is a continuous function.
a) If g is a differentiable function and if
F(x) = \int_{0}^{x}g(x)f(t)dt
find F ' (x).
b) Show that
\int_{0}^{x}(x - t)f(t)dt = \int_{0}^{x} [\int_{0}^{u}f(t)dt] du
Suggestion: Use a) and the racetrack...