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    Point source of light, opaque screen with hole. photodiode problem

    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...
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    Meaning of representations of groups in different dimensions

    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...
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    Transfer function of this simple circuit

    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...
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    Conservation of angular momentum (electromagnetism)

    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...
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    Really tricky integral

    Homework Statement $$ I=\int_{0}^{\infty} \frac{x^3}{e^x-1}\ln(e^x-1)\,dx $$ Homework Equations Any identity involving \zeta(s). The Attempt at a Solution I noted that: $$ \ln(e^x - 1) = \ln(x) + \sum_{n=0}^{\infty} \frac{B_{n+1}(1)}{(n+1)^2n!}x^{n+1} $$ Therefore, the integral...
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    Constants of motion in Lagrangian

    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...
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    Negative energy eigenvalues of Hamiltonian

    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...
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    Mathematica Curve Fitting With Uncertainties

    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...
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    What shape will a bent wire take?

    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...
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    Error bounds on series approximation

    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...
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    Archived Linear Drag Coefficient

    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...
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    Op-Amp Frequency Response Theory

    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...
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    Infinite series wizardry

    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...
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    Green's Functions (for ODE)

    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...
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    Laplace Transforms for improper integrals?

    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...
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    Series Solution to ODE

    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...
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    Courses Suitable Math Course For 2nd year Astrophysics?

    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...
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    Moment of Inertia (Disk)

    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...
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    Series Convergence

    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...
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    Partial Derivative Chain Rule

    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...
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    Electric Potential From Rod

    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...
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    Electric Potential Expression

    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...
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    Linear Algebra DE

    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...
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    Function Composition

    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...
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    Linear Transformations Confusion

    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...
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    Integrate [1/(x^4 + 4)] dx

    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...
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    Linear Algebra Standard Matrix

    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}...
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    Linear Algebra Dimension Proof

    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...
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    Need Linear Algebra Proofs Proof-read

    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}...
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    Definite Integrals of Definite Integrals

    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...
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