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0
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2answers
27 views

Non-uniform circular motion, computing the angle

There is an object moving in circle, with this law: $$ \alpha = -k^2 \theta $$ With $ \alpha = \frac{ d \omega }{dt} $, and $ \omega= \frac{d \theta}{dt} $, $\theta$ = angle, $k$ positive ...
0
votes
1answer
27 views

Movement with non-constant acceleration [duplicate]

Suppose we have a material point. If it is moving from position $X_0$ with initial velocity $V_0$ and constant acceleration $A$, then from elementary physics course I remember that its movement is ...
1
vote
2answers
210 views

Position dependent speed, how to compute position

I can't solve a problem: $ A= 0.5 (ms)^{-1}$, $ x_0 = 0.5 m $, $v(t)= A \cdot x^2 $, I have to compute the position at $t=3$ ($x_0$ is the initial position). So my guess is that I should be able ...
5
votes
4answers
511 views

How are electric flux calculations not double integrals?

A disk of radius 0.10 m is oriented with its normal unit vector $\hat{n}$ at 30$^{\circ}$ to a uniform electric field $\vec{E}$ of magnitude 2000 N/C. What is the electric flux through the disk? ...
1
vote
3answers
155 views

Physical motivation for differentiation under the integral

I am thinking about the mathematical process of "differentiating underneath the integral", i.e. applying the theorem $$\partial_s \int_{-\infty}^\infty f(x,s)\,dx=\int_{-\infty}^\infty \partial_s ...
2
votes
2answers
215 views

Finding the illuminance from a triangular light source

Since most light sources in games are point-like, it's pretty difficult to approximate area light sources with point sources. As triangles are a universal form to represent 3D models (thus area light ...
1
vote
1answer
26 views

Moment of inertia of a sphere

I'm looking at sample calculations of moment of inertia of a sphere here. In the first example (disc method), it has the integral as $dI = \frac{1}{2}r^2 \,dm$, while in the second example (shell ...
0
votes
0answers
14 views

Self-inductance of a toroidal inductor

I am trying to determine the self-inductance of a toroidal coil of mean radius $R$ with $N$ loops of radius $a$ with a current $I$ flowing within them. I have calculated the magnetic field by noting ...
4
votes
3answers
130 views

Integral form of Gauss's law for magnetism from Stokes' theorem?

How can the integral form of Gauss's law for magnetism be described as a version of general Stokes' theorem? How does it follow?
1
vote
1answer
92 views

Free particle propagator - Evaluating Integral

In path integral formalism, when evaluating the free particle propagator, we obtain the functional integral of the form, $$ K_0 = \lim_{n\rightarrow\infty} \bigg( \frac{m}{2\pi ...
1
vote
1answer
36 views

Integrating for velocity [duplicate]

Trying to determine velocity of a falling body with respect to traveled distance and initial speed. I've been provided with the following equation for acceleration as a function of distance and the ...
0
votes
1answer
46 views

Line Integral Parameterization

In math, I was taught to parameterize a scalar/vector line integral. In physics, I remember doing problems where I didn't parameterize the problem and it still came out correct. So, by ...
3
votes
1answer
293 views

Relationship between irradiance and radiance

A question related to radiometry: Irradiance $E$ at a point $x$ can be written as: $E = \int_\Omega L(x, \omega) cos(\theta) d\omega$ I understand this formula and where it comes from. The equation ...
3
votes
1answer
164 views

Three integrals in Peskin's Textbook

Peskin's QFT textbook 1.page 14 $$\int_0 ^\infty \mathrm{d}p\ p \sin px \ e^{-it\sqrt{p^2 +m^2}}$$ when $x^2\gg t^2$, how do I apply the method of stationary phase to get the book's answer. ...
1
vote
1answer
36 views

When is the speed specified for an object experiencing an exponential force?

So this is the question given in my text book: A particle of mass m is at rest at the origin at time $t = 0$. It is subjected to a force $F (t) = F_0e^{–bt}$ in the $x$ direction. Its speed ...
1
vote
2answers
121 views

Integration constants in Maxwell's equations (ambiguousness?)

In classical electrodynamics, if the electric field (or magnetic field, either of the two) is fully known (for simplicity: in a vacuum with $\rho = 0, \vec{j} = 0$), is it possible to unambiguously ...
2
votes
3answers
79 views

Calculating the Potential from the E-Field

I find that often times I'll be tripped up by questioning whether or not I can do something mathematically, and be unable to come up with a satisfying answer. This is, unfortunately, one of those ...
3
votes
0answers
27 views

The question about MTW 4-momentum integral expression and lorentz nature

In section 5.8 of Misner, Thorne, and Wheeler's "Gravitation" there is a proof that 4-momentum determined as $$ \tag 1 p^{\mu} = \int T^{\mu 0}\,\mathrm{d}^{3}\mathbf r , \quad \partial^{\mu}T_{\mu ...
1
vote
1answer
53 views

Prefactor for phase space integration

When calculating the canonical partition sum, we had the following: $$ Z_\text C = \sum_{\vec p} \sum_{\vec x} \exp(-\beta H(\vec p, \vec x)) $$ Now, since $\vec p$ and $\vec x$ are pretty much ...
5
votes
2answers
184 views

Evaluate $1$-loop contribution to the $4$-point Green's function

I am trying to evaluate the following integral \begin{equation} I = \int \frac{d^d p_\text{E}}{(2 \pi)^d} \frac{1}{(p_\text{E}^2+m^2)((q_\text{E}-p_\text{E})^2 + m^2)} \tag{1} \end{equation} where ...
2
votes
2answers
5k views

Determining the center of mass of a cone

I'm having some trouble with a simple classical mechanics problem, where I need to calculate the center of mass of a cone whose base radius is $a$ and height $h$..! I know the required equation. But, ...
6
votes
1answer
189 views

Integral in $n$−dimensional euclidean space

I've asked this question in Mathematics Stack Exchange, but unfortunately there is no answer yet. I repost it because this integral comes from QFT and maybe someone here did it before or could help ...
1
vote
2answers
132 views

Is there a physically motivated “trick” to evaluate this convolution?

I'm working on adapting some of the formalism in this paper to a system I'm working with. The part I'm interested in amounts to convolving a density profile $\rho(r)$ with a smoothing kernel ...
3
votes
2answers
222 views

Gaussian type integral with negative power of variable in integrand

How can we compute the integral $\int_{-\infty}^\infty t^n e^{-t^2/2} dt$ when $n=-1$ or $-2$? It is a problem (1.11) in Prof James Nearing's course Mathematical Tools for Physics. Can a situation ...
0
votes
1answer
85 views

Making a cut trough a center of mass, can the masses of the pieces be equal?

Let's say point $P$ is the center of mass of an irregularly shaped object. If I make a straight cut trough point $P$ and split the object in two, is it possible for the two pieces to have the same ...
7
votes
4answers
582 views

Integrating radial free fall in Newtonian gravity

I thought this would be a simple question, but I'm having trouble figuring it out. Not a homework assignment btw. I am a physics student and am just genuinely interested in physics problems involving ...
3
votes
1answer
87 views

Correction to Period of a Pendulum

In one derivation of the corrected period of a pendulum, we started off like so: The mass has a height $y$ given by $l(1-\cos \theta )$. $E = K + E \rightarrow \frac{1}{2}ml^2 \dot{\theta}^2 + ...
9
votes
1answer
524 views

Why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$?

I'm preparing for my exam, but I have difficulties in perceiving why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$?
6
votes
1answer
244 views

Using $\frac{1}{A+i\epsilon} = PV\frac{1}{A}-i\pi\delta(A)$ in Feynman Integrals

Are the following operations O.K.? This is related to the Feynman parameter trick. $$F:= \int_0^1 \mathrm{d}x\int_0^{1-x}\mathrm{d}y \frac{1}{f(x,y)+\mathrm{i}\epsilon}.$$ Now using ...
4
votes
2answers
188 views

Integrating equations with units

I was looking through an old copy of Barron's AP Physics and found this problem relating to impulse which I was initially confused about how to integrate. Example 6.1 During a collision with a ...
2
votes
1answer
130 views

Integrals over grassmann numbers

I want to prove an identity from Peskin&Schroeder, namely that $$\left(\prod\limits_i^{} \int d \theta^*_i d\theta_i\right) \theta_m^* \theta_l \exp(\theta_j^* B_{jk} \theta_k)=\det(B) ...
5
votes
0answers
186 views

The commutator of scalar field [closed]

I have a real scalar field which is given by the propagator as: $$[\phi(x),\phi(y) ] =\int \frac{d^3 p}{(2\pi)^3} \frac{1}{2E_0} (\exp(-ip\cdot (x-y)) -\exp (ip\cdot (x-y)))$$ And I am asked to show ...
2
votes
1answer
85 views

Integration on a general equation for instantaneous angular acceleration

An equation for instantaneous angular acceleration is given as: $$ \alpha \equiv \lim_{\Delta t\to0}\frac{\Delta \omega}{\Delta t} = \frac{d\omega}{dt} $$ The text I am reading says writing this ...
1
vote
1answer
593 views

Gravitational force exerted by a rod on a point mass

I have doubts with the solution of a certain problem. I will give the entire solution below and will lay out my doubts as well. A point mass $m_1$ is separated by a distance $r$ from a long rod of ...
0
votes
1answer
82 views

Integration with Grassmann variables

How to show that $$ \int d\Psi d\bar {\Psi}e^{i \int d^{4}x\bar {\Psi} \hat {A} \Psi} = det (\hat {A})? $$ $\Psi , \bar {\Psi}$ refers to Dirac spinors (the second is $\bar {\Psi} = ...
0
votes
2answers
67 views

Number of 2-electron integrals

Consider 2-electron integrals over real basis functions of the form $$ (μν|λσ)=∫dr⃗_1dr⃗_2ϕ_μ(r⃗_1)ϕ_ν(r⃗_1)r^{−1}_{12}ϕ_λ(r⃗_2)ϕ_σ(r⃗_2) $$ I am told that for a basis set of size $K=100$, there are ...
4
votes
1answer
74 views

Question about exterior derivatives

I know from Carroll that the integration in GR is basically a mapping from n-form to the real number. And it's given that $$d^nx=dx^0\wedge\ldots\wedge ...
4
votes
1answer
302 views

Number of unique 2-electron integrals

Consider 2-electron integrals over real basis functions of the form $$(\mu\nu|\lambda\sigma) = \int d\vec{r}_{1}d\vec{r}_{2} \phi_{\mu}(\vec{r}_{1}) \phi_{\nu}(\vec{r}_{1}) r_{12}^{-1} ...
0
votes
1answer
382 views

Derivation of Moment of Inertia and centre of mass?

In the equation above for the MI for a rod, why are we taking the limits from -l/2 to l/2? And why doesn't the integral doesn't include the centre of mass?
0
votes
2answers
103 views

Function with poles/singularities; Polynomial approximant has no poles

I don't know if i should ask this question or if it makes too much sense. My knowledge of this topic is quite incomplete, so please bear through with me. Any insights are appreciated. A function ...
7
votes
5answers
11k views

How to get distance when acceleration is not constant?

I have a background in calculus but don't really know anything about physics. Forgive me if this is a really basic question. The equation for distance of an accelerating object with constant ...
0
votes
2answers
49 views

How to get $t$ from $a(v)$?

I read what if we have acceleration given as a function of velocity we can calculate time as $$t(v) = t_0 + \int_{v_0}^{v} \frac{dv}{a(v)}.$$ Why?
2
votes
1answer
446 views

Momentum variance in momentum space for particle in a box

My assignment asks me to compute the momentum space wavefunction of the nth energy eigenstate of the particle in a one-dimensional infinite square well, then "show that your result is in agreement ...
7
votes
2answers
185 views

A problematic integral in calculating the entanglement entropy in 1+1 D free massive bosonic field theory

I encountered a curious integration identity when I was reading the paper by Pasquale Calabrese and John Cardy on the entanglement entropy of 1+1D quantum field theory (arXiv). The identity is given ...
2
votes
2answers
78 views

Integration of 3-momentum

During a lecture that I missed, I was trapped when the lecturer uses the relation $$dp_x~ dp_y ~dp_z ~=~d^3\mathbf{p} ~=~ 4\pi p^2 dp.$$ Can I know how is this relation derived please?
0
votes
1answer
81 views

Vehicle acceleration

What I'm essentially doing is Kalman Filter. If anyone is familiar with (but it doesn't really matter in this case). Consider the following formulas: $$x_k=x_{k-1}+v_{k-1}dt+a_{k-1}\frac{dt^2}{2}$$ ...
2
votes
3answers
270 views

Electrostatic energy integral for point charges

The electric energy stored in a system of two point charges $Q_1$ and $Q_2$ is simply $$W = \frac{1}{4\pi\epsilon_0}\frac{Q_1Q_2}{a}$$ where $a$ is the distance between them. However, the total ...
4
votes
2answers
419 views

About an electrostatics integral and a delta-function kernel

I'm having trouble with an integral and I would like some pointers on how to "take" it: $$ \int \limits_{-\infty}^{\infty}\frac{3\gamma a^{2}d^{3}\mathbf r}{4 \pi \left( r^{2} + ...
0
votes
0answers
177 views

Making a 3D physics engine, realistic? If so, where do I begin my research?

As a programming (technology), physics and math project in school, I'm considering programming my own 3D physics engine as a learning exercise. The physics engine should then be able to be used in a ...
1
vote
0answers
17 views

Hemisphere irradiance

How do I calculate sky irradiance from radiance (L) from a hemisphere above a surface which is tilted relative to the normal (x=0,y=0,z=1). I have L as a function of zenith (0 to 180deg) and azimuth ...