Derivation Definition and 1000 Threads

I am extremely engaged in studying the different ways of derivation of the equation of motion in General Relativity. On the way, I found a very general form of the equation of motion that no standard books have done (to my knowledge). Although the process is implemented in deriving the...

We want to relate acceleration in two frames, an inertial frame S, and the instantaneous inertial reference frame of the particle on which it is being accelerated, S', which is moving in the ##x## direction at the moment. Let the acceleration in S be ##(a_x,a_y)## and in S' be ##(a_x',a_y')##...

Hi, Unfortunately, I can't quite work out the terms in the task The expression in the integral (4) corresponds to the Lagrangian and since ##t## and ##\varphi## are cyclic variables, the following follows: $$-\frac{d}{dt} \frac{d L}{d \dot{t}}=0 \quad \rightarrow \quad -\frac{d L}{d...

Hi. I'm trying to derive Snell's law from a mechanical model: An axle of length ##d## with two wheels that run at speeds ##v_1## or ##v_2##, depending on the medium they are in (##v_1>v_2## for the moment). The ##x##-axis is the interface, ##v_1## for ##y>0## and ##v_2## for ##y<0##. The wheels...

Assuming I push a 1 kg block on a level surface, with no energy lost to friction, and I have an equation F(x) for force in terms of position, how would I derive an equation v(t) for velocity in terms of t? Specifically the function F(x) = sqrt(x) * sin(x^2), for 0 <= x <= sqrt(pi), if possible...

This question refers to Doppler Effects observed in circular motion (at non relativistic speeds, so ##v\ll c##, ignoring transversal Doppler shifts). Suppose there is a source emitting a frequency, ##f_s##. An observer at the center will experience no shift in observed frequency (##f_r##). As...

Hi Guys Please refer to my attached derivation, do you think it is acceptable? There is a torsional spring at the base of the column. in reality the column is going to be a complex structure however the torsional stiffness can be approximated using dummy loads on the structural model.

I was looking at the derivation of the lens maker's formula and I have a minor confusion which does not seem to go away. So it is derived from the same principle for the refraction of a light ray at a spherical convex/concave surface, except that it undergoes refraction twice due to two...

Here is the solution: From vector identity, $$\nabla (\vec A \times \vec B) = \vec B \cdot \nabla \times A - \vec A \cdot \nabla \times \vec B $$ If ##\vec B = 0##, then $$\nabla \vec S = \nabla \frac{\vec E \times \vec B}{\mu_0} = \frac{1}{\mu_0} (\nabla \times E \cdot \vec B - \cdot E \cdot...

This page is Quantum mechanics by bransden. My homework is explain why there is no regulation of quantum number n in selection rule. Also explain that by solving that integral of radial part is always non-zero. ∫∞0[rRnl(r)]Rn′l′(r)r2dr (n is different with n') I tried to solve it by just...

I tried to use square box with two perpendicularly oscillating photons to derive length contraction formula. Photons are emitted, reflected from box walls and absorbed simultaneously from the box's point of view. If the box have some velocity v, then assuming photons are still emitted and...

Okay, so we have that $$dU = \left( \frac{\partial U}{\partial V} \right)_S dV + \left( \frac{\partial U}{\partial S} \right)_V dS$$ And comparing that to the first law, we get that $$T=\left(\frac{\partial U}{\partial S}\right)_V$$. Comparing expressions of ##T##...

I can't see what is the problem with my derivation but the answer is incorrect. Please help. We assume here that ##\omega_0 = 0## and ##v_0 = 0##, hence it immediately rolls without slipping without any transitional phase. Hence ##v = \omega R##. Thus, ##v(L) = R\omega(L)##. Since our...

for over a century the weak field limit of general relativity is Newtonian, which when applied to galaxy rotation curves requires cold dark matter as Newtonian limit is too weak this recent paper arXiv:2408.00358 (gr-qc) [Submitted on 1 Aug 2024] Quasilocal Newtonian limit of general...

Derive formula for surface brightness The formula in question we wanna derive is: S(mag/arcsec^2) = M⊙ + 21.572 -2.5 log_{10} S(L⊙/pc^2). Best regards.

By the First Law, Definition of an Adiabatic Process, and Definition of Work: ##\Delta E = Q - W = - W = - P \Delta V ## (because ##Q = 0##) (Equation 1) By the Equipartition Theorem: ##\Delta E = \frac{3}{2} Nk \Delta T ## (Equation 2) By Setting Equation 1 equal to Equation 2 ## \Delta T...

I'm not sure where the equation E_k=(gmR^2)/4h comes from & I also don't really know where to start either :(

Given that ##P = ρgh##, there's obviously a problem with the following derivation of fluid pressure under gravity. Can someone spot the flaw? $$W = mgh$$ $$W = ρVgh$$ $$F \cdot dh = ρVgh$$ $$F \cdot dh = ρ(Ah)gh$$ $$F \cdot dh = ρgAh^{2}$$ $$\frac{d(F \cdot dh)}{dh} = \frac{d(ρgAh^{2})}{dh}$$...

Hi, I'm looking at this relativistic rocket equation on Wikipedia. Something doesn't make sense here, and I can't find a derivation for this equation in the linked source, so I'm trying to derive it myself with limited success...

I've worked out how to derive the formulas for a solid cylinder and a solid sphere rolling down a hill. E.g., for a cylinder: Emech = KE + PE mgh = 1/2 mv^2 + 1/2 Iw^2 gh = 1/2 v^2 + 1/2 (1/2r^2) v^2/r^2 gh = 3/4 v^2 v^2 = 4/3 gh I then performed a derivative with respect to time and found a...

I know that the formula qqk/r applies to a system (two charges), but where is the flaw in my derivation? Thanks!

This question is not crucial, but I'd like to understand better the equation (14.35) in this derivation: Here ##\Phi## is an eigenvalue of ##\hat \phi##, i.e., ##\hat \phi (\vec x ) |\Phi \rangle = \Phi (\vec x) |\Phi \rangle##. First, I think that there is a typo in (14.35): the Hamiltonian...

My question is about this step in the derivation: When the ##\partial_\nu \mathcal L## in 3.33 moves under the ##\partial_\mu## in 3.34 and gets contracted, I'd expect it to become ##\delta_{\mu \nu} \mathcal L##. Why is it rather ##g_{\mu \nu} \mathcal L## in the 3.34? (In this text, ##g_{\mu...

In this derivation, a basis of one-particle states ##\langle x|=\langle \vec x,t|## is expressed with the field operator, $$\langle x|=\langle 0| \phi (\vec x, t)$$ "Then, a Schrodinger picture wavefunction is $$\psi (x)=\langle x| \psi \rangle$$ which satisfies $$i \partial _t \psi (x) = i...

I have been trying to solve this problem for hours using the mentioned equations but no matter what I do I cannot get the correct answer, that is v = 22.4 m/s. I thought that maybe if I could get an expression where v is a function of time I could solve the problem but I don't know how to do...

It is from [ Class 11th] SL Arora, pg no. 11.3, the heading is [11.7] Absolute Scale of Temperature.

I was trying to show that ##sin(x-y) = sin(x)cos(y)-cos(x)sin(y)## using Pythagoras' theorem and ##cos(x-y)=cos(x)cos(y)+sin(x)sin(y)##. I have: $$sin^2(x-y)=1-cos^2(x-y)$$ $$sin^2(x-y)=1-(cos(x)cos(y)+sin(x)sin(y))^2$$...

Hello, I am looking for a detailed derivation of the equations used to generate the modified sine curve. I found one in Cam design handbook by Harold A. Rothbart but I didn't understand how we get certain equations. My end goal is to combine the modified sine curve with constant velocity and get...

I was / am trying to derive the energy shift resulting from the normal Zeeman-Effect by coupling the Hamiltonian to the external field ##\vec{A}##, that carries the information about the field ##\vec{B}## via ##\vec{B} = \nabla \times \vec{A}##. Let ##q = -e## be the charge of the electron and...

I want to calculate ##\int \vec{P}\left(\overrightarrow{r^{\prime}}\right) \cdot \vec{\nabla}_{\overrightarrow{r^{\prime}}} \frac{1}{\left|\vec{r}-\overrightarrow{r^{\prime}}\right|} d^{3} \overrightarrow{r^{\prime}}## with macroscopic polarization...

TL;DR Summary: I am Highschool student writing a 4000 word research paper on Bernoulli's principle and the coanda effect. I need help with derivation of a formula that connects flow rate of water and distance moved by the sphere in my experiment. I am a high school student writing a 4000 word...

Hi, I'm trying to follow the derivation of the Zeno time from two sources and am struggling. I think I'm missing some sort of algebraic trick and any tips would be appreciated. A bit more detail below. In the attached paper \citep{Facchi_2008}, the Zeno time (equation (6)) is derived from...

I can derive it for a circular loop: $$dF=BI\sin\phi\ dl=BIr\sin\phi\ d\phi$$ Torque on quarter circle when field is parallel to plane of loop=$$\tau=\int^{(\pi/2)}_0 BI \ dl \sin\phi (r\sin\phi)$$$$=\int^{(\pi/2)}_0 BIr^2 \sin^2\phi\ d\phi$$ Net torque=##4\tau=BIA## If magnetic field is at any...

I need help to understand how equation (27) in this paper has been derived. The definition of P(k) (I discarded in the question ##\eta## or the integration with respect for it) is given by (26) and the definition of h(k) and G(k) are given by Eq. (25) and Eq. (24) respectively. In my...

I know that ##∇(A⋅v)=(A⋅∇)⋅v+(v⋅∇)⋅A+v×(∇×A)+A×(∇×v)## The third term ##v×(∇×A)## simplifies to ##v×B##. I'm just now sure how to "get rid" of the other terms. I tried checking for some vector identities but couldn't make any headways. Any guidance?

Hello, I am following the paper: https://www.yumpu.com/en/document/read/42212557/exact-exchange-in-density-functional-calculations and I am confused on page 14 where the generalized Kohn-Sham equations are derived. I follow that the ground state energy is The minimization of this step leads...

TL;DR Summary: I'm stuck trying to find the equation for time period T of a physical pendulum without any calculus using torque. Hello all. I am currently writing my IB Physics HL IA (high school physics lab report). I am investigating the effect of length on the time period of a uniform rod...

a) We have ## S[y+\epsilon h]=\int_{1}^{2}[3(y'+\epsilon h')^2-2(y+\epsilon h)^2]dx ##. Note that ## \frac{d}{d\epsilon}S[y+\epsilon h]=\int_{1}^{2}[6(y'+\epsilon h')h'-4(y+\epsilon h)h]dx=2\int_{1}^{2}[3(y'+\epsilon h')h'-2(y+\epsilon h)]dx ##. Then ## \bigtriangleup S[y...

I'm new to this. I would like to know how we derive the earth's yearly mean temp. I assume there are thermometers all over...

I have several questions relating to electrostatics: first of all, in this derivation for the formula of the electric potential energy: work is being done against the electric field right, so the work should be negative, but in this case it's positive. I'm wondering if it's because the direction...

By conservation of mechanical energy: $$ E(r_0)=-\frac{GMm}{r_0}+\frac{1}{2}\mu \left (\dot{r_0}^2+r_0^2 \omega_0^2 \right) $$ where R0 =Rmax. Because our body is located at the apoapsis the radial velocity is 0. Hence: $$ E(r_0)=-\frac{GMm}{r_0}+\frac{1}{2}\mu (r_0\omega_0)^2 $$ By the...

To derive ##\vec r (t)=(−Rsin(ωt),Rcos(ωt)) ## I start by integrating ##ω=\frac{dθ}{dt}## to get ##θ_f=θ_i+ωt##. Therefore since ##Δθ=θ## by definition since the angular displacement is always taken with respect to some initial reference line, then ##θ_f−θ_i=θ## , thus, ##\theta = \omega t##...

Recollections of a late Spring semester's lesson describing the derivation of Lorentz's Transformation often solicit many unanswered questions. The textbook used has been secured; however, it is unknown. Whether, that secondary school instructor provided the premises used for the derivation from...

Let say we have two quantities, A and B. A = a ± Δa and B = b ± Δb, where a and b are the value of A and B and Δa and Δb are their absolute uncertainty respectively. Now we have a formula of C, where C = A + B. The absolute uncertainty is Δc = Δa + Δb. How to derive this formula? Is the...

The text derives C_p-C_v=nR for ideal gasses. They start with $$H = U + PV = U + nRT$$ for ideal gas. Since U is only a function of temperature for an ideal gas, the right-hand side is only a function of temperature so $$\frac{dH}{dT} = \frac{dU}{dT} + nR$$. Now the text does something I...

Hello, Please see this part of the article. I need to obtain the ##\rho (\phi)## value after obtaining the c0 to c5 constants of the ##\sigma (\phi)##. But as you can see after finding the coefficients, solving Eq.(1) could be a demanding job(I wasn't able to calculate the integral of Eq(1)...

For this, Does someone please know where they got that ##f'## is number of waves fronts received per unit time from? Also could we write the equation highlighted as ##f' = \frac{n\lambda}{t}## where ##n## is the number of wavefronts in a time ##t##? I derived that from ##\frac{vt}{\lambda} =...

Can someone please help derive the relations below from first principles? Also does someone please know what happens when ##ρ_{object} = p_{fluid}##? Many thanks!

I already solved w x x/|x| For (w1,w2,w3) and (x1,x2,x3)