Questions tagged [approximations]

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99 views

How is Einstein's postulate about the invariance of the laws of physics justified? [duplicate]

According to one of Einstein's postulates related to special relativity, > "the laws of physics remain invariant in their form and nature in all inertial frames". But global inertial frames don't ...
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0answers
85 views

Is there any approximation in which the double pendulum has an exact solution?

We know that the double pendulum isn't an integrable system, since only energy is conserved versus two degrees of freedom. My question is, does a physical approximation exist, in which the total time ...
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Existence of solid mechanics problems that cannot be solved through Lax-Milgram approaches

Very often, solid mechanicians employ finite-element analyses to solve problems in linear solid mechanics. This approach is guaranteed to work because the Lax-Milgram theorem, along with some ...
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3answers
91 views

What are point objects?

I can't seem to get the idea of point mass into my head. Why are equations of physics applicable on only point masses and should be altered while dealing with object that has a collection of points? ...
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1answer
68 views

Can the jerk diverge?

In other words can the acceleration change instantly? In direction and/or magnitude. There are two aspects to this question. In a problem, can you treat acceleration as changing instantly? (when in ...
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3answers
102 views

How strict are the boundaries that divide dimensions? Is a single-layer sheet of graphene 2D or 3D? [closed]

I would like to know if there is any theory that describes a set of rules that define the boundaries of dimensions. For example, does a single layer sheet made of graphene considered a two or a three ...
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1answer
1k views

Can Hooke's law be derived?

Can we derive Hooke's law from the theory of elasticity? I know it is not a fundamental law and therefore can be derived from more basic considerations.
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1answer
78 views

Particle trapped in a double well potential and “slowly” increasing the height of the potential barrier

Today I was musing over the following problem. Consider a non-relativistic particle confined in an one-dimensional double well potential of the form $V(x)=\kappa(x^2-a^2)^2$ where both $a$ and $\kappa$...
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1answer
174 views

How can I find the metric in weak field limit for specific theory?

What is the general approach to finding a modified version of Poisson equation by means of the weak field limit of a specific gravitational theory? What is the first step? Can you introduce the main ...
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1answer
247 views

Eigenkets of degenerate perturbation theory

Suppose the original Hamiltonian is $H$ and we perturb it by a small potential $V$. The basis kets of the original hamiltonian $H$ contains some degeneracy. Since there's some degeneracy, we take ...
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1answer
91 views

Approximating sums as integrals and divergent terms

I have the following sum (notice that the sum starts from 2, i.e. there's no divergence): $$\sum_{i=2}^{N}C_i\dfrac{\exp{\left(-k| \mathbf{R}_i-\mathbf{R}_1| \right) }}{| \mathbf{R}_i-\mathbf{R}_1|}$$...
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1answer
58 views

How to deal with negative potential in the WKB approximation?

I'm trying to model a system as being inside an infinite potential well with $V(x)=-ax^v$ where $a$ and $v$ are some positive real numbers. However I'm a bit confused: if I take the - sign inside ...
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2answers
2k views

Why don't the Navier-Stokes equations simplified for hydrodynamics contain gravitational acceleration?

The incompressible Navier-Stokes equations widely used in hydrodynamics don't have the gravitational acceleration. $$ \begin{align} \frac{\partial u_i}{\partial x_i} & = 0, \\ \frac{\partial u_i}{\...
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1answer
48 views

How do you expand $\langle x'-\Delta x'\rvert \alpha\rangle$?

In my textbook (Sakurai) the following identity is often used: $$ \left< x'-\Delta x' \, \middle| \, \alpha\right>~=~\left< x' \, \middle| \, \alpha \right> - \Delta x'\frac{\partial}{\...
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1answer
103 views

In what physical situations is the weak-field limit invalid?

in the weak-field limit gravitation is described by a symmetric tensor field $h_{μν}(x)$ in flat spacetime. Linear theory suffices for nearly all experimental applications of general relativity ...
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2answers
40 views

Is there anything wrong with my Euler's method equations for a pendulum outside of small angles?

I'm trying to write a program to calculate the angle, angular speed and energy of a pendulum at different times using Euler's method. The equation I started with was:$${\rm d}^2θ/{\rm d}t^2 = - g\sin(...
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1answer
81 views

Why does mass increase when gravitational potential energy increases?

I saw a solved example in a book (Concepts of Physics by H.C. Verma, volume 2), where there is a body near surface of the earth, the problem is to calculate the increase in mass of the body when it is ...
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0answers
63 views

Derivation of rocket equation [duplicate]

Let $m$ be the mass of a rocket in free space (including fuel) at the time $t$. Now suppose that the rocket ejects a mass $\Delta m$ during the interval $\Delta t$ with velocity $v_e$ relative to the ...
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1answer
40 views

Approximating mechanical systems with high friction

I remember to have read (a long time ago) that if I have a mechanical system $\ddot x=\frac1mF(x)$ with a high friction, then I can instead study the other system $$\dot x\sim F(x)$$ to get an ...
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Validity of the thin wall approximation

Inspired by the question How can I understand the tunneling problem by Euclidean path integral where the quadratic fluctuation has a negative eigenvalue?, I decided to come back to the first paper by ...
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2answers
241 views

How the last digit in significant figures is considered doubtful?

If a reading of a length on meter rod is 44.6cm with least count of 1mm And last point of the length is exactly on 44.6 not in between of 44.6 or 44.7 Then how is it doubtful?
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1answer
163 views

Nearly circular orbits (and their angular frequency)

In 'nearly' circular motion, the radius is not constant. If the force is central the and the angular momentum is still conserved, we have a central force: where $r_0$ is equilibrium position. For ...
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2answers
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The possiblity of one more raindrops drop the ground exact same time? [closed]

Think there is a simple rain and we have got a very very sensitive clock. Is it possible to at least two raindrops hit to ground exact in same time? I mean the date time when they hit the ground, not ...
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1answer
55 views

Strong force gets weaker at small distances yet approximated by -1/r potential

My particle physics textbook (by Martin and Shaw) has confused me, it states in ch.7 that the strong force gets weaker at small distances, and that it can be approximated by $V(r) = -\frac{4 \alpha_s}{...
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1answer
73 views

How to “derive the leading correction” to an energy expression?

The kinetic energy of motion of a particle is the relativistic total energy minus the rest energy. (a) A particle has rest mass $M$ and speed $v$. If $v \ll c$, then show that the kinetic energy ...
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78 views

System of $N$ bosons

What precaution needs to be observed in writing down an expression for the total number of bosons $N$ valid at low temperatures?
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1answer
251 views

Why is the area of an elemental ring with radius $r$ and width $dr$ is $2πr dr$?

As shown below the topic is to calculate Electric Potential due to a disk whose surface charge density is $ \sigma $. To do this we considered an elemental ring with inner radius = $r$, outer radius =...
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55 views

What is the significance of the plot of $E_y/E_0$ versus $y/\ell$?

In the problem as shown below The electric field comes out to be $$ E_y = \frac{1}{4\pi\epsilon_0} \frac{2\lambda\sin\theta}{y} = \frac{1}{4\pi\epsilon_0}\frac{2\lambda}{y}\frac{\ell/2}{\sqrt{y^2 +...
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5answers
622 views

Mean field theory Vs Gaussian Approximation?

I am getting confused about the distinction between Mean-field theory (MFT) and the Gaussian approximation (GA). I have being told on a number of occasions (in the context of the Ising model) that the ...
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1answer
55 views

Why is it intuitively unreasonable for this transition probability to grow quadratically in $t$?

In Sakurai's "Modern Quantum Mechanics" section 5.6, there is a seemingly simple statement made that I do not understand the logic of. The author is considering a physical situation in which we "turn-...
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0answers
26 views

Maximum intensification by refraction

Suppose that a beam of light in a medium with index of refraction $n$ reaches the surface of the medium, with vacuum on the outside. Its incident angle with respect to the normal is $\theta$. Only a ...
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1answer
157 views

Gauss' law: Infinitely long cylinder approximation

Why approximating a cylinder as infinitely long works when $R << L$? Where $R$ is the radius and $L$ is the length of the cylinder. I was able to prove for a line of charges when r << L (...
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1answer
64 views

Type-I seesaw: order of magnitude of the eigenvalues of effective $M_\nu$

Consider a square matrix $C$ constructed out of two other square matrices $A$ and $B$ as $$C=-A^TB^{-1}A.$$ Suppose all the elements of $B$ are very large compared to those of $A$. In such a case, is ...
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1answer
411 views

A Problem With Deriving Ideal Gas Entropy From Multiplicity

To derive the entropy of an ideal gas via the ergodic hypothesis, we first find the density of states function: $$g(E)=\frac{V^{N}}{h^{3N}}\frac{(2\pi m E)^{\frac{3N}{2}}}{\left(\frac{3N}{2}-1\right)!}...
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1answer
169 views

Scattering amplitude with Born series [closed]

I didn't understand the step before similar sign in the image u can refer image for more elaboration.
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1answer
75 views

is it mathematically accurate to simply objects to point masses when calculating gravitational forces between them? or is it just an approximation?

i tried searching for the exact mathematical proof that validates this assumption, but couldn't find any. also, is this assumption still accurate if the density of the object resembles a planet (...
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102 views

Is there an easy way to understand why doubly excited helium does not exist?

1) A method for evaluating the energy of helium states To understand why doubly excited helium is not allowed in nature, we should show, to my eyes, that if we try to build this state we'll find a ...
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2answers
930 views

Why does distance vs period graph look linear for a pendulum?

Question/problem In my introductory physics class in college, we were supposed to find a relationship between the length of a pendulum and the period. Period=$2\pi*\sqrt{l/g}$ So thus i believed ...
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46 views

pNRQCD at high pT

pNRQCD is an effective field theory for heavy Quarkonium, where the velocities are non-relativistic due to large mass. But is pQCD applicable when the Quarkonium is moving at high velocities? The ...
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29 views

Gravitational potential energy on the Earth's surface [duplicate]

We assume that gravitational potential energy at a height $h$ from the Earth's surface is $mgh$. Is that accurate or only approximately correct ? Here is my approach. On the surface of the Earth, $...
0
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1answer
91 views

Why can the equation of adiabatic process $P_1 V_1^\gamma=P_2 V_2^\gamma$ be written as $\Delta P/P=-\gamma \Delta V/V$?

Why can the equation of adiabatic process $P_1 V_1^\gamma=P_2 V_2^\gamma$ be written as $$\frac{\Delta P}{P}=-\gamma \frac{\Delta V}{V},$$ where $P_2-\Delta P=P_1$ and $V_2+\Delta V=V_1$ as in ...
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1answer
146 views

Two-source interference in the perpendicular direction and small-angle approximations

In the following setup, we have two point sources of light producing monochromatic, spherical light waves in-phase of wavelength $\lambda$, and a screen positioned in a plane prependicular to the line ...
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1answer
37 views

Working on Newton's Gravitational Law and cannot understand this approximation

Working through some questions in a text book and came across one which I did not quite understand how to find the solution. I looked up the solution in the manual and was unable to understand how ...
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2answers
121 views

Why can large objects at greater distance be treated as a point particle?

Why can large objects at greater distance be treated as a point particle? "The bodies of our solar system are so far apart compared with their diameters that they can be treated as particles to an ...
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1answer
60 views

Taylor approximation vs setting to zero

I was recently solving a problem where I had to find the velocity of a particle. The correct result is the following: \begin{equation}v^2 = v_0^2 + \frac{k\delta^2}{m}\left(1-e^{-\frac{\Delta^2}{\...
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1answer
60 views

Is it possible to approximate the motion under a V-shaped potential as harmonic motion?

Let's assume an $xy$ plane and let there be a force field defined by the potential $$V=F_0|x|$$ Though the potential is not differentiable still its a perfectly realisable system. If we solve the ...
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Elliptical Trajectory, or Parabolic?

Discuss whether this statement is correct: “In the absence of air resistance, the trajectory of a projectile thrown near the earth’s surface is an ellipse, not a parabola.” Is the above statement ...
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Mean field scaling introduce a factor of $1/N$ instead of $1/N^2$?

In this arxiv paper (pg2) they introduce the concept of mean field scaling, which keeps the overall charge constant. To do this they introduce a factor of $1/N$ in the force term: $$\dot V_i =\frac{1}{...
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1answer
52 views

Approximations in general

In analysis, a statement like $f(x) \ll g(x)$ (as $x\to x_0)$, has a very precise meaning: $$ \lim_{x\to x_0}\dfrac{f(x)}{g(x)}=0. $$ I was wondering, when physicists write $L_1 \ll L_2$, for, say, ...
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94 views

Why do we use differential equations in physics instead of $h$-difference ones?

Since we don't know whether space and time are discrete or continuous wouldn't it be a better idea to use $h$-difference equations where the derivative is $$f'(x) =\frac{f(x+h)-f(x)}{h},$$ since they ...