Questions tagged [approximations]

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19
votes
8answers
3k views

Why do physicists believe that particles are pointlike?

String theory gives physicists reason to believe that particles are 1-dimensional strings because the theory has a purpose - unifying gravity with the gauge theories. So why is it that it's popular ...
4
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1answer
6k views

Is acceleration due to gravity constant?

I was taught in school that acceleration due to gravity is constant. But recently, when I checked Physics textbook, I noted that $$F = \dfrac{G m_1 m_2}{r^2}. $$ So, as the body falls down, $r$ ...
12
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1answer
2k 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.
26
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3answers
5k views

Why does a simple pendulum or a spring-mass system show simple harmonic motion only for small amplitudes?

I've been taught that in a simple pendulum, for small $x$, $\sin x \approx x$. We then derive the formula for the time period of the pendulum. But I still don't understand the Physics behind it. Also, ...
4
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2answers
2k views

Why are continuum fluid mechanics accurate when constituents are discrete objects of finite size?

Suppose we view fluids classically, i.e., as a collection of molecules (with some finite size) interacting via e&m and gravitational forces. Presumably we model fluids as continuous objects that ...
7
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7answers
32k views

Do we take gravity = 9.8 m/s² for all heights when solving problems? Why or why not?

Do we take gravity = 9.8 m/s² for all heights when solving problems?
29
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1answer
2k views

What different approximations yield Gravitoelectromagnetism and Weak Field Einstein Equations?

This question is inspired by this answer, which cites Gravitoelectromagnetism (GEM) as a valid approximation to the Einstein Field Equations (EFE). The wonted presentation of gravitational waves is ...
2
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3answers
1k views

Two bodies of finite size treated as two point masses in Newtonian gravity

When discussing gravitation between two bodies of finite size, for instance Earth around the Sun, we suppose the mass of Earth and the Sun to be perfectly localized at the center of each body. Is this ...
13
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5answers
1k views

Near Earth vs Newtonian gravitational potential

Newton's Law of Universal Gravitation tells us that the potential energy of object in a gravitational field is $$U ~=~ -\frac{GMm}{r}.\tag{1}$$ The experimentally verified near-Earth gravitational ...
13
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6answers
3k views

If the solar system is a non-inertial frame, why can Newton's Laws predict motion?

Since there is no object in the universe that doesn't move, and the solar system likely accelerates through space, how did Newton's Laws work so well? Didn't he assume that the sun is the acceleration-...
5
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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 ...
3
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2answers
2k views

Is gravitational potential energy proportional or inversely proportional to distance?

We know that if an object has been lifted a distance $h$ from the ground then it has a potential energy change: $$\Delta U = mgh $$ so $h$ is proportional to $\Delta U$. However, we have also the ...
6
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3answers
5k views

Is Torricelli's law “wrong” for big holes? - Tank draining problem

Consider a tank full of water with a constant cross-sectional area A1 placed vertically on the ground. Now someone drills a hole of an area A2 in the bottom of the tank, and the liquid starts escaping ...
3
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1answer
231 views

Why is it OK to use electrostatics in Bohr's Model if the electron is moving?

In the Bohr's Atomic model, we have assumed the centripetal force to be provided by the electrostatic force between the proton and electron and derived the radius, energy of orbit and the velocity of ...
15
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1answer
1k views

Could Navier-Stokes equation be derived directly from Boltzmann equation?

I know how to derive Navier-Stokes equations from Boltzmann equation in case where bulk and viscosity coefficients are set to zero. I need only multiply it on momentum and to integrate it over ...
3
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2answers
152 views

A problem of approximation [duplicate]

Possible Duplicate: Why are continuum fluid mechanics accurate when constituents are discrete objects of finite size? When we apply differentiation on charge being conducted with respect to time,...
5
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2answers
3k views

Small oscillations of the double pendulum

From the Lagrangian I've got the following equations of motion for the double pendulum in 2D. (The masses are different but the lengths of the two pendula are equal.) Let $m_2$ be the lowest-hanging ...
3
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1answer
1k views

WKB Quantization Condition - negative?

In deriving the quantization condition for a bound state in a potential with "no verticle walls" we start with the WKB connection formulas to find the wavefunction in the interior of the well ($x_1<...
-1
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3answers
7k views

Why is acceleration due to gravity a constant? [duplicate]

I just learned of Newton's law of gravitation and that distance between two bodies is a factor in the gravitational force. My question is if that's true why is the Earth's gravitational acceleration a ...
36
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3answers
20k views

How is the Saddle point approximation used in physics?

I am trying to understand the saddle point approximation and apply it to a problem I have but the treatments I have seen online are all very mathematical and are not giving me a good qualitative ...
19
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1answer
2k views

Is the existence of a sole particle in an hypothetical infinite empty space explicitly forbidden by QM?

Suppose the universe is completely empty with one sole particle trapped in it. To simplify, I will only be looking at the one dimensional case. However, all arguments are applicable for three ...
15
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5answers
630 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 ...
12
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3answers
980 views

Validity of mean-field approximation

In mean-field approximation we replace the interaction term of the Hamiltonian by a term, which is quadratic in creation and annihilation operators. For example, in the case of the BCS theory, where $...
2
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2answers
3k views

Massless string Paradox

If we introduce the notion of a massless string to denote the fact that net force on a massless string will always be $0$, since it is massless. How can these massless strings ever accelerate when ...
7
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1answer
331 views

Is drag force on an oscillating sphere an effective model for a swimmer?

I saw the latest video from Sixty Symbols Little Swimmers. At the end of the videos he says that we do not know how to calculate the movement of the little swimmers. He says(6:14-6:40 in video) that ...
5
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1answer
1k views

What is the range of validity of Fermi's Golden Rule?

It is well known that to calculate the probability of transition in the scattering processes, as a first approximation, we use the Fermi golden rule. This rule is obtained considering the initial ...
4
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2answers
380 views

Relation between perturbation theory and Taylor expansion in QM

So I am looking at non-degenerate perturbation theory. The idea is that the perturbing term in the Hamiltonian is small so you somehow expand the energies and wave functions in this small term and ...
3
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1answer
1k views

WKB approximation for multiple turning points

I'm working on a numerical program which approximates the eigenvalues of a Schrödinger equation by making use of the WKB approximation formulas. For example, if the Schrödinger equation is $$ y''(x) = ...
8
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4answers
959 views

Special Relativistic approximation to GR

Some time ago I was talking to a professor in college about some of the fundamental aspects and origin of General Relativity. I was surprised to learn, in fact, that a pretty good approximation to GR ...
4
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1answer
160 views

Coulomb's Law modified in general relativity?

It seems difficult to track down a clear explanation of this statement: So although the Coulomb law was discovered in a supporting frame, general relativity tells us that the field of such a charge ...
1
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1answer
2k views

How to apply the WKB approximation in this case?

I'm trying to learn how to apply the WKB approximation. Given the following problem: An electron, say, in the nuclear potential $$U(r)=\begin{cases} & -U_{0} \;\;\;\;\;\;\text{ if } r < ...
0
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3answers
1k views

Is my derivation of the potential energy formula $m*g*h$ correct?

I've just wondered where the formula $E_{pot} = mgh$ you learn at school comes from so I've tried to work it out - is my reasoning correct? The change in energy is given by $$\Delta E=\int_{e}^{e+h}G\...
10
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1answer
2k views

Why do physicists say that elementary particles are point particles?

For example, an electron, it has mass and charge, but is considered to have point mass and point charge, but why? Why are they assumed to have charge and mass in a single infinitely small point in ...
9
votes
1answer
600 views

How to justify RPA (random phase approximation)?

The Random Phase Approximation (RPA) is a technical method used in field theory to account for interactions when calculating correlation functions. It consists of only keeping a certain class of ...
3
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1answer
812 views

Can a very small portion of an ellipse be a parabola?

We can show that when a particle is projected from a certain height (from the surface of the Earth) with a speed lesser than the orbital speed and in a direction tangential to the surface of the Earth ...
1
vote
1answer
133 views

conservation of momentum? [closed]

At hyperphysics I got this image, with the same description in text as is in this image It says that when a massive particle (say $A$) moving with a velocity collides with an object having a ...
0
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2answers
226 views

How is the uniform gravitational field approximation $F_g\approx mg$ near Earth's surface derived from Newton's law $F_g=GMm/r^2$ of gravitation?

I am really bothered about how we can derive the equation of projectile motion. Suppose a point mass will move in the gravitational field of the Earth according to the equation $$\ddot R =-\frac{...
0
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1answer
2k views

When to use ideal gas law in fluid mechanics?

The ideal gas law (aka the equation of state) is given by $$ p/\rho_N = k_BT, $$ where $\rho_N$ is number density. When am I allowed to use this to describe a fluid?
5
votes
1answer
130 views

Approximate cloning of a qubit, given multiple starting copies

Suppose I'm given several clones of a qubit in a pure unentangled state. That is to say, I'm given the state $(a \left|0\right\rangle + b \left|1\right\rangle)^{\otimes n}$. My goal is to make $d$ ...
3
votes
1answer
330 views

Evaluating low-temperature dependence of the BCS gap function

How does one go about evaluating the behavior of the BCS gap $ \Delta = \Delta(T) $ for $ T \to 0^+ $ under the weak coupling approximation $ \Delta/\hbar\omega_D \ll 1 $? In Fetter & Walecka, ...
0
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2answers
2k views

Significant Digits for different units [closed]

I read, From (iv), 12.3 has three significant figures. And from (v) we can infer that 12.30 has four significant figures. So let's say it's meters, then 12.30 m = 1230 cm = 12300 mm But 1230 and ...
0
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1answer
148 views

About linearisation of equations of motion rigid body

I am modeling a system of solid bodies. Consider $\theta \approx 0$ and $\chi \approx 0$. At a certain moment I get the following formula for the angular velocity: $$ \omega = \begin{bmatrix} 0 &-...
0
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2answers
74 views

Conservative $E$-field and Kirchoff rule in practice

In undergrad physics, when analyzing an LR circuit, it is often considered that Kirchoff rule holds. However, as far as I understand, Kirchoff rule only holds when E field is conservative (curl of E ...
0
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2answers
166 views

Is the geodesic equation independent of an initial condition?

The following argument is used to determine the unknown factors (e.g., $A(r)$ and $B(r)$) in the Schwarzschild metric. $$ \lim_{r \to ∞}A(r) = \lim_{r \to ∞}B(r) = 1 \space\space\space\space\space\...