Expansion Definition and 1000 Threads

I'm a bit frustrated at the moment, as this minor problem should be fairly easy. But I seem to go wrong at some point... So I've got to do a 1st order expansion of the function \begin{equation} f=\frac{\cos(\theta)}{\sin(\theta)}\ln(\frac{L\sin(\theta)}{d\cos( \theta)}+1) \end{equation}...

A glass bottle is filled with salted water, and a pipette is inserted in the top through the cork. ( Leaving the bottle sealed). A thermometer is also inserted to keep record of the temperature. The glass bottle is then inserted in ice until the water reaches Zero Celsius. Finaly, the...

Homework Statement A standard mercury thermometer consists of a hollow glass cylinder, the stem, attached to a bulb filled with mercury. As the temperature of the thermometer changes, the mercury expands (or contracts) and the height of the mercury column in the stem changes. Marks are made on...

Hello, I started this account to ask you about that, if the place of big bang was in the middle and we are on its right side, we can't observe that what was on the left side? My train of thought: << (Left side, object) <<<(Big Bang)>>> (Right side, we) >>

Hi all, I'm reading Kinney's lectures on inflation: http://arxiv.org/abs/0902.1529 and got stuck trying to show that for some comoving length scale \lambda, the quantity \left(\frac{\lambda}{d_h}\right)^2 |\Omega -1| is conserved, if w is constant in the equation of state. Here d_h is...

Ideal gas behavior of super heated steam Why is super heated steam said to exhibit ideal gas behavior?

How far is it correct to ask, "What is universe expanding into?". How to solve this paradox?

Inflation is often referred to as a period of 'superluminal' or 'faster-than-light' expansion (e.g. see article on Wikipedia and hundreds of research papers on the subject). This has always bugged me. What exactly is superluminal about an inflating universe that does not apply to a non-inflating...

exp (hv/kT) - 1 For hv<<kT exp (hv/kT) - 1 is approximately equal to hv/kT thanks for any ideas.

Consider the integral \begin{equation} I(x)= \frac{1}{\pi} \int^{\pi}_{0} sin(xsint) dt \end{equation} show that \begin{equation} I(x)= 4+ \frac{2x}{\pi}x +O(x^{3}) \end{equation} as x\rightarrow0. => I Have used the expansion of McLaurin series of I(x) but did not work. please help...

Consider the integral \begin{equation} I(x)= \frac{1}{\pi} \int^{\pi}_{0} sin(xsint) dt \end{equation} show that \begin{equation} I(x)= \frac{2x}{\pi} +O(x^{3}) \end{equation} as $x\rightarrow0$. => I Have used the expansion of McLaurin series of $I(x)$ but did not work. please help me.

Homework Statement A circular steel plate of radius 15 cm is cooled from 350 C to 20 C. By what percentage does the plates area decorate ? Homework Equations A=∏r^2 Af = Ai (1+2∂ΔT) specific heat of steel = 12 x 10^-6 The Attempt at a Solution r = 15 cm = .15 m Ai = .070685 m^2...

Homework Statement Find the partial fractions expansion in the following form, G(s) = \frac{1}{(s+1)(s^{2}+4)} = \frac{A}{s+1} + \frac{B}{s+j2} + \frac{B^{*}}{s-j2} Homework Equations The Attempt at a Solution I expanded things out and found the following, 1 = A(s^{2} + 4)...

Homework Statement Part(a): Show dL/dT can be expressed as: Part(b): Show L = L0 + ΔCT for an indeal gas Part(c): Show the following condition holds for an adiabatic expansion, when some liquid condenses out. Homework Equations The Attempt at a Solution Finished parts (a)...

Homework Statement Problem: Find the first eight coefficients (i.e. a_0, a_1, a_2, ..., a_7) of the power series expansion y = ##Σ_{n = 0}^{∞}## [##a_n## ##x^n##] of the solution to the differential equation y'' + xy' + y = 0 subject to the initial-value conditions y(0) = 0, y'(0)...

Homework Statement 2.1E5 J of heat enters an ideal gas as it expands at a constant T = 77°C to four times its initial volume. How many moles of gas are there? T=350K, Q=2.1E5 J, Vi=x, Vf=4x Homework Equations ΔU=Q-W W=\intpdV U=nCvT The Attempt at a Solution I'm not sure if I'm even on the...

Homework Statement Taken from 'Concepts in Thermal Physics': Homework Equations The Attempt at a Solution For the step highlighted in red, why does the '-1' go into the integrand?

Homework Statement Taken from Concepts in thermal Physics: Homework Equations The Attempt at a Solution Shouldn't the van der waal's equation be: p = \frac{RT}{V_m -b} - \frac{a}{V_m^2} pV_m = \frac{VRT}{V_m -b} - \frac{a}{V_m}

Homework Statement f = \frac{1}{z(z-1)(z-2)} Homework Equations Partial fraction The Attempt at a Solution R1 = 0 < z < 1 R2 = 1 < z < 2 R3 = z > 2 f = \frac{1}{z(z-1)(z-2)} = \frac{1}{z} * (\frac{A}{z-1} + \frac{B}{z-2}) Where A = -1 , B = 1. f = \frac{1}{z} *...

Problem statement: The Bragg angles of a certain reflection from copper is 47.75◦ at 20◦C but is 46.60◦ at 1000◦C. What is the coefficient of linear expansion of copper? (Note: the Bragg angle θ is half of the measured diffraction (deflection) angle 2θ). Attempt at solution: Using...

I'm confused by problem 2.31 in mathematical tools for physics. Problem: 2.31 The Doppler effect for sound with a moving source and for a moving observer have different formulas. The Doppler effect for light, including relativistic effects is different still. Show that for low speeds they are...

Homework Statement expand f(x) = x^4 - 3x^3 + 9x^2 +22x +6 in powers of (x-2) Hence evaluate integral, (limits 2.2 - 2) f(x) dx Homework Equations Taylor expansion for the first part integral f(x) dx with limits 2.2-2 The Attempt at a Solution Expansion of the function...

https://www.physicsforums.com/attachments/68247 I had been assigned this problem, I worked out the expansions (for practice) so they could have errors in them! I got to a point (in the photograph) where I could take out a common factor of 1/x but I'm pretty stumped although via other methods...

It is said that in an isobaric expansion of a gas pressure remains constant throughout the expansion process. Suppose we have a quantity of gas at initial pressure P1 and volume V2 in a piston cylinder arrangement.We heat it slowly such that it expands to obtain a state with pressure P2 and...

Hi There, I came across the following passage in Sam Glasstone's 'Nuclear Reactor Engineering' See where I underlined in red that taylor series expansion? I don't understand how (dt/dx)_(x+dx) is equal to that. I know it's a Taylor series expansion, but where did the x+dx go?

NOTE: I am not a cosmologist, so if any of my statements are not correct please tell me. When we observe distance galaxies we can measure how fast they move away using the red-shifting of their light. So how do we know space itself is expanding vs the galaxies are just moving away relative to...

Homework Statement Actually this is not from homework. It occurs in my brain this afternoon. Is it possible to derive the analytic expression of a function by its Taylor series expansion? For example, given the following expansion, how to derive the analytic expression of it? f(x) =...

Wasn't exactly sure where to post this. Wanted to see if I did this correctly.Can someone check my work please? Problem: Consider f defined below. Apply Shannon's expansion theorem (also given below) with respect to input y as if you were implementing this function using a 2:1 MUX. Find the...

Hello. Apparently for SU(N) gauge theories there is a perturvative approach call "large N expansion". I need a working knowledge of this method but I can not find it in any textbook only in research papers that are rather for the expert. Someone has any suggestion where should I get started...

Hello, my colleague and I are having a disagreement about the amount of thermal expansion in a particular part we are working on (manufacturing environment). It is a piece of structural steel, which has a thermal expansion value of 12 (10^-6/K according to the chart at...

In this section, they derive the Sommerfeld formula. In the first step it seems like they have expanded ##\frac{1}{(1+e^x))^2}##. I'm not sure why does the series taylor expand as ##e^{-nx}##? Also how did they get from the 2nd to the 3rd step? Simply by comparing terms we see they are...

Any boolean function on n variables can be thought of as a function f : \mathbb{Z}_2^n \rightarrow \mathbb{Z}_2 which can be written as f(x) = \sum_{s \in \mathbb{Z}_2^n} \hat{f}(s) \prod_{i : x_i = 1} (-1)^{x_i} where \hat{f}(s) = \mathbb{E}_t \left[ f(t) \prod_{i : s_i = 1}...

Homework Statement Find the n+1 and n-1 order expansion of \stackrel{df}{dy}Homework Equations (n+1)Pn+1 + nPn-1 = (2n+1)xPn ƒn = \sum CnPn(x) Cn = \int f(x)*Pn(u)The Attempt at a Solution I know you can use the recursion relation for Legendre Polynomials once you combine Cn with the...

Are there any theories or thoughts that view spacetime as 'having' a coefficient of thermal expansion... analogous to the CTE of water? An inflection with density in regards to temperature?

Show that for small positive $x$, $$\left( \sin x \right)^{\cos x} = x -\left( 3 \log x + 1\right) \frac{x^{3}}{3!} + \Big( 15 \log^{2} x + 15 \log x + 11 \Big) \frac{x^{5}}{5!} + \mathcal{O}(x^{7})$$

I have the following expression: $$\frac{1}{p} \ln\left(1+\frac{p^1}{1!n}\sum_{i=1}^n x_i + \frac{p^2}{2!n} \sum_{i=1}^n x_i^2 + \frac{p^3}{3!n} \sum_{i=1}^n x_i^3 + \frac{p^4}{4!n} \sum_{i=1}^n x_i^4 + \cdots \right)$$ Now let $$Y = \frac{p^1}{1!n}\sum_{i=1}^n x_i + \frac{p^2}{2!n}...

I was reading an article about the recent type Ia supernova in the big dipper when the question popped into my head that I've forgotten about until now. Since we use the Ia supernova as our candles to measure the expansion of the universe, how do we know that everything is accelerating...

in order to explain the big bang theory and the expansion of space itself, people often draw upon the analogy of blowing air into a balloon and the 2 dimensional surface of the balloon expanding. isn't the 2-D balloon surface expanding in the third dimension, since the volume of the balloon is...

Consider the following charge distribution:A positive charge of magnitude Q is at the origin and there is a charge -Q on each of the x,y and z axes a distance d from the origin. I want to expand the potential of this charge distribution using spherical coordinates.Here's how I did it...

Is there a simple way to series expand a function of the form $$ \frac{1}{\sum_{n=0}^{\infty} a_n x^n} $$ about the point ##x=0##, such that it can be expressed as another sum ##\sum_n c_n x^n##? I tried doing it by taylor expansion but I end up with a sum of sums of products of sums :)...

Hey fellas please consider helping me with this.. There is a bar of some material that is heated from state 1 to 2 to 3. If l1 is the length at 1 then we have, l2 - l1 = l1(1 + a(t2-t1)) l3 - l1 = l1(1 + a(t3-t1)) If, t3-t2 = t2-t1, Then l3-l2 = l2-l1 But if it is written l3 - l2 =...

I want to create a mechanical vacuum (semi not necessarily complete), I've heard the easiest way to create one is by expansion (basically expanded a pump). But is this also the easiest way physically, as in takes the least amount of energy? Or is there any other 'mechanical' way possible for...

You probably know that for two commutative quantities x and y,we have: (x+y)^n=\sum_{r=0}^n \left( \begin{array}{c} n \\ r \end{array} \right) x^{n-r} y^r Now I want to know is there a similar formula for the case when x and y don't commute and we have [x,y]=c and [x,c]=[y,c]=0 ? Thanks

Hello, this is probably one of those shoot yourself in the foot type questions. I am going through Landau & Lifshits CM for fun. On page 7 I do not understand this step: L' = L(v'^2) = L(v^2 + 2 \vec{v} \cdot \vec{\epsilon} + \epsilon^2) where v' = v + \epsilon . He then expands the...

Homework Statement 1 mole of an ideal gas initially at 100° C and 10 atm is expanded adiabatically against a constant pressure of 5 atm until equilibrium is re-established. Given that the temperature dependence of the heat capacity is CV = 18.83 + 0.0209T calculate deltaU, deltaH and deltaS...

Homework Statement You are building a device for monitoring ultracold environments. Because the device will be used in environments where its temperature will change by 211°C in 2.99s, it must have the ability to withstand thermal shock (rapid temperature changes). The volume of the device is...

Homework Statement A clock based on a simple pendulum is situated outdoors in Anchorage, Alaska. The pendulum consists of a mass of 1.00kg that is hanging from a thin brass rod that is 2.000m long. The clock is calibrated perfectly during a summer day with an average temperature of 19.5°C...

Homework Statement Use Maclaurin’s theorem to derive the first five terms of the series expansion for ##(1+x)^{r}##, where -1<x<1. Assuming the series, obtained above, continues with the same pattern, sum the following infinite series ##1 + \frac{1}{6} - \frac{(1)(2)}{(6)(12)} +...

Homework Statement When the temperature of liquid mercury increases by one degree Celsius (or one kelvin), its volume increases by one part in 550,000. The fractional increase in volume per unit change in temperature (when the pressure is held fixed) is called the thermal expansion coefficient...

Homework Statement Given that ##f(x)=(1+x) ln (1+x)##. (a) Find the fifth derivative of f(x), (b) Hence, show that the series expansion of f(x) is given by ##x+\frac{x^{2}}{2} -\frac{x^{3}}{6} + \frac{x^{4}}{12} - \frac{x^{5}}{20}## (c) Find, in terms of r, an expression for the rth term...