Expansion Definition and 1000 Threads

  1. K

    What is the 1st Order Expansion Problem for a Function with a Logarithm?

    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}...
  2. M

    Does Material Expansion Impact Accuracy in Thermal Expansion Experiments?

    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...
  3. N

    Volume Expansion: Answers to Homework Questions

    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...
  4. I

    Question: Expansion of the universe

    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) >>
  5. M

    Conserved charge in FRW expansion

    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...
  6. Soumalya

    Law of expansion for super heated steam

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

    Can we ask such a question about expansion of universe?

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

    Inflation and superluminal expansion

    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...
  9. R

    Which expansion is used for this result?

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

    How Does Integration by Expansion Work for Sine Integrals?

    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...
  11. R

    MHB Is McLaurin Expansion the Key to Solving Integration by Expansion?

    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.
  12. A

    Thermal Expansion of a circular steel plate

    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...
  13. jegues

    How to Solve Partial Fractions Expansion?

    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)...
  14. U

    Liquid-Vapour Interface: Adiabatic Expansion

    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)...
  15. S

    Find the first eight coefficients of the power series expansion.

    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)...
  16. C

    Thermodynamics - Ideal gas expansion

    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...
  17. U

    Virial Expansion, Van Der Waals Gas

    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?
  18. U

    Van der Waals Equation, Virial Expansion

    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}
  19. T

    Partial Fractions in Laurent Series Expansion

    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} *...
  20. I

    What Is the Coefficient of Linear Expansion of Copper Based on Bragg Angles?

    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...
  21. T

    You can edit the title of the first post and add [solved] at the beginning.

    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...
  22. C

    Integrating a Taylor Expansion with Limits: Finding the Exact Value

    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...
  23. KiNGGeexD

    Using Taylor expansion to find the limits of a 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...
  24. Soumalya

    Constant pressure or isobaric expansion of gas

    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...
  25. T

    One Dimensional Slab Heat Transfer Taylor Expansion in Glasstone

    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?
  26. I

    Expansion of space vs stuff just moving away

    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...
  27. K

    Derive the analytic expression of a function by its Taylor expansion

    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) =...
  28. S

    MHB Applying Shannons Expansion Theorem

    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...
  29. M

    Large N Expansion: Getting Started

    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...
  30. A

    Manufacturing question for thermal expansion disagreement

    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...
  31. U

    Why does the series Taylor expand as e^-nx?

    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...
  32. D

    Fourier expansion of boolean functions

    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}...
  33. A

    How do I find the n+1 and n-1 order expansion of a Legendre series?

    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...
  34. R

    A thermal expansion coefficient of spacetime?

    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?
  35. polygamma

    MHB Series Expansion: Show Sin^Cos x = x + O(x^3)

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

    MHB Manipulating Taylor Expansion for Sample Mean, Variance, Skewness & Kurtosis

    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}...
  37. ViperSRT3g

    How Do We Know the Universe's Expansion Is Accelerating?

    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...
  38. P

    The expansion of the universe, evidence of a 4th spatial dimension?

    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...
  39. ShayanJ

    Point charges and multipole expansion

    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...
  40. Hepth

    Series Expansion of 1/Polynomial

    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 :)...
  41. K

    Linear thermal expansion problem

    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 =...
  42. M

    Is Expansion the Most Efficient Method for Creating a Mechanical Vacuum?

    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...
  43. ShayanJ

    Newton's expansion for non-commutative quantities

    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
  44. B

    Lagrangian for a free particle expansion problem

    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...
  45. M

    Adiabatic expansion with temperature dependent heat capacity.

    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...
  46. C

    What is the maximum volume expansion coefficient of ?

    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...
  47. C

    How Does Temperature Affect a Pendulum Clock in Alaska?

    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...
  48. S

    What is the Relationship Between Maclaurin Series and Infinite Series?

    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)} +...
  49. B

    Thermal Physics Expansion Coefficient Problem

    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...
  50. S

    Finding the Maclaurin Series Expansion of (1+x)ln(1+x)

    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...
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