Length Definition and 1000 Threads

Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the International System of Units (SI) system the base unit for length is the metre.
Length is commonly understood to mean the most extended dimension of a fixed object. However, this is not always the case and may depend on the position the object is in.
Various terms for the length of a fixed object are used, and these include height, which is vertical length or vertical extent, and width, breadth or depth. Height is used when there is a base from which vertical measurements can be taken. Width or breadth usually refer to a shorter dimension when length is the longest one. Depth is used for the third dimension of a three dimensional object.Length is the measure of one spatial dimension, whereas area is a measure of two dimensions (length squared) and volume is a measure of three dimensions (length cubed).

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  1. A

    Are length contraction and time dilation conventions?

    Since many authors call simultaneity between events a convention, and that under the specific set of rules we may choose a convention or a coordinate system relative to an IRF (or non-inertial) to describe space time, I wonder what's the relation between this and the effects of length...
  2. E

    Length of Pendulum with Variables Only

    Homework Statement A grandfather clock has a pendulum that consists of a thin brass disk of radius r and mass m that is attached to a long thin rod of negligible mass. The pendulum swings freely about an axis perpendicular to the rod and through the end of the rod opposite the disk, as shown in...
  3. E

    Can a dot product be negative in case of length?

    Let's say A and B are 2 vectors with length in cm and the angle between them is 170°. Obviously, the dot product of A and B will give cm2 as unit but since the value of cos(170) is negative, will the dot product be negative (something)cm2?
  4. F

    Path length difference and Diffraction

    Homework Statement A double slit experiment is set up using a helium-neon laser (wavelength 633 nm). Suppose we add a small piece of glass (n = 1.50) over one of the slits. Then, the central point on the screen is occupied by what had been the m = 10 dark fringe. Determine the thickness t of...
  5. R

    Plotting the graph of pendulum period versus length

    why do we square the value of T ( time period) while plotting the graph of effect on time period of a pendulum with change in its effective length ? also while deriving the formula of T=2∏√(l/g) why do we take T^2 = 1/g. Whats the need for squaring the time period ?
  6. N

    Difficult simplification for Arc length integral

    Homework Statement Find the length of the curve x = 3 y^{4/3}-\frac{3}{32}y^{2/3}, \quad -64\le y\le 64Homework Equations Integral for arc length (L): L = \int_a^b \sqrt{1 + (\frac{dy}{dx})^{2}} dx The Attempt at a Solution Using symmetry of the interval and the above integral for arc length...
  7. S

    How Does Length Contraction Affect Markings on a Moving Conveyor Belt?

    Homework Statement A relativistic conveyor belt is moving at speed 0.5c relative to frame S.Two observers standing beside the belt, 10 ft apart as measured in S, arrange that each will paint a mark on the belt at exactly the same instant (as measured in S). How far apart will the marks...
  8. PsychonautQQ

    Finite well penetration depths dependence on Well Length

    So in the infinite well Energy is proportional to 1/L^2, so I'm assuming in the finite well there is some sort of similar relation. So as the L decreases, the energy increases, so the wavelength decreases. Decreasing the wavelength means more energy, so it should penetrate further, but also if...
  9. A

    Adjusting length and period using (inverse) transformations

    Trying to see the logic in deriving length contraction and time dilation using the Lorentz transformations and inverse Lorentz transformations. In the following treatise it leads to ambiguities. Given ##Δ\acute{t}=\gamma(Δt-\beta c^{-1}Δx)## (1) ##Δ\acute{x}=\gamma(Δx-\beta c Δt)##...
  10. E

    Solving Arc Length Integral with Trigonometric Substitution

    ∫sqrt(x^4/4 + 1/(x^4) + 1/2) dx from x = 1 to 4 Could someone help me solve this? I can't seem to find a substitution that works, or find the square root of (x^4/4 + 1/(x^4). Any help would be very appreciated. Thanks in advance!
  11. X

    Optics: Focal length of Koenig eyepiece

    Homework Statement A Koenig eyepiece with a focal length of 100mm is constructed in the following way: The first lens is made of 755276 glass which is 10 mm thick on axis. The radius of curvature of its left hand surface is -225mm, the radius of curvature of it right hand surface is 83.6...
  12. U

    Spherical coordinates length from differential length

    is it logical to ask this question in Spherical coordinates: Using the differential length dl , find the length where r=1 0<Θ<∏/4 ∏/2< θ <∏/4 where Θ is the azimuthal angle. What I mean by ∏/2< θ <∏/4 is that the line is a "diagonal" line which has an ascention of ∏/4 from the xy...
  13. S

    MHB Is Finding the Arc Length of a Curve the Same as Finding the Length?

    This may seem like a dumb question, but is finding the "arc length" of a curve and finding the "length" of a curve the same thing? Just worded differently?
  14. A

    Time dilation, length contraction and relative simultaneity on Earth

    The three main effects of SR occur in inertial frames and change the description of space-time relative to a particular observer. My question here is how do these effects occur on Earth, since we know that motion on Earth is non-inertial. I know that we travel at small speeds and that we can't...
  15. V

    Length of cooling including latent heat

    Hello, I would like to know how to calculate how long the phase transition when cooling lasts. So basically how long the latent transition lasts. I really don't have any ideas how this could be calculated, although I would assume it is function of the mass. Any help is much appreciated.
  16. A

    Arc Length Units: Explained & Solved Problem

    Hello, I solved the arc length for a particular problem. However, what is the unit of arc length if the units of the velocity vs time graph are m/s vs s? I am really confused.
  17. S

    Exploring the Smallest Measurable Units: Planck Length & Pi

    Hi all. This is my first time posting so forgive me If I am doing something wrong. I am a year 7 student interested in all types of physics and my question is, if nothing can be smaller than Planck length then wouldn't past a certain point the digits of pi become obsolete? Simply because the...
  18. L

    Analyzing Two Converging Lenses of the Same Focal Length

    Homework Statement Two converging lenses of the same focal length f are separated by distance 2f. The axis of the second lens is inclined at angle θ = 60º with respect to the axis of the first lens. A parallel paraxial beam of light is incident from left side of the lens. Then: (A)...
  19. J

    Using a length() function in a loop's condition

    I see on StackOverflow people doing stuff like for (int k = 0; k < something.length(); k++) where the body of the for loop never has any chance of modify something's length. I thought that was considered bad coding practice. The implementation of something's class might not return a private...
  20. MarkFL

    MHB Minimize Total Length of Cables: Estimate Minimum Value?

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  21. Y

    Why Does the Magnitude of r'(s) Equal 1 When s is the Arc Length Parameter?

    1. The problem statement, all variables and given/known If C is a smooth curve given by r(s)= x(s)i + y(s)j + z(s)k Where s is the arc length parameter. Then ||r'(s)|| = 1. My professor has stated that this is true for all cases the magnitude of r'(s) will always equal 1. But he wants me...
  22. marcus

    Length and curvature operators in Loop gravity

    Most of us are familiar with the fact that in Loop gravity the area and volume observables have discrete spectrum. The discrete spectrum of the area operator, leading to a smallest positive measurable area, has lots of mathematical consequences that have been derived in the theory. It helps...
  23. K

    Length of wire wrapped around a solenoid?

    Homework Statement Hi everyone, the problem I'm stuck on reads: " Imagine a solenoid 3m in diameter, 5m long having 1164 turns of super conducting cable. What length of super conducting cable is used to wind this solenoid?"Homework Equations Circumference = 2pi*radiusThe Attempt at a Solution...
  24. A

    Is Virtual Length Contraction Influenced by Changes in Refractive Index?

    What should it be, {sin(ωt-\frac{k}{n}x)} or {sin(ωt-nkx)}? I am contemplating this with respect to the proper waveform in a medium, following specific rules, particularly when it comes to the dispersion relation and ultimately what this would mean in the context of the Lorentz transformations...
  25. S

    MHB Finding the exact length of the curve (II)

    Find the exact length of the curve $0 \le x \le 1$ y = 1 + 6x^{\frac{3}{2}} <-- If you can't read this, the exponent is \frac{3}{2} \therefore y' = 9\sqrt{x} \int ^1_0 \sqrt{1 + (9\sqrt{x})^2} \, dx = \int ^1_0 \sqrt{1 + 81x} \, dx = \int^1_0 1 + 9\sqrt{x} \, dx Now can't I just...
  26. S

    MHB Find the exact length of the curve

    A little bit confused. Find the exact length of the curve y = \frac{1}{4}x^2 - \frac{1}{2}\ln x 1 \le x \le 2 Using the formula: y = \sqrt{1 + (\frac{dy}{dx})^2} \, dx I obtained this: \int ^2_1 \sqrt{ \frac{1}{2} + \frac{x^2}{4} + \frac{1}{4x^2}} Now my problem is I'm stuck. If I...
  27. E

    Mean ray length from apex to base of an oblique circular cone

    Consider an oblique circular cone of altitude h, base radius R, with apex directly above a point on the base circumference. What is the mean length (& variance) for the set of all rays from the apex to points on or within the base circumference?
  28. M

    Energy of a pendulum (variable length, Lyapunov)

    Hello, question about the energy of a variable length pendulum. Suppose you have a pendulum in the standard sense where θ is the angle, and we let the length r be a function of time r = r(t). What is the energy of the pendulum? So far, I have determined that kinetic energy is =...
  29. T

    A closed organ pipe has a length of 2.40 m?

    A closed organ pipe has a length of 2.40 m. a.) What is the frequency of the note played by the pipe? Use 343 m/s as the speed of sound. b.) When a second pipe is played at the same time, a 1.40 Hz beat note is heard. By how much is the second pipe too long? The a.) problem's solution is...
  30. S

    Confusion about time dilation and length contraction

    Hi all, I tried searching for this but failed to find an answer to my question. I am having an issue with properly interpreting the equations for time dilation and length contraction. Let's assume that I am standing still and a train is passing by next to me (moving with uniform velocity). Let...
  31. B

    Calculating String Tension: Mass, Length, and Frequency

    Homework Statement What is the tension of the string? Mass of piano string- 3.5g Lenght of piano string- 75cm Fundamental frecuency- 469Hz Homework Equations λ=2L/n ? Dont know where to start.
  32. L

    Superconductor coherence length and penetration depth

    Homework Statement I have a lot of information about 2 different superconductor materials; indium and lead. The indium is pretty much 100% indium with no impurities, and the lead is unknown purity. I have the temperatures and magnetic fields at which they are superconducting and the...
  33. J

    Varying thermal conductivity with length

    I'm interested in modeling a system where the material varies along its length, thus the conductivity coefficient would be a function of both T, and x. k(T,x). For starters, if I assume negligible change w.r.t T, then he heat diffusion equation would be d/dt(k(x)dT/dx)=0. Correct? What if k just...
  34. S

    Finding arc length using integration

    Find the length of the positive arc of the curve y=cosh^{-1}(x) (for which y≥0) between x=1 and x=\sqrt{5}. My attempt: x=cosh(y) → \frac{dx}{dy} = sinh(y) → (\frac{dx}{dy})^{2}=sinh^{2}(y), so ds=dy\sqrt{1+sinh^{2}(y)}, therefore the arc length is S=\int_{y=0}^{y=cosh^{-1}(\sqrt{5})} cosh(y)...
  35. O

    Derive length contraction formula given a two system experiment.

    Homework Statement Show that the experiment depicted in Figure 2.11 and discussed in the text leads directly to the derivation of length contraction. Figure 2.11: Homework Equations d=v*t Requested result: L=L0\sqrt{1-\frac{v^2}{c^2}} The Attempt at a Solution In K the...
  36. T

    How do you calculate the length of this piece?

    http://chestofbooks.com/home-improvement/woodworking/Handicraft-For-Boys/images/Making-the-Model-Engine-292.png I want to know how you would determine how long the rocker arm in this picture is...
  37. U

    Diffraction Grating, focal length of lens

    Homework Statement Light from infinity strikes a diffracting grating normally with 600 lines per mm. The emitted light then passes through a lens to project visible light (400-800 nm) spectrum onto a screen that is 50mm, just enough to cover it. Find the focal length of the lens. Homework...
  38. R

    MHB Find arc length given chord, radius

    The solution to this question (whose answer is pi) is eluding me: The radius of a circle is 3 feet. Find the approximate length of an arc of this circle, if the length of the chord of the arc is 3 feet also.
  39. A

    Relationship between frequency and length of pendulum

    Homework Statement We calculated times of the periods of varying pendulum lengths. (20cm, 40cm, 60cm, 80cm). Then the frequency was calculated for each length and then a frequency-length graph was made. Since the graph is an exponential relationship we graphed our values on a log-log chart...
  40. M

    Time Dilation and Length Contraction

    I had a quick question about Time Dilation and Length Contraction. Are the two just different ways of measuring/describing the same effect? Or rather they both follow as a consequence from one another? i.e. I can find how much a length is contracted by finding the dilated time interval and...
  41. A

    MHB Integration question (obtained from arc length question)

    how do i integrate the function sqrt(1 + 1/2(y^1/2 - y^(-1/2))^2) from 0 to 1??
  42. D

    Calculating Wave Speed and Wavelength for a Vibrating String

    Homework Statement A long string with a mass/length of 500g/m is placed under a tension of 400N. The string is then vibrated up and down with a period of .425sec. What is the wave speed? What is the wavelength of the resulting wave? I have no idea where to beginI would really...
  43. Saitama

    Shortening the length of pendulum

    Homework Statement Homework Equations The Attempt at a Solution I tried with the conservation laws. Angular momentum conservation won't work. To use energy conservation, I need the force pulling up the string but I don't have it. The force in the given case is tension but the tension...
  44. X

    Length of coaxial cable, based on signal reflection

    Homework Statement We measured the time between a signal source, and it's reflection coming back through our probe after going through an open-ended coaxial cable. My teacher told us this: the cable has a polyethylene insulator between central wire and the grounding web, which has a...
  45. C

    The product of a vector and the length of a polar coordinate

    Homework Statement So I am not sure how to multiply these two (A*R^2) together. Homework Equations A=( x^2 + y^2 + z^2 ) (xe + y e + z e ) Where x represents the three vector compones I also have R^2=x^2+y^2+z^2 The Attempt at a Solution Is the product of A (x^3e + y^3 e + z^3...
  46. S

    MHB Finding Mode for Grouped Frequency Distribution with Unequal Class Length

    How to find mode for grouped frequency distribution with unequal class length? I have to find the mode for the following problem: Marks No of students 0-20 32 20-50 45 50-70 15 70-100 8For equal class length, we use the formula Mode=...
  47. M

    Random sequence - full alphabet run length

    Hi, Suppose we're looking at a random sequence of digits from 0 to 9. We start off reading the digits until every digit from 0 to 9 has been seen at least once and we mark the count of digits read up to that point (run length). We then reset the run length and continue until the whole random...
  48. maajdl

    Why no electric field from a current in a wire? Length contraction.

    I found a funny video on youtube, but I am not totally convinced by the argument. It says that a pure magnetic field caused by a current in a wire can lead to a combined electric + magnetic field in a moving frame. It explains more or less convincingly that this can be understood as a...
  49. T

    Do the distance between two objects contract in length contraction

    Imagine two asteroids which are separed by 1Ly distance. They are in uniform velocity (.9c) with respect to an observe in space. So from the observers point of view (rest frame), does the distance between the asteroids (ie. 1Ly) appear contracted?
  50. adjacent

    How can we create an image at infinity using a converging lens?

    https://dl.dropboxusercontent.com/u/260388836/index.html Let arrow height be 1cm. focal length:1cm So if I move the arrow to the focal point,image is not formed.(i.e formed at infinity) But, Lets move it to say 0.999 Now the image is virtual and is magnified to about 1000X If I make it...
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