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. J

    Sign of differential length in work

    Homework Statement [/B] Calculate the work done by a force, against an electric field, to bring a charged particle (2 coulomb) from the point (2,0,0) to (0,0,0). Also calculate the work from (0,0,0) to (0,2,0). Finally calculate the work done going directly from (2,0,0) to (0,2,0) and...
  2. B

    I Ave length of intersection of all lines through a unit cube

    I'm having trouble even beginning to figure out how to approach solutions for this. I begin with a unit cube, and imagine all the possible lines that intersect the cube. I am assuming there must be an average length of these intersections; I want to find that average length. Another way to...
  3. T

    Sound & Music - Mass per unit length

    Homework Statement Suppose a harp string is tuned to middle C (C4) and is 0.56m long. If I want the tension to be 193.5 N, what mass per unit length do I need the harp string to be? Calculate your answer in kg.m Homework Equations Vs = sqrt(T/μ) V = fλ The Attempt at a Solution Having a very...
  4. M

    Prove the shortest distance between two points is a line

    Homework Statement Let γ : [0, L] → Rn be arclength parametrized. Show that the distance between the endpoints of the curve can at most be L, and equality can only hold when γ is a straight line segment. Thus, the shortest path between two points is the straight line segment connecting them...
  5. S

    Finding length of an arc produced by projectile

    So I decided to try deriving a general formula for fun. Being a high school student, the calculus got scary very fast. At this point, I'm just curious as to what the best approach to this might be. The approach I used was finding y as a function of x and then inputting it into the arc length...
  6. Sharkey4123

    I Does the Debye Length Change with Applied Potential Strength in a Plasma?

    The debye length is the effective length over which electrostatic potential disturbances are "screened out" in a plasma. So if I drop a charged point particle in a plasma, then I expect after some Debye Length, D, from this point charge, I can see no difference between any other point in the...
  7. R

    Strength of springs - length of travel vs tension weight

    I'm trying to design some novel cross bow designs and also trebuchet's with the trebuchet (similar to catapults to those who don't know) using springs or a combination of weight and springs. I'm looking into using linear springs and pullies for the crossbow depending on whether they can be...
  8. M

    A Velociraptor is pursuing you....

    Homework Statement [/B] This is a problem from my Differential Geometry course A velociraptor is spotting you and goes after you. There is a shelter in the direction perpendicular to the line between you and the raptor when he spots you. So you run in the direction of the shelter at a...
  9. Pouyan

    Distance and Time for a Moving Muon

    Homework Statement A muon is created in the atmosphere 3 km above Earth's surface, heading downward at speed 0.98c. It survives 2.2 * 10-6s in its own frame of reference before decaying. Relativistitically, according to the muon, what is the distance from the point in the atmosphere where the...
  10. M

    Invariance of length of curve under Euclidean Motion

    Homework Statement Show that the length of a curve γ in ℝn is invariant under euclidean motions. I.e., show that L[Aγ] = L[γ] for Ax = Rx + a Homework Equations The length of a curve is given by the arc-length formula: s(t) = ∫γ'(t)dt from t0 to tThe Attempt at a Solution I would imagine I...
  11. M

    Verify Unit length to y-axis from Tractrix Curve

    Homework Statement The problem is described in the picture I've attached. It is problem number 6. Homework Equations Tangent line of a curve Length of a curve The Attempt at a Solution I don't know why I'm so confused on what seems like it should be a relatively straightforward problem, but I...
  12. M

    Proving length of Polygon = length of smooth curve

    Homework Statement The problem statement is in the attached picture file and this thread will focus on question 7 Homework Equations The length of a curve formula given in the problem statement Take a polygon in R^n as an n-tuple of vectors (a0,...,ak) where we imagine the vectors, ai, as the...
  13. G

    B A supplementary length contraction question

    Sorry community, I know I have another outstanding relativity question but something else is currently troubling me. What got me thinking was the question of muons moving towards earth. Now according to their half lives very few of them should make it through 12Km of atmosphere... ANALYSIS...
  14. G

    B Time dilation and length contraction question

    Hi All, Bob and Alice are both 21. I'm imagining a scenario where Alice starts on a journey at 0.8c towards a distant planet. Bob (stays still) says this planet is 16 light years away and that it takes Alice 16/0.8 = 20 years to get there (so Bob is 41 years old when Alice gets there). So Bob...
  15. D

    MHB Calculating the Length of an Angle Bisector: Is My Solution Correct?

    How to find the length of an angle bisector (BK) in a triangle A(1;4), B(7;8), C(9;2). I calc BK: \frac{x-7}{\frac{1+\frac{\sqrt{13}}{\sqrt{10}} \cdot 9}{1+\frac{\sqrt{13}}{\sqrt{10}}} - 7} - \frac{y-8}{\frac{4+\frac{\sqrt{13}}{\sqrt{10}} \cdot 2}{1+\frac{\sqrt{13}}{\sqrt{10}}} - 8}=0 Is it right?
  16. F

    I Length Contraction & Time Dilation Beyond Planck Scale: Correct?

    If an object was traveling fast enough relative to an observer such that it's length is contracted down to the Planck scale (as with time), I would imagine that any further increase in speed would result in no more observable relativistic effects? Please correct me if I'm wrong.
  17. U

    Constant string length question with pulleys

    Homework Statement Given Ma = Mb =Mc and muk between c and the surface is 0.30 Homework Equations (rope length) = constant Maaa = Mag - T Mbab = Mbg - 2T Mcac = T - (mu)Mcg The Attempt at a Solution The equations I came up on my own and the answer key did as well. The problem is I also need...
  18. joeyc

    Force per unit length acting on four parallel wires

    Homework Statement An arrangement of four wires, in cross section, with θ=30° and u=5cm. Wire 2 and wire 4 carry the same magnitude of current (I2=I4=10mA) directed out of the page. Wire 3 carries current I3=25mA directed into the page. If the resultant force per unit length on wire 1 is zero...
  19. A

    Inductance per unit length of coaxial transmission line

    1. Homework Statement An air coaxial transmission line has a solid inner conductor of radius a and a very thin outer conductor of inner radius b. Determine the inductance per unit length of the line. Homework Equations the book states the methodology to find the inductance as follows: 1)...
  20. N

    A Can strings increase length without changing their energy?

    For a string in string theory, Energy = string tension * length So I'm wondering if it can increase its length with increasing its energy.
  21. N

    I Does the length of a string depend on it's energy?

    Does the length of a string depend on its energy, or does its energy depend on its length? Similarly, does the area of a d brane depend on its energy, or does its energy depend on its energy? Are these stupid questions?
  22. J

    I Solving Length Contraction Paradox

    Hello there, Suppose there are two inertial frames of reference ##S## and ##S'## with coordinates ##(x,ct)## and ##(x',ct')## such that ##S'## is moving relative to ##S## with velocity ##v##. Suppose ##v>0##, that implies ##\gamma >1##. We know that a Lorentz boost is given by: $$ x' = \gamma...
  23. G

    What is the length described in surface tension?

    https://web.mst.edu/~numbere/CP/CHAPTER%203.htm I have trouble understanding why the length used in 3.1.1 is pir^2. My notes don't explain what length we use either.
  24. O

    Schools George Washington University Statement of Purpose Length

    Long shot, but has anyone applied to the GWU Physics PhD program? If so, how long was your Statement of Purpose? There is no information about it on the application, I ran across this page: https://www.programs.gwu.edu/graduate/physics. This says 250-500 words, but asks for "your academic...
  25. cathal84

    Finding length of vector with unknown variable

    Finding length of vector with unknown variable. Purely for study purposes. Find the smallest possible length of the vector →v . Let vector V = (-2/3,b,16/7) Equation for finding length of vector : Sqrt(a^2+b^2+c^2)Question would be quite straight forward had there been no unknown variable but...
  26. Inderjeet

    B Length Contraction equation derivation

    I recently saw the derivation of length contraction in Special Relativity . At the end , it said x' = (x - vt) γ(gamma) x = (x' + vt') γ(gamma) Where γ(gamma) is Lorentz transformation . It is = 1/√(1- v²/c²) Then derivation continued , with expansion of x' = (x + vt)γ As t = 0 in this case...
  27. J

    A The de Sitter group and minmal length?

    The de Sitter group is often used as an extension of the Poincaré group, because its a simple group and preserves, in addition to a velocity c, a length L. A natural candidate for this length scale is the Planck length. Thus it seems to make sense to think about the invariant Planck length as...
  28. R

    Experimentally finding focal length of small biconvex lens?

    I have fabricated a few biconvex lens (12mm diameter) for an experiment and am trying to find their focal lengths. My though is to set up three parallel lasers and find where the beams converge. However, even with the smallest laser diodes I could find, three beams surpass the diameter of the...
  29. Kara386

    Dimensions of interaction/radiation length

    I've been given that in an electromagnetic calorimeter the radiation length of a material consisting of a single nucleus is: ##X_0 = \frac{716.4A}{Z(Z+1)\ln(287/\sqrt{74}}## Where ##X_0## is in g ##cm^{-2}##. How can it be in those units when everything in that expression is dimensionless? It's...
  30. F

    Error propagation (tape shorter than measured length)

    Homework Statement The problem is about calculating the error on the area of a rectangular field. What is known is that the sides of the rectangle are 120 m and 180 m, and they have been measured with a 10 m measuring tape. The tape has a sensitivity of 2 cm.Homework Equations Since the area...
  31. superkraken

    How does the length of a rod change when it moves fast?

    Homework Statement how does the length of a rod change when it moves fast? Homework EquationsThe Attempt at a Solution
  32. R

    B If the pedal crank arm length of a bicycle increases....

    You have 2 bikes. They only differ in 1 way. The length of the crank arm (where pedal attaches) https://en.wikipedia.org/wiki/Crankset How will these bikes differ when riding? Is one harder to pedal than the other? Why?
  33. T

    Experiment: Measuring the focal length of a converging lens

    Homework Statement You are required to set up a light source, an object grid, a converging lens and a screen so that a quantitative investigation of object and image distances can be performed. Tabulate corresponding object and image distances (p and p' respectively). Also tabulate values of...
  34. A

    Finding the change in length of the rope

    Homework Statement a= 0.3 m / s2 k = 10 000 N / m m = 1800 kg ( of the car that is being pulled) Ignore friction. A car is being pulled by another car by a rope ( stiffness = 10 000 N/m) with acceleration of 0.3 m/s2.. Find the change in length of the rope. Homework Equations F = ma F = kx (x...
  35. F

    MHB Arc length & similar questions

    Hello! I really don't understand this concept, and I have an example problem that I am working on that I just CANNOT figure out! Any help? Thanks so much in advance! A group of people get on a pirate ship ride at the fair. This ride is a swinging pendulum with a maximum swing angle of 65 degrees...
  36. F

    I Correlation functions and correlation length

    I thought I understood the concept of a correlation function, but I having some doubts. What exactly does a correlation function quantify and furthermore, what is a correlation length. As far as I understand, a correlation between two variables ##X## and ##Y## quantifies how much the two...
  37. J

    Average path length through a box

    Hi, I would like to try and calculate the average distance traveled through a box of length ##w## by any straight path possible. As shown in the attached figure. Where the path through the box to an adjacent face is given by $$r_{adj}^2=\left(\frac{w}{2}-x\right)^2+y^2+z^2.$$ The average...
  38. P

    B Measuring Rest Length While Moving

    I don't remember seeing this solution to the question. Imagine an object of length d with a reflector on the far end (x=4). A moves past the object at speed a, emitting a signal at the origin in the positive x direction. A makes one pass in the +x direction, reverses and makes one pass in the...
  39. P

    B What is the formula for the length of the graph of e^x in the first quadrant?

    Hi. I derived a formula for the length of the graph of e^x in the first quadrant between x=a and x=b. It is: sqrt(1+e^(2b)) - sqrt(1+e^(2a)) + (a-b) + log [(sqrt(1+e^(2b)) - 1) / (sqrt(1+e^(2a)) - 1)] I think it works because it gave a value of approx 2.003 units for a=0 and b=1. For a=0 and...
  40. D

    High Capillary Length Materials: 1cm+ Easily Found

    Which materials have high capillary length, and can be found easily? (above 1cm)
  41. F

    I Length contraction vs expansion

    I'm coming across a contradiction that I can't resolve. Two points on the ground are separated by 30km. If you're in a moving frame at velocity +v relative to the ground and you view those two points on the ground, would those two points be greater or less than 30km apart? At first, I thought it...
  42. bananabandana

    General proof of Arc Length For Parametrised Coodrdinates

    Homework Statement Prove that, given a metric ##g_{ij}## such that ##ds^{2}=g_{ij}dx^{i}dx^{j}##, where ##x^{r} = x^{r}(\lambda)## , we have the following result for the arc length: $$ L(p,q) = \int_{p}^{q} ds = \sqrt{ g_{ij} \frac{dx^{i}}{d \lambda} \frac{ dx^{j}}{d \lambda} } d \lambda $$...
  43. Nipuna Weerasekara

    Can a triangle be formed with these length constraints?

    Homework Statement There is a triangle with sides $$ 3,3r,3r^2 $$ such that 'r' is a real number strictly greater than the Golden Ratio. Is this statement true or false...? Homework Equations $$Golden \space Ratio = \phi = 1.618... $$ The Attempt at a Solution Actually I have no clue at all...
  44. kdunkzz

    What would be the relaxed length of the spring?

    Homework Statement A ball of mass 620 g hangs from a spring whose stiffness is 115 Newtons per meter. A string is attached to the ball and you are pulling the string to the right, so that the ball hangs motionless, as shown in the figure. In this situation the spring is stretched, and its...
  45. G

    Calculate the focal length of a planoconvex lens?

    Homework Statement Calculate the focal length of a planoconvex lens that produces a real image 2.5 times as big as the object if the object is 2.5 cm from the lens? Homework Equations The only one I can think of is 1/f = 1/v + 1/u The Attempt at a Solution [/B] My 'attempt' is 1/f = 1/62.5...
  46. J

    B Variable focal length TV / screen

    Recently I have had a lot of eye strain from excessive computer use. The problem is because your eye lenses re stuck in a fixed position. I wanted to know if it would be possible to make a screen that is 1m away look like it is 4m away for example. Would it be possible by making the screen out...
  47. F

    I Definition and measurement of proper length

    As I understand it, the proper length, ##L## of an object is equal to the length of the space-like interval between the two space-time points labelling its endpoints, i.e. (in terms of the corresponding differentials) $$dL=\sqrt{ds^{2}}$$ (using the "mostly plus" signature). Furthermore, this is...
  48. Albert1

    MHB Length of $\overline{CE}$ in Square $ABCD$ with $45^o$ Angle

    A square $ABCD$ , each side with length 13, if points $E,F$ on $\overline{BC} ,\overline{AB} $ respectively,$\angle EDF=45^o,$and $\overline{EF}=11$ find length of $\overline{CE}=?$
  49. I

    I Leyden Jars increase spark length?

    Good Evening/Morning/Afternoon Everyone, Question: Why is it that demonstrations of Van de Graff generators use Leyden Jars in series to get bigger sparks? If a spark gap's size determined by voltage, and Leyden jars are simply to be treated as capacitors which wouldn't increase voltage, why...
  50. A

    "ANSYS Workbench" Contact tool results plotting vs length?

    Hello ansys users in ANSYS workbench, how can I plot the contact tool results (Frictional stress, sliding distance...etc) with length (500mm for example) in the "x-axis" instead of the default "by time" option?. the path method can not be used with this type of results "Contact tool results"...
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