1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Lenght in a circle

  1. Jan 10, 2005 #1

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Hi,

    I'd like to know the name for the lenght in a circle that I will describe:

    Consider 2 arbitrary points lying on the perimeter of a circle. I'm talking about the shortest distance distance between these those points. (i.e. the straight line joining them.)

    Thx
     
  2. jcsd
  3. Jan 10, 2005 #2

    jamesrc

    User Avatar
    Science Advisor
    Gold Member

    It's called a chord.
     
  4. Jan 10, 2005 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Yep, a chord.
     
  5. Jan 10, 2005 #4

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Ok, I have a confusion involving a cord. In physics, we often make the transformation from an infinitesimal cord to an infinitesimal arc: d(cord) = d(arc). And I think it's used in some math proofs too.

    For exemple, imagine a position vector [itex]\vec{r}(t)[/itex] of fixed norm, rotating around the z axis (i.e. [itex]\vec{\omega} = \omega \hat{z}[/itex]). Let's say the radius of the circle described by the motion of its tip/head/arrow is R. After a time [itex]\Delta t[/itex], it has rotated an angle [itex]\omega \Delta t[/itex] and then they (the manuals) say that it can be seen that when [itex]\Delta t[/itex] is small, [itex]||\Delta \vec{r}||[/itex] (which is a cord), is very near the lenght of the arc [itex]\Delta s[/itex] subtended by the angle [itex]\omega \Delta t[/itex], and thus, poof, [itex]||d\vec{r}|| = ds[/itex].

    And while this seems to be true intuitively, I have never seen a proof of this statement. And when I try to do it, here's what I get:

    I start from a cercle of radius R and a cord [itex]\delta[/itex] subtended by an angle [itex]\theta[/itex]. I find that the lenght of the cord is given by

    [tex]\delta = 2Rsin\left(\frac{\theta}{2}\right)[/tex]

    Therefor,

    [tex]d\delta=Rcos\left(\frac{\theta}{2}\right)d\theta[/tex]

    While

    [tex]ds=rd\theta[/tex]

    So

    [tex]d\delta=cos\left(\frac{\theta}{2}\right)ds[/tex]

    A result indicating that even the differential version is just an approximation because only true for a principal angle [itex]\theta=0[/itex].
     
    Last edited: Jan 10, 2005
  6. Jan 10, 2005 #5

    StatusX

    User Avatar
    Homework Helper

    it follows from the identity

    [tex] \lim_{\theta \rightarrow 0} \frac{\sin \theta}{\theta} = 1 [/tex]

    Also, the mistake in your derivation is that theta is a function of the chord length.
     
    Last edited: Jan 10, 2005
  7. Jan 11, 2005 #6

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Thanks for your reply StatusX

    At which point does that identity fit in?


    Could you elaborate?
     
  8. Jan 11, 2005 #7

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    Sure,if "y" is a function of "x",then "x" is a function of "y",right??So he basically didin't say anything new...You were right,though...Your calculations were corrrect.I guess u knew that,but u have my confirmation...

    Daniel. :smile:
     
  9. Jan 11, 2005 #8

    StatusX

    User Avatar
    Homework Helper

    A chord of length 2 R sin(theta) is subtended by an angle of 2 theta (in radians) and so has an arclength of 2 R theta. The identity means d(chordlength)/d(arclength) = 1.

    I'm sorry, I just glanced at your last step and assumed you had forgotten that if arclength is 0, then theta must be as well. You did forget that, but for a different reason. What you derived is how an chord of finite length changes with an infinitessimal change in arclength. What you wanted was the ratio of a differential chord to the differential arclength it subtends. To get the differential you're looking for, just take theta=0.
     
    Last edited: Jan 11, 2005
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Lenght in a circle
  1. A circle of circles (Replies: 3)

Loading...