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Re: Viewing the spokes

Message-ID <57C4E181.17E2@ix.netcom.com> (permalink)
Date 2016-08-29 18:29 -0700
From The Starmaker <starmaker@ix.netcom.com>
Newsgroups sci.physics
Subject Re: Viewing the spokes
References <nq1q63$bdd$1@gioia.aioe.org>

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Sergio wrote:
> 
> Viewing the spokes
> 
> A wheel with spokes rolls on the ground. A stationary camera takes a
> picture of the wheel. Due to the nonzero exposure time of the camera,
> the spokes will generally appear blurred. At what location(s) in the
> picture does (do) the spoke(s) not appear
> blurred?
> 
> answer,
 
 The contact point on the ground does not look blurred, because it is
 instantaneously
 at rest. But although this is the only point on the wheel that is at
 rest, there will
 be other locations in the picture where the spokes do not appear blurred.
 The relevant property of a point in the picture where a spoke does not
 appear
 blurred is that the point lies on the spoke throughout the duration of
 the camera’s
 exposure. (The point in the picture need not, however, actually
 correspond to the
 same atom on the spoke.)
 At a given time, consider a spoke in the lower half of the wheel. A
 short time
 later, the spoke will have moved (via both translation and rotation),
 but it will
 intersect its original position. The spoke will not appear blurred at
 this intersection
 point. We must therefore find the locus of these intersections. We can
 do this in
 two ways.
 First method: Let R be the radius of the wheel. Consider a spoke that
 makes an
 angle of µ with the vertical at a given time. If the wheel then rolls
 through a small
 angle dµ, the center moves a distance Rdµ. The spoke’s motion is a
 combination
 of a translation through this distance Rdµ, plus a rotation through the
 angle dµ
 (around the top end).
 Let r be the radial position of the intersection of the initial and
 final positions of
 the spoke. In the following figure, we can write down two expressions
 for the short
 segment drawn perpendicular to the initial position of the spoke.
 
 Rdq
 dq
 q
 q
 r
 The two expression are (Rdµ) cos µ, and r dµ (to first order in dµ).
 Equating these
 gives r = Rcos µ. This describes a circle whose diameter is the lower
 vertical radius
 of the wheel, as shown below.
 1
 q
 r=Rcosq
 wheel
 Second method: Since the wheel’s contact point with the ground is
 instantaneously
 at rest, the wheel may be considered to be instantaneously rotating around
 this point. This means that every atom in the wheel (both in the spokes
 and the
 rim) instantaneously traces out the arc of a circle centered at the
 contact point. For
 the point P in the figure below, this circle is the dotted one drawn
 (with Q being
 the contact point). A spoke will not appear blurred at the point where
 this circular
 motion is along the direction of the spoke. That is, a spoke will not
 appear blurred
 at the point where the dotted circle is tangent to the spoke, as shown.
 R
 q
 P wheel
 O
 QWe are therefore concerned with the locus of all points P such that the
 segments
 PQ and PO are perpendicular. As seen in the previous diagram above, this
 locus
 is the circle whose diameter is the lower vertical radius of the wheel.
 2

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Viewing the spokes Sergio <invalid@invalid.com> - 2016-08-29 12:08 -0500
  Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-08-29 18:00 +0000
    Re: Viewing the spokes benj <benj@nobody.net> - 2016-08-29 21:00 -0400
      Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-08-30 01:14 +0000
        Re: Viewing the spokes benj <benj@nobody.net> - 2016-08-29 22:45 -0400
          Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-08-30 04:14 +0000
            Re: Viewing the spokes The Starmaker <starmaker@ix.netcom.com> - 2016-08-29 21:59 -0700
              Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-08-30 05:16 +0000
                Re: Viewing the spokes The Starmaker <starmaker@ix.netcom.com> - 2016-08-29 23:00 -0700
    Re: Viewing the spokes noTthaTguY <abu.kuanysh05@gmail.com> - 2016-08-29 18:29 -0700
  Re: Viewing the spokes The Starmaker <starmaker@ix.netcom.com> - 2016-08-29 18:29 -0700
  Re: Viewing the spokes The Starmaker <starmaker@ix.netcom.com> - 2016-08-29 18:46 -0700
    Re: Viewing the spokes benj <benj@nobody.net> - 2016-08-30 19:40 -0400
      Re: Viewing the spokes The Starmaker <starmaker@ix.netcom.com> - 2016-08-30 22:07 -0700
        Re: Viewing the spokes The Starmaker <starmaker@ix.netcom.com> - 2016-08-30 23:48 -0700
          Re: Viewing the spokes The Starmaker <starmaker@ix.netcom.com> - 2016-08-31 09:54 -0700
            Re: Viewing the spokes The Starmaker <starmaker@ix.netcom.com> - 2016-09-02 10:00 -0700
              Re: Viewing the spokes noTthaTguY <abu.kuanysh05@gmail.com> - 2016-09-02 13:23 -0700
              Re: Viewing the spokes The Starmaker <starmaker@ix.netcom.com> - 2016-09-03 11:30 -0700
                Red, Blue and Green The Starmaker <starmaker@ix.netcom.com> - 2016-09-03 13:18 -0700
                Re: Red, Blue and Green The Starmaker <starmaker@ix.netcom.com> - 2016-09-03 14:57 -0700
                Re: Red, Blue and Green The Starmaker <starmaker@ix.netcom.com> - 2016-09-03 16:01 -0700
                Re: Red, Blue and Green The Starmaker <starmaker@ix.netcom.com> - 2016-09-04 12:46 -0700
                Re: Red, Blue and Green Ergios <invalid@invalid.com> - 2016-09-07 12:32 -0500
                Re: Red, Blue and Green noTthaTguY <abu.kuanysh05@gmail.com> - 2016-09-08 12:12 -0700
                Red, Blue and Green The Starmaker <starmaker@ix.netcom.com> - 2016-09-07 23:22 -0700
                Re: Red, Blue and Green The Starmaker <starmaker@ix.netcom.com> - 2016-09-03 16:03 -0700
                Re: Red, Blue and Green The Starmaker <starmaker@ix.netcom.com> - 2016-09-03 16:06 -0700
                Re: Red, Blue and Green The Starmaker <starmaker@ix.netcom.com> - 2016-09-20 00:18 -0700
                Re: Red, Blue and Green noTthaTguY <abu.kuanysh05@gmail.com> - 2016-09-20 12:52 -0700
                Re: Red, Blue and Green The Starmaker <starmaker@ix.netcom.com> - 2016-09-07 10:01 -0700
                Re: Red, Blue and Green Gutless Umbrella Carrying Sissy <taustinca@gmail.com> - 2016-09-07 10:12 -0700
                Re: Red, Blue and Green The Starmaker <starmaker@ix.netcom.com> - 2016-09-20 09:47 -0700
        Re: Viewing the spokes noTthaTguY <abu.kuanysh05@gmail.com> - 2016-08-31 14:32 -0700
  Re: Viewing the spokes "reber g=emc^2" <herbertglazier0@gmail.com> - 2016-09-04 11:39 -0700
    Re: Viewing the spokes "hanson" <hanson@quick.net> - 2016-09-04 20:36 -0700
  Re: Viewing the spokes "nuny@bid.nes" <Alien8752@gmail.com> - 2016-09-09 09:08 -0700
    Re: Viewing the spokes noTthaTguY <abu.kuanysh05@gmail.com> - 2016-09-09 12:32 -0700
    Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-09 17:51 -0400
      Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-10 00:10 +0000
        Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-09 20:43 -0400
          Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-10 00:57 +0000
            Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-09 21:07 -0400
              Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-10 01:33 +0000
                Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-12 20:59 -0400
                Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-13 02:41 +0000
      Re: Viewing the spokes "nuny@bid.nes" <Alien8752@gmail.com> - 2016-09-09 19:43 -0700
        Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-12 21:29 -0400
          Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-13 02:43 +0000
          Re: Viewing the spokes noTthaTguY <abu.kuanysh05@gmail.com> - 2016-09-13 13:28 -0700
          Re: Viewing the spokes "nuny@bid.nes" <Alien8752@gmail.com> - 2016-09-13 17:14 -0700
            Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-13 21:52 -0400
              Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-14 05:11 +0000
                Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-15 17:49 -0400
                Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-16 00:50 +0000
                Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-15 21:52 -0400
                Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-16 04:11 +0000
                Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-16 16:58 -0400
                Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-16 21:40 +0000
                Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-16 20:56 -0400
                Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-17 01:47 +0000
                Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-17 17:24 -0400
                Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-18 00:07 +0000
              Re: Viewing the spokes noTthaTguY <abu.kuanysh05@gmail.com> - 2016-09-14 14:26 -0700
                Re: Viewing the spokes noTthaTguY <abu.kuanysh05@gmail.com> - 2016-09-14 15:04 -0700
                Re: Viewing the spokes noTthaTguY <abu.kuanysh05@gmail.com> - 2016-09-16 12:54 -0700
              Re: Viewing the spokes "nuny@bid.nes" <Alien8752@gmail.com> - 2016-09-14 19:47 -0700
                Re: Viewing the spokes benj <nobody@gmail.com> - 2016-09-14 23:24 -0400
                Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-15 05:12 +0000
                Re: Viewing the spokes "nuny@bid.nes" <Alien8752@gmail.com> - 2016-09-15 15:45 -0700
                Re: Viewing the spokes benj <benj@nobody.net> - 2016-09-15 21:36 -0400
                Re: Viewing the spokes jimp@specsol.spam.sux.com - 2016-09-16 02:58 +0000
                Re: Viewing the spokes "nuny@bid.nes" <Alien8752@gmail.com> - 2016-09-18 12:15 -0700
    Re: Viewing the spokes Wally W. <ww84wa@aim.com> - 2016-09-15 09:15 -0400
      Re: Viewing the spokes "nuny@bid.nes" <Alien8752@gmail.com> - 2016-09-15 15:48 -0700
        Re: Viewing the spokes noTthaTguY <abu.kuanysh05@gmail.com> - 2016-09-18 15:33 -0700

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