Physics Fun with Tethers and Catenaries

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Physics Fun with Tethers and Catenaries

• William H. Ingham
• Department of Physics
• James Madison University
• Harrisonburg, VA 22807

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Abstract

• This presentation will examine some of the
  physics and mathematics involved in analyzing the
  equilibrium of a buoyant but tethered (
  rigid-walled) spherical balloon in a steady
  breeze.

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Calm-day Equilibrium of a Tethered Balloon 1

• Tether is vertical.
• Tension at upper end of tether equals difference
  between the buoyant force on the balloon and the
  true weight of the balloon
• If we treat the tether as having zero mass and
  zero volume, then the tension does not vary along
  the line, so the tension at the bottom end is the
  same as at the top

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Calm-day Equilibrium of a Tethered Balloon 2

• If we treat the tether as a flexible but
  inextensible string with a known radius (r) and
  mass per unit length (m), then the tension
  decreases along the string from the previously
  given value at the top. We use s to denote
  distance measured downward along the line from
  the top end. By applying Newtons 2nd Law to the
  equilibrium of each segment of the tether, we
  find out how the tension varies

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Calm-day Equilibrium of a Tethered Balloon 3

• Since a string can pull but not push, only
  positive values of the tension are physically
  meaningful. This allows us to set an upper limit
  to the length of the tether

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Calm-day Equilibrium of a Tethered Balloon 4

• The mass per unit length of the tether is
  determined by the strings (volume) density and
  its radius
• Typically the string density is much greater
  than the air density, so that the buoyant force
  on the string can be neglected. Then
• This equation for the maximum tether length just
  says that the balloon cannot carry a hanging
  weight greater than its (net) lift.

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Calm-day Equilibrium of a Tethered Balloon 5

• Lets do an example. Using 1.23 kg/m3 as the
  desnity of air, if we consider a balloon of
  radius 1.00 m and a total mass of 2.50 kg, the
  (net) lift works out to about 26 newtons. If the
  tether is nylon cord one-quarter inch in
  diameter, the weight per unit length of tether is
  about 0.32 N/m. Dividing the lift by the weight
  per unit length, we find
• Thus, the maximum altitude for the bottom of
  this balloon on this tether is less than the
  length of a football field.

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What about a breezy day? (1)

• On a breezy day, of course, a tethered balloon in
  equilibrium is not directly above the anchor
  but is some distance downwind.
• The slope of the tether at its upper end (s0)
  equals the ratio of the (buoyant) lift L to the
  air drag D
• The tension at upper end (s 0) has magnitude

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Breezy Day 2

• If we ignore air drag on the tether itself, then
  applying Newtons 2nd Law to the tether yields
  the famous catenary shape. The horizontal
  component (t cos q) of the tension is constant
  along the string, while the vertical component
• (t sin q) decreases with distance s down along
  the (curved) string

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Breezy Day 3

• These equations imply that
• Careful thinking about this equation reveals
  that the maximum altitude for the balloon still
  occurs when the length of the tether line equals
• which is just the same as on a calm day!

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Breezy Day 4

• If we adopt Cartesian coordinates (x,y) with
  origin at the upper end of the tether and with x
  downwind and y vertically up, then a kite flying
  at maximum altitude has its bottom end at (xg,yg)
  given by

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Breezy Day 5

• It is not difficult to calculate the shape of the
  tether numerically. The dimensionless length
  variable s used in the above equations is
  convenient.
• Lets do a numerical example based on the earlier
  calm-day example. For a balloon of radius 1.0
  meter in a wind of 10 mph (v 4.5 m/s), the
  Reynolds number is about 6 x 105, for which the
  drag coefficient CD 0.50. The drag force is
  then

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Breezy Day 6

• This gives L/D 26/19 1.37. The angle at the
  top of the tether is thus about 54 degrees.
• The shape of the tether has been numerically
  computed using MATLAB.
• The computed altitude for the balloon is about 42
  meters (as opposed to the 82 meters altitude on a
  calm day). The profile of the tether is shown on
  the next slide

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Plans for Future Work (er, Fun)

• Take account of wind drag on the tether
• Estimate effects of winds altitude profile
• Animations
• Follow the problem where the wind takes me!