1 / 15

Physics Fun with Tethers and Catenaries

Physics Fun with Tethers and Catenaries. William H. Ingham Department of Physics James Madison University Harrisonburg, VA 22807. Abstract.

melosa
Download Presentation

Physics Fun with Tethers and Catenaries

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Physics Fun with Tethers and Catenaries William H. Ingham Department of Physics James Madison University Harrisonburg, VA 22807 CS-AAPT Meeting Geroge Mason University

  2. 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. CS-AAPT Meeting Geroge Mason University

  3. 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: CS-AAPT Meeting Geroge Mason University

  4. 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 Newton’s 2nd Law to the equilibrium of each segment of the tether, we find out how the tension varies: CS-AAPT Meeting Geroge Mason University

  5. 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: CS-AAPT Meeting Geroge Mason University

  6. Calm-day Equilibrium of a Tethered Balloon: 4 • The mass per unit length of the tether is determined by the string’s (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. CS-AAPT Meeting Geroge Mason University

  7. Calm-day Equilibrium of a Tethered Balloon: 5 • Let’s 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. CS-AAPT Meeting Geroge Mason University

  8. 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 (s=0) equals the ratio of the (buoyant) lift L to the air drag D: • The tension at upper end (s = 0) has magnitude CS-AAPT Meeting Geroge Mason University

  9. Breezy Day: 2 • If we ignore air drag on the tether itself, then applying Newton’s 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: CS-AAPT Meeting Geroge Mason University

  10. 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! CS-AAPT Meeting Geroge Mason University

  11. 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 CS-AAPT Meeting Geroge Mason University

  12. 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. • Let’s 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 CS-AAPT Meeting Geroge Mason University

  13. 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 CS-AAPT Meeting Geroge Mason University

  14. CS-AAPT Meeting Geroge Mason University

  15. Plans for Future Work (er, Fun) • Take account of wind drag on the tether • Estimate effects of wind’s altitude profile • Animations • Follow the problem where the wind takes me! CS-AAPT Meeting Geroge Mason University

More Related