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!

Homework Help: Cable Taper

  1. Feb 24, 2006 #1
    I'm doing some research on space elevators and have found a site (http://www.zadar.net/space-elevator/#transverse") which gives some insight into the maths behind the elevator cable. If you click on the link and scroll down, up, whichever direction to equation (2) and a picture of a blue trapezium the guy has found the difference in the volume of the tapering cable with what looks like volume of revolution, but I cant seem to figure out what he's done. He says that it's 'easy to show...'. Am I missing something? I could use a little help. thanks.
    Last edited by a moderator: Apr 22, 2017
  2. jcsd
  3. Feb 24, 2006 #2

    Tom Mattson

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Quick way to show this: You are dealing with finding the volume of a frustum of a cone. The volume [itex]V[/itex] of a cone is [itex]V=\frac{1}{3}Ah[/itex], where [itex]A[/itex] is the area of the base and [itex]h[/itex] is the height.

    You can get the volume of the frustum illustrated in the figure by repeatedly applying the above formula and subtracting volumes, as follows:

    [tex]V=V_{frustum+missing top}-V_{missing top}[/tex]

    Long way to show this: You can calculate the volume with a single integral because the frustum is a surface of revolution. Find an equation for the right boundary of the figure (it's a straight line segment, so that's easy) and solve it for x. Then revolve that line segment about the y-axis and write down an expression for the differential volume. Then integrate over y, and you should get the same expression.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook