Work done by the gravitational force

  1. Hi there!
    I'd like to calculate the work done by the gravitational force. I know the work is defined by the integration of a 1-form:
    [tex]L=\int_\gamma \omega[/tex]

    This works fine in cartesian coordinates and I know how to integrate it, but what if I want to use spherical coordinates?
    Then I'd have:
    Suppose [tex]\gamma[/tex] is a curve defined in spherical coordinates (i.e. [tex]\vec\gamma=R(t)\hat r+\Theta(t)\hat\theta+\Phi(t)\hat\phi[/tex]),
    how do I integrate the 1-form along [tex]\gamma[/tex]?
  2. jcsd
  3. jtbell

    Staff: Mentor

    No, you need to use the line element in spherical coordinates:

    [tex]d \vec l = dr \hat r + r d\theta \hat \theta + r \sin \theta d\phi \hat \phi[/tex]

    so that

    [tex]\omega = F_r dr + F_\theta r d\theta + F_\phi r \sin \theta d\phi[/tex]

    Now, what are [itex]F_r[/itex], [itex]F_\theta[/itex], and [itex]F_\phi[/itex]?
    Last edited: Feb 1, 2011
  4. Then the integral is like this?

    [tex]\int_\gamma \omega = \int_{t_0}^{t_1} \vec F \cdot\frac{d\vec l}{dt}dt=\int_{t_0}^{t_1} \left( F_r\frac{dr}{dt}+F_\theta r \frac{d\theta}{dt}+F_\phi rsin\theta\frac{d\phi}{dt}\right)dt[/tex]
  5. Wouldn't the work done when moving between two points in a gravitational field just be the difference between the potential energies at those two points? You'd really only need to worry about the up direction...or r in spherical polar coordinates...if the coordinate origin is the earth's center.
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