The Hubble Space Telescope (HST) is a scientific instrument of extreme sensitivity. Because of its finite size, the telescope is subject to a tidal force from the planet's gravitational field (much like the way the world's oceans experience tidal forces due to the moon's gravity). If this tidal force is too large, the stress may affect the desired operation of the telescope. In this problem, treat the HST like a uniform cylinder with length and diameter orbiting with its center of mass at an altitude ℎ above the surface of Earth, which has radius and mass . Find an approximate expression, to first order, in the satellite's dimensions ( or , or both) for the ratio of the maximum tidal acceleration of the satellite to the centripetal acceleration of its center of mass.
Note that your approximation scheme should involve the use of the binomial theorem which, to first order in , gives
(1+)≈1+
as long as is a real number and ||<1.
tidal, max/c=