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Unformatted text preview: tidal friction. You should neglect
any eﬀects due to the rotation of the Moon.
(a) Write down expressions for the total energy E and total angular momentum J of the EarthMoon system. Some useful symbols will be the Earth’s angular rotation frequency ω ; the Moon’s
(Keplerian) orbital frequency Ω; the masses of the Earth and Moon, Me and Mm ; the Earth’s
moment of inertia I ; and the mean separation of the Earth and Moon, a.
(b) Use the equation for J to eliminate ω from the energy equation.
(c) Show that the energy equation can be cast into the dimensionless form
= − + α(j − s1/2 )2 ,
where is the total energy in units of (Gme mm /2a0 ), j is the total angular momentum in units
of (µa2 Ω0 ), s = a/a0 is the dimensionless separation, µ is the reduced mass, and the subscript
“0” refers to values at the present epoch in history.
(d) Find numerical values for α and j . Look up the masses of the Earth and Moon, and take the
Earth’s moment of inertia to be (2/5)Me Re and a0 = 3.84 × 105 km.
(e) Graph the dimensionless energy equation to ﬁnd the two values of s for which is an extremum. (f) Find the same two values of s quantitatively by diﬀerentiating the energy equation and solving the
resulting nonlinear equation numerically by the Newton-Raphson method or some other scheme.
Show that ω = Ω at these orbital separations. Find the corresponding orbital period of the Moon
and rotation period of the Earth.
(g) Find the diﬀerence in energy ∆E between the current epoch and the time in the future when the
Earth’s rotation and the Moon’s orbit will be synchronous.
(h) Estimate the rate of energy dissipation due to tidal friction by assuming that, twice per day, the
top 1-m layer of the oceans is lifted by 1-m and then lowered. Further assume that a few percent
of this mechanical energy is dissipated as heat.
(i) From the energy dissipation rate and total energy ∆E that must be lost in order for the...
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