- #1

Sergio Rodriguez

- 14

- 4

## Homework Statement

A satellite of mass [itex]m_s[/itex] orbits the Earth in a circular orbit of radius [itex]r_0[/itex]. If the satellite orbits at the upper part of the atmosphere and the friction force f is constant, it would trace an spiral and fall to the earth. but if we suppouse that the friction force is small, so in every moment the orbit would be circular. Calculate the change of the radius at every revolution.

## Homework Equations

$$ W_f = ΔME $$

$$ME = \frac {-Gm_sM_e}{2r}$$

## The Attempt at a Solution

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The work done by the friction force is: f ⋅ 2πr, where 2πr the length of the orbit with radius r.

The change in mechanical energy:

[itex] ΔME = \frac {-Gm_sM_e} {2r} + \frac {Gm_sM_e} {2(r-Δr)} [/itex]

because Δr is diference between the radius of one lap and the next one.

$$2πrf =\frac {-Gm_sM_e} {2r} + \frac {Gm_sM_e} {2(r-Δr)} $$

but when I try to solve for Δr is: [itex] Δr = \frac {4πr^2}{4πrf - 2Gm_sM_t}[/itex] very different from the the solution of the book: [itex]Δr = \frac {fr^{\frac {3} {2} } }{m_s\sqrt{M_tG} }[/itex]

I have spent sveral hours with this exercise and don't know where is the error. Plese, help.

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