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Physics / Physics: Principles with Applications 6 / Chapter 5 / Problem 55P

Physics: Principles with Applications | 6th Edition | ISBN: 9780321569837 | Authors: Douglas C. Giancoli

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(III) (a) Show that if a satellite orbits very near the surface of a planet with period

Chapter 5, Problem 55P

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QUESTION:

(III) (a) Show that if a satellite orbits very near the surface of a planet with period T, the density (mass/volume) of the planet is $p=m/V=3\pi{}/G{T}^{2}$ . (b) Estimate the density of the Earth, given that a satellite near the surface orbits with a period of about 85 min.

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QUESTION:

(III) (a) Show that if a satellite orbits very near the surface of a planet with period T, the density (mass/volume) of the planet is $p=m/V=3\pi{}/G{T}^{2}$ . (b) Estimate the density of the Earth, given that a satellite near the surface orbits with a period of about 85 min.

ANSWER:

Solution 55P:

Step 1 of 2:-

Here we have to show that, if the satellite orbits very near to a planet, the density of the planet must be $3\pi{}/G{T}^{2}$ .

The mass of the planet is m.

The period of the satellite is: T.

a)

We know that, the orbital velocity is defined as,

${v}_{o}=\sqrt{Gm/R}$

The speed of a satellite around the circular path is: $v=\frac{2\pi{}R}{T}$ .

So, the se two must be equal.

$\sqrt{GM/R}=2\pi{}R/T$

$\Rightarrow{}\frac{Gm}{R}=\frac{4{\pi{}}^{2}{R}^{2}}{{T}^{2}}$ .

$\Rightarrow{}m/{R}^{3}=\frac{4{\pi{}}^{2}}{G{T}^{2}}$ .

By dividing both sides by $\frac{4}{3}\pi{}$ ,

$\frac{m}{\frac{4}{3}\pi{}{R}^{3}}=\frac{3\pi{}}{G{T}^{2}}$

$\Rightarrow{}m/volume\ of\ planet=\frac{3\pi{}}{G{T}^{2}}$

$\Rightarrow{}\rho{}=\frac{3\pi{}}{G{T}^{2}}$ . proved

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