Part A. If a satellite orbits very near the surface of a planet with period T, derive an...

Question:

Part A. If a satellite orbits very near the surface of a planet with period T, derive an algebraic expression for the density (mass/volume) of the planet. Express your answer in terms of the variable T and the gravitational constant G.

Part B. Estimate the density of the Earth, given that a satellite near the surface orbits with a period of about 85 min.

Circular Orbit:

If an object is a circular motion around the heavier object in the sky, the orbital period squared is proportional to the orbital radius cubed. The orbital velocity can be obtained by the circumference of the circular path divided by the orbital period.

Answer and Explanation:

(A) Let the orbital speed be v, the orbital radius be b, and the orbital period be T. Then, the centripetal acceleration is expressed as

{eq}\dfrac{v^2}{r}=\dfrac{GM}{r^2} {/eq}

Here, G is gravitational constant and M is the mass of the planet.

Also, the orbital speed v is written as

{eq}v=\dfrac{2\pi r}{T} {/eq}

Solving for M, we have

{eq}\begin{align} M&=\dfrac{r^2}{G}\dfrac{(2\pi r)2}{rT^2}\\\\ &=\dfrac{4\pi^2 r^3}{T^2G} \end{align} {/eq}

The density of the planet is

{eq}\rho=\dfrac{M}{V} {/eq}

Here, V is the volume of the planet and it is given as {eq}\frac{4}{3}\pi r^3 {/eq} if we assume that the planet is a sphere.

Thus,

{eq}\begin{align} \rho&=\dfrac{M}{V}\\\\ &=\dfrac{4\pi^2 r^3}{T^2 G \frac{4}{3} \pi r^3}\\\\ &=\boxed{\dfrac{3\pi}{GT^2}} \end{align} {/eq}

(B)

When T is 85 min, the density of the Earth is

{eq}\begin{align} \rho&=\dfrac{3\pi}{GT^2}\\\\ &=\dfrac{3\pi}{ 6.67\times10^{-11} \times (85\times 60)^2}\\\\ &=\boxed{5.43\times 10^3\, kg/m^3} \end{align} {/eq}

Please note, 1 min has 60 seconds. The value of G is {eq}6.67\times10^{-11} N m^2/kg^2 {/eq}


Learn more about this topic:

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Stable Orbital Motion of a Satellite: Physics Lab

from Physics: High School

Chapter 8 / Lesson 16
548

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