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Chapter 8: Problem 44

Satellites A and B are in circular orbits, with A twice as far from Earth's center as B. How do their orbital periods compare?

Short Answer

Expert verified
The orbital period of satellite A (A) is \(2\sqrt{2}\) times longer than that of satellite B (T).

Step by step solution

01

Understand and apply Kepler's Third Law

Kepler's third law of planetary motion states that the square of the period of a planet orbiting an object (in this case Earth), is proportional to the cube of its average distance from the object. This can be expressed algebraically as:\(T^2 \propto r^3\)where \(T\) is the orbital period and \(r\) is the average distance from the center of Earth.
02

Set up the proportions

Since satellite A is twice as far from the Earth's center than satellite B, denote the distance of satellite B from Earth's center as \(r\) and that of satellite A as \(2r\). Also, let's denote the orbital period of satellite B as \'T' and that of satellite A as 'A'. According to Kepler's third law, we then have:\(T^2 = kr^3\)\(A^2 = k(2r)^3\) where \(k\) is the constant of proportionality.
03

Solve for A in terms of T

Next, we solve the equation of satellite A in terms of the equation of satellite B. We'll have to divide the second equation by the first one. This gives:\(\frac{A^2}{T^2} = \frac{k(2r)^3}{kr^3} = 8\)Taking the square root of both sides, we get:\(A = 2\sqrt{2} T.\)

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Most popular questions from this chapter

Neglecting Earth's rotation, show that the energy needed to launch a satellite of mass \(m\) into circular orbit at altitude \(h\) is $$\left(\frac{G M_{\mathrm{E}} m}{R_{\mathrm{E}}}\right)\left(\frac{R_{\mathrm{E}}+2 h}{2\left(R_{\mathrm{E}}+h\right)}\right).$$

Does the gravitational force of the Sun do work on a planet in a circular orbit? In an elliptical orbit? Explain.

The asteroid Pasachoff orbits the Sun with period 1417 days. Find the semimajor axis of its orbit from Kepler's third law. Use Earth's orbital radius and period, respectively, as your units of distance and time.

Determine escape speeds from (a) Jupiter's moon Callisto and (b) a neutron star, with the Sun's mass crammed into a sphere of radius \(6.0 \mathrm{km} .\) See Appendix E for relevant data.

A friend who knows nothing about physics asks what keeps an orbiting satellite from falling to Earth. Give an answer that will satisfy your friend.
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