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<h2 class="hd hd-2 unit-title">Circular Orbits</h2>
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<h2>4.2.1 Circular Orbits</h2>
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<p>The most elementary type of orbital motion is that of one object moving about another, much larger object, as in the way that the Earth moves around the Sun. Although such orbits are elliptical in shape, in this segment we will consider the simplifying case of circular orbits to illustrate the relationship between gravity, the masses of the orbital bodies, the distance between them, as well as orbital velocity and period. As you watch the video below, pay close attention to the expressions for circular velocity and period, which you will need later to answer the embedded questions.</p>
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<h3 class="hd hd-2">Video: Circular Orbits</h3>
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<h2 class="hd hd-2 unit-title">Newton's Law of Universal Gravitation</h2>
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<h2>4.2.2 Newton's Law of Universal Gravitation</h2>
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Apply Newton's Law of Universal Gravitation to describe the behavior of objects in circular orbits" type="button" id="LD">MO 4.1</button></p>
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<p>Newton's Law of Universal Gravitation states that any two bodies attract each other with a force \(F\) that is directly proportional to the product of their masses \(M_1\) and \(M_2\), and inversely proportional to the square of the distance \(R\) separating them (note that \(R\) is measured from the centers of the objects). The quantity \(G\) is the gravitational constant which is approximately equal to \(6.674 \times 10^{-11}\, \text{m}^3/\text{kg}/\text{s}^2\).</p>
<p>\[ F = \frac{G M_1 M_2}{R^{2}}\]</p>
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<h2>Interactive Simulation: Circular Orbits</h2>
How do the size, mass, and distance of orbiting bodies affect the period and velocity of a circular orbit? Explore these effects by changing various parameters in the following interactive simulation.
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<p style="text-align: center;"><em><span style="font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;">Simulation credit: Boston University, <a href="http://www.bu.edu/astronomy/visualizations/AlienWorlds/">http://www.bu.edu/astronomy/visualizations/AlienWorlds/</a><br> Used with permission.</span></em> </p>
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<h2 class="hd hd-2 unit-title">Review of Concepts: Circular Orbits</h2>
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<h2>4.2.3 Review of Concepts: Circular Orbits</h2>
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Circular Velocity
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<p>Consider two celestial objects with masses \(M_1\) and \(M_2\) as shown above. Assume that object 2 is orbiting object 1 in a circular orbit. The circular velocity \(V_c\) of object 2 depends on which of the following quantities? Check all that apply.</p>
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<input type="checkbox" name="input_e57d351b06f0443489ca684104543b48_2_1[]" id="input_e57d351b06f0443489ca684104543b48_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="e57d351b06f0443489ca684104543b48_2_1-choice_0-label" for="input_e57d351b06f0443489ca684104543b48_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_e57d351b06f0443489ca684104543b48_2_1"> \(M_1\), the mass of object 1
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<input type="checkbox" name="input_e57d351b06f0443489ca684104543b48_2_1[]" id="input_e57d351b06f0443489ca684104543b48_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="e57d351b06f0443489ca684104543b48_2_1-choice_2-label" for="input_e57d351b06f0443489ca684104543b48_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_e57d351b06f0443489ca684104543b48_2_1"> \(R\), the distance between the centers of objects 1 and 2
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Low Mars Orbit
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<p>Consider a satellite in a circular low Mars orbit, 300 km above the planetary surface. Use Newton's Law of Universal Gravitation and the concepts introduced in this section to answer the questions below. Use the following quantities in your calculations and pay close attention to unit conversions.</p>
<ul>
<li>Radius of Mars: \(R = 3396 \,\text{km}\)</li>
<li>Mass of Mars: \(M = 6.419 \times 10^{23} \,\text{kg}\)</li>
<li>Universal gravitational constant: \(G = 6.674 \times 10^{-11}\, \text{m}^3/\text{kg}/\text{s}^2\)</li>
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Mars Synchronous Orbit
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<p>Suppose that instead of a low Mars orbit, the satellite is in a synchronous orbit around Mars (i.e., has the same rotational period). In your calculations, use the constants provided in the "Low Mars Orbit" problem, as well as the fact that the rotational period of Mars is equal to 24 hours, 37 minutes, and 22 seconds, or \(P = 24.6229\,\text{hours}\). Hint: Recall the relation from the video in Subsection 4.2.1 that \(P^2 =\frac{ 4 \pi^2 R^3}{GM}\).</p>
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