{"id":27904,"date":"2025-06-20T07:51:43","date_gmt":"2025-06-20T07:51:43","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=27904"},"modified":"2025-06-20T07:51:47","modified_gmt":"2025-06-20T07:51:47","slug":"two-moons-orbit-a-planet-in-nearly-circular-orbits","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/two-moons-orbit-a-planet-in-nearly-circular-orbits\/","title":{"rendered":"Two moons orbit a planet in nearly circular orbits"},"content":{"rendered":"\n<p>Two moons orbit a planet in nearly circular orbits. Moon A has orbital radius r, and moon B has orbital radius 4r. Moon A takes 20 days to complete one orbit. How long does it take moon B to complete one orbit? Multiple Choice 160 days 80 days 20 days 320 days<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\"><strong>The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n<p>Let&#8217;s analyze the problem step-by-step:<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Given:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Moon A orbits at radius rr with period TA=20T_A = 20 days.<\/li>\n\n\n\n<li>Moon B orbits at radius 4r4r with unknown period TBT_B.<\/li>\n\n\n\n<li>Orbits are nearly circular.<\/li>\n\n\n\n<li>Question: Find TBT_B.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Understand the relationship between orbital radius and period<\/h3>\n\n\n\n<p>For moons orbiting a planet under gravity, <strong>Kepler\u2019s Third Law<\/strong> applies: T2\u221dr3T^2 \\propto r^3<\/p>\n\n\n\n<p>This means the square of the orbital period TT is proportional to the cube of the orbital radius rr.<\/p>\n\n\n\n<p>We can write the ratio for the two moons: TB2TA2=(4r)3r3=43=64\\frac{T_B^2}{T_A^2} = \\frac{(4r)^3}{r^3} = 4^3 = 64<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Solve for TBT_B<\/h3>\n\n\n\n<p>TB2202=64\u2005\u200a\u27f9\u2005\u200aTB2=64\u00d7400=25600\\frac{T_B^2}{20^2} = 64 \\implies T_B^2 = 64 \\times 400 = 25600 TB=25600=160&nbsp;daysT_B = \\sqrt{25600} = 160 \\text{ days}<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Answer: Moon B takes 160 days to complete one orbit.<\/strong><\/h3>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation in 300 words:<\/h3>\n\n\n\n<p>Two moons orbiting the same planet follow <strong>Kepler&#8217;s Third Law<\/strong>, which states that the square of the orbital period TT is proportional to the cube of the orbit\u2019s radius rr. Mathematically, this is: T2\u221dr3T^2 \\propto r^3<\/p>\n\n\n\n<p>This relationship means if you know the orbital period and radius of one moon, you can find the orbital period of another moon by comparing their radii.<\/p>\n\n\n\n<p>In this problem, Moon A orbits at radius rr with a period of 20 days. Moon B orbits farther out, at a radius 4r4r. Since the period depends on the radius raised to the power of 32\\frac{3}{2} (because T\u221dr3\/2T \\propto r^{3\/2}), increasing the radius by a factor of 4 increases the period by 43\/24^{3\/2}.<\/p>\n\n\n\n<p>Calculate this step: 43\/2=(43)1\/2=64=84^{3\/2} = (4^3)^{1\/2} = \\sqrt{64} = 8<\/p>\n\n\n\n<p>Thus, Moon B&#8217;s period is 8 times that of Moon A: TB=8\u00d720=160&nbsp;daysT_B = 8 \\times 20 = 160 \\text{ days}<\/p>\n\n\n\n<p>Therefore, Moon B takes 160 days to complete one orbit around the planet. This result illustrates how orbital periods increase significantly with orbital radius due to the gravitational dynamics dictated by Kepler\u2019s laws.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><strong>Final answer:<\/strong> 160 days<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Two moons orbit a planet in nearly circular orbits. Moon A has orbital radius r, and moon B has orbital radius 4r. Moon A takes 20 days to complete one orbit. How long does it take moon B to complete one orbit? Multiple Choice 160 days 80 days 20 days 320 days The correct answer [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-27904","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/27904","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=27904"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/27904\/revisions"}],"predecessor-version":[{"id":27905,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/27904\/revisions\/27905"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=27904"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=27904"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=27904"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}