{"id":26160,"date":"2025-06-19T10:43:00","date_gmt":"2025-06-19T10:43:00","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=26160"},"modified":"2025-06-19T10:43:02","modified_gmt":"2025-06-19T10:43:02","slug":"the-planet-earth-orbits-around-the-sun-and-also-spins-around-its-own-axis","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/the-planet-earth-orbits-around-the-sun-and-also-spins-around-its-own-axis\/","title":{"rendered":"The planet Earth orbits around the Sun and also spins around its own axis"},"content":{"rendered":"\n<p>The planet Earth orbits around the Sun and also spins around its own axis. Part (a) Calculate the angular momentum of the Earth in its orbit around the Sun in kg m^2\/s. L_orb = 2.76 \u00c3\u2014 10^40 kg m^2\/s Part (b) Calculate the angular momentum of the Earth spinning on its axis in kg m^2\/s. Part (c) How many times larger is the angular momentum of the Earth in its orbit than the angular momentum of the Earth around its axis? Grade Summary: L_orb Deductions: 000<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-light-green-cyan-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p><strong>Correct Answer:<\/strong><\/p>\n\n\n\n<p><strong>Part (a)<\/strong><br>Given:<br>Lorb=2.76\u00d71040\u2009kg\u22c5m2\/sL_{\\text{orb}} = 2.76 \\times 10^{40} \\, \\text{kg} \\cdot \\text{m}^2\/\\text{s}Lorb\u200b=2.76\u00d71040kg\u22c5m2\/s<\/p>\n\n\n\n<p><strong>Part (b): Angular momentum of the Earth spinning on its axis<\/strong><\/p>\n\n\n\n<p>To calculate the Earth&#8217;s rotational angular momentum:Lspin=I\u22c5\u03c9L_{\\text{spin}} = I \\cdot \\omegaLspin\u200b=I\u22c5\u03c9<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>III is the moment of inertia of Earth, approximated as a solid sphere: I=25MR2I = \\frac{2}{5} M R^2I=52\u200bMR2<\/li>\n\n\n\n<li>M=5.97\u00d71024\u2009kgM = 5.97 \\times 10^{24} \\, \\text{kg}M=5.97\u00d71024kg (mass of Earth)<\/li>\n\n\n\n<li>R=6.37\u00d7106\u2009mR = 6.37 \\times 10^6 \\, \\text{m}R=6.37\u00d7106m (radius of Earth)<\/li>\n\n\n\n<li>\u03c9=2\u03c0T\\omega = \\frac{2\\pi}{T}\u03c9=T2\u03c0\u200b is the angular velocity<\/li>\n\n\n\n<li>T=86400\u2009sT = 86400 \\, \\text{s}T=86400s (rotation period of Earth)<\/li>\n<\/ul>\n\n\n\n<p>Step 1: Calculate moment of inertiaI=25\u22c5(5.97\u00d71024)\u22c5(6.37\u00d7106)2=25\u22c55.97\u00d71024\u22c54.06\u00d71013=9.69\u00d71037\u2009kg\u22c5m2I = \\frac{2}{5} \\cdot (5.97 \\times 10^{24}) \\cdot (6.37 \\times 10^6)^2 = \\frac{2}{5} \\cdot 5.97 \\times 10^{24} \\cdot 4.06 \\times 10^{13} = 9.69 \\times 10^{37} \\, \\text{kg} \\cdot \\text{m}^2I=52\u200b\u22c5(5.97\u00d71024)\u22c5(6.37\u00d7106)2=52\u200b\u22c55.97\u00d71024\u22c54.06\u00d71013=9.69\u00d71037kg\u22c5m2<\/p>\n\n\n\n<p>Step 2: Calculate angular velocity\u03c9=2\u03c086400\u22487.27\u00d710\u22125\u2009rad\/s\\omega = \\frac{2\\pi}{86400} \\approx 7.27 \\times 10^{-5} \\, \\text{rad\/s}\u03c9=864002\u03c0\u200b\u22487.27\u00d710\u22125rad\/s<\/p>\n\n\n\n<p>Step 3: Calculate rotational angular momentumLspin=9.69\u00d71037\u22c57.27\u00d710\u22125\u22487.05\u00d71033\u2009kg\u22c5m2\/sL_{\\text{spin}} = 9.69 \\times 10^{37} \\cdot 7.27 \\times 10^{-5} \\approx 7.05 \\times 10^{33} \\, \\text{kg} \\cdot \\text{m}^2\/\\text{s}Lspin\u200b=9.69\u00d71037\u22c57.27\u00d710\u22125\u22487.05\u00d71033kg\u22c5m2\/s<\/p>\n\n\n\n<p><strong>Part (c): Compare the two angular momenta<\/strong>LorbLspin=2.76\u00d710407.05\u00d71033\u22483.91\u00d7106\\frac{L_{\\text{orb}}}{L_{\\text{spin}}} = \\frac{2.76 \\times 10^{40}}{7.05 \\times 10^{33}} \\approx 3.91 \\times 10^6Lspin\u200bLorb\u200b\u200b=7.05\u00d710332.76\u00d71040\u200b\u22483.91\u00d7106<\/p>\n\n\n\n<p><strong>Final Answers:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(a) Lorb=2.76\u00d71040\u2009kg\u22c5m2\/sL_{\\text{orb}} = 2.76 \\times 10^{40} \\, \\text{kg} \\cdot \\text{m}^2\/\\text{s}Lorb\u200b=2.76\u00d71040kg\u22c5m2\/s<\/li>\n\n\n\n<li>(b) Lspin\u22487.05\u00d71033\u2009kg\u22c5m2\/sL_{\\text{spin}} \\approx 7.05 \\times 10^{33} \\, \\text{kg} \\cdot \\text{m}^2\/\\text{s}Lspin\u200b\u22487.05\u00d71033kg\u22c5m2\/s<\/li>\n\n\n\n<li>(c) Orbital angular momentum is about <strong>3.91 million times larger<\/strong> than rotational angular momentum<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><strong>Explanation<\/strong><\/p>\n\n\n\n<p>Angular momentum is a physical quantity that describes how much rotation an object has. It depends on both how mass is distributed and how fast it is spinning. For Earth, we can study angular momentum in two major ways: from its motion around the Sun (orbital angular momentum) and from its rotation about its own axis (spin angular momentum).<\/p>\n\n\n\n<p>The angular momentum of Earth in its orbit around the Sun is already given as 2.76\u00d71040\u2009kg\u22c5m2\/s2.76 \\times 10^{40} \\, \\text{kg} \\cdot \\text{m}^2\/\\text{s}2.76\u00d71040kg\u22c5m2\/s. This is extremely large because the Earth travels a vast distance each year and its orbital radius is about 150 million kilometers.<\/p>\n\n\n\n<p>To calculate Earth&#8217;s spin angular momentum, we model it as a solid sphere. Its moment of inertia depends on its mass and radius. The angular velocity tells us how fast Earth spins each day, and multiplying these values gives the rotational angular momentum, which we found to be about 7.05\u00d71033\u2009kg\u22c5m2\/s7.05 \\times 10^{33} \\, \\text{kg} \\cdot \\text{m}^2\/\\text{s}7.05\u00d71033kg\u22c5m2\/s.<\/p>\n\n\n\n<p>By dividing the orbital angular momentum by the rotational one, we find that the Earth\u2019s orbital angular momentum is about 3.91 million times greater. This huge difference is mainly due to the Earth\u2019s large distance from the Sun and the speed of its orbital motion. While the Earth spins quickly, the sheer scale of its orbit leads to much greater angular momentum. This comparison helps highlight the significance of mass distribution and radius in rotational motion.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"722\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner10-64.jpeg\" alt=\"\" class=\"wp-image-26164\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner10-64.jpeg 722w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner10-64-212x300.jpeg 212w\" sizes=\"auto, (max-width: 722px) 100vw, 722px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>The planet Earth orbits around the Sun and also spins around its own axis. Part (a) Calculate the angular momentum of the Earth in its orbit around the Sun in kg m^2\/s. L_orb = 2.76 \u00c3\u2014 10^40 kg m^2\/s Part (b) Calculate the angular momentum of the Earth spinning on its axis in kg m^2\/s. [&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-26160","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/26160","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=26160"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/26160\/revisions"}],"predecessor-version":[{"id":26166,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/26160\/revisions\/26166"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=26160"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=26160"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=26160"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}