{"id":198,"date":"2023-02-15T15:01:12","date_gmt":"2023-02-15T20:01:12","guid":{"rendered":"https:\/\/opentextbc.ca\/foundationsofphysics\/?post_type=chapter&#038;p=198"},"modified":"2023-09-14T18:51:01","modified_gmt":"2023-09-14T22:51:01","slug":"gravitational-forces-and-fields","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/foundationsofphysics\/chapter\/gravitational-forces-and-fields\/","title":{"raw":"Gravitational Forces and Fields","rendered":"Gravitational Forces and Fields"},"content":{"raw":"<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Resources<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ul>\r\n \t<li>Video to Watch: <a href=\"https:\/\/www.youtube.com\/watch?v=f7MTOb8GUwk\">Mechanical Universe - Episode 8 - Apple and the Moon<\/a><\/li>\r\n \t<li>Extra Help: <a href=\"ttp:\/\/www.a-levelphysicstutor.com\/index-field.php\">A-Level Physics Tutor<\/a><\/li>\r\n \t<li>Extra Help: <a href=\"https:\/\/www.physicsclassroom.com\/class\/circles\/Lesson-3\/Gravity-is-More-Than-a-Name\">Gravity is More Than a Name<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<ul>\r\n \t<li>Article to Read: <a href=\"https:\/\/phys.org\/news\/2018-12-scientists-easier-cheaper-gravity.html\">University of Otago (2018) Scientists invent easier, cheaper way to measure gravity<\/a><\/li>\r\n \t<li>Article to Read: <a href=\"https:\/\/phys.org\/news\/2018-12-most-distant-solar.html\">Carnegie Institution for Science (2018) Outer solar system experts find \u2018far out there\u2019 dwarf planet<\/a><\/li>\r\n \t<li>Article to Read: <a href=\"https:\/\/phys.org\/news\/2018-12-nasa-reveals-saturn-worst-case-scenario.html\">NASA\u2019s Goddard Space Flight Center (2018) NASA research reveals Saturn is losing its rings at \u2018worst-case-scenario\u2019 rate<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\nEquations Introduced and Used in this Topic:\r\n<ul class=\"threecolumn\" style=\"list-style-type: none;\">\r\n \t<li>[latex]F_g=\\dfrac{Gm_1m_2}{d^2}[\/latex]<\/li>\r\n \t<li>[latex]a_g \\text{ or }g = \\dfrac{Gm}{{d}^2}[\/latex]<\/li>\r\n \t<li>[latex]\\Delta V=-\\dfrac{Gm}{d}[\/latex]<\/li>\r\n<\/ul>\r\n<p style=\"text-align: center;\">G = 6.67408(31) \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup> (round to needed accuracy)<\/p>\r\nWhere\u2026\r\n<ul>\r\n \t<li>[latex]F_g[\/latex] is the Gravitational Force of attraction, measured in newtons (N)<\/li>\r\n \t<li>[latex]G[\/latex] is Newton\u2019s Gravitational Constant, currently estimated to be 6.67408(31) \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup><\/li>\r\n \t<li>[latex]m_1[\/latex] &amp; [latex]m_2[\/latex] are the Masses of the two interacting bodies, measured in Kilograms (kg)<\/li>\r\n \t<li>[latex]a_g[\/latex] or [latex]g[\/latex] is the Gravitational Field Strength of a body, measured in metres\/second squared (m\/s2) or newtons per Kilogram (N\/kg)<\/li>\r\n \t<li>[latex]d[\/latex] is the Distance away from the Mass Centre of a Body (gravitational field) or the Distance between Mass Centres of Two Bodies (gravitational fields), measured in metres (m)<\/li>\r\n \t<li>[latex]\u2206V[\/latex] is the potential difference measured in J\/kg<\/li>\r\n<\/ul>\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Solar System Data<\/caption>\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 25%;\" scope=\"col\">Body<\/th>\r\n<th style=\"width: 25%;\" scope=\"col\">Mass (kg)<\/th>\r\n<th style=\"width: 25%;\" scope=\"col\">Size (radius ... m)<\/th>\r\n<th style=\"width: 25%;\" scope=\"col\">Orbit (radius ... m)[footnote]The radius of orbits are given around the Sun, except the moon, which is given as its orbit around the Earth.[\/footnote]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">Sun<\/td>\r\n<td style=\"width: 25%;\">1.9891 \u00d7 10<sup>30<\/sup><\/td>\r\n<td style=\"width: 25%;\">6.980 \u00d7 10<sup>8<\/sup><\/td>\r\n<td style=\"width: 25%;\">n\/a<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">Mercury<\/td>\r\n<td style=\"width: 25%;\">3.302 \u00d7 10<sup>23<\/sup><\/td>\r\n<td style=\"width: 25%;\">2.439 \u00d7 10<sup>6<\/sup><\/td>\r\n<td style=\"width: 25%;\">5.791 \u00d7 10<sup>10<\/sup><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">Venus<\/td>\r\n<td style=\"width: 25%;\">4.869 \u00d7 10<sup>24<\/sup><\/td>\r\n<td style=\"width: 25%;\">6.052 \u00d7 10<sup>6<\/sup><\/td>\r\n<td style=\"width: 25%;\">1.082 \u00d7 10<sup>11<\/sup><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">Earth<\/td>\r\n<td style=\"width: 25%;\">5.974 \u00d7 10<sup>24<\/sup><\/td>\r\n<td style=\"width: 25%;\">6.371 \u00d7 10<sup>6<\/sup><\/td>\r\n<td style=\"width: 25%;\">1.496 \u00d7 10<sup>11<\/sup><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">Mars<\/td>\r\n<td style=\"width: 25%;\">6.419 \u00d7 10<sup>23<\/sup><\/td>\r\n<td style=\"width: 25%;\">3.390 \u00d7 10<sup>6<\/sup><\/td>\r\n<td style=\"width: 25%;\">2.279 \u00d7 10<sup>11<\/sup><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">Jupiter<\/td>\r\n<td style=\"width: 25%;\">1.899 \u00d7 10<sup>27<\/sup><\/td>\r\n<td style=\"width: 25%;\">6.991 \u00d7 10<sup>7<\/sup><\/td>\r\n<td style=\"width: 25%;\">7.783 \u00d7 10<sup>11<\/sup><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">Saturn<\/td>\r\n<td style=\"width: 25%;\">5.685 \u00d7 10<sup>26<\/sup><\/td>\r\n<td style=\"width: 25%;\">5.832 \u00d7 10<sup>7<\/sup><\/td>\r\n<td style=\"width: 25%;\">1.429 \u00d7 10<sup>12<\/sup><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">Uranus<\/td>\r\n<td style=\"width: 25%;\">8.685 \u00d7 10<sup>25<\/sup><\/td>\r\n<td style=\"width: 25%;\">2.536 \u00d7 10<sup>7<\/sup><\/td>\r\n<td style=\"width: 25%;\">2.871 \u00d7 10<sup>12<\/sup><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">Neptune<\/td>\r\n<td style=\"width: 25%;\">1.028 \u00d7 10<sup>26<\/sup><\/td>\r\n<td style=\"width: 25%;\">2.462 \u00d7 10<sup>7<\/sup><\/td>\r\n<td style=\"width: 25%;\">4.504 \u00d7 10<sup>12<\/sup><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">Moon<\/td>\r\n<td style=\"width: 25%;\">7.349 \u00d7 10<sup>22<\/sup><\/td>\r\n<td style=\"width: 25%;\">1.737 \u00d7 10<sup>6<\/sup><\/td>\r\n<td style=\"width: 25%;\">3.844 \u00d7 10<sup>8<\/sup><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ul>\r\n \t<li>Solar Mass (m<sub>s<\/sub>) is the mass of the Sun. 1 Solar mass = 1.98855 \u00b1 0.00025 \u00d7 10<sup>30<\/sup> kg<\/li>\r\n \t<li>Astronomical Unit (AU) is the distance from Earth to the Sun. 1 AU = 1.49597870700 \u00d7 10<sup>11<\/sup> m<\/li>\r\n<\/ul>\r\n<h1>17.1 Gravitational Forces<\/h1>\r\n<div class=\"textbox\">\r\n<ul>\r\n \t<li>Article to Read: <a href=\"https:\/\/phys.org\/print449228257.html\">Physicists set limits on size of neutron stars<\/a><\/li>\r\n \t<li>Article to Read: <a href=\"https:\/\/www.newsweek.com\/astronomers-detect-giant-rogue-planet-floating-near-our-solar-system-1058270?spMailingID=3887126&amp;spUserID=MTI0NzM2NzgwNjYS1&amp;spJobID=1090235283&amp;spReportId=MTA5MDIzNTI4MwS2\">Astronomers Detect Giant Rouge Planet Floating Near Our Solar System<\/a><\/li>\r\n \t<li>Article to Read: <a href=\"https:\/\/www.nature.com\/articles\/d41586-018-07591-8?utm_source=briefing-dy&amp;utm_medium=email&amp;utm_campaign=briefing&amp;utm_content=20181205\">Boyle, R. (2018) These dusty young stars are changing the rules of planet-building<\/a><\/li>\r\n \t<li>Article to Read: <a href=\"https:\/\/www.theguardian.com\/science\/2019\/jan\/04\/nearby-galaxy-large-magellanic-cloud-set-to-collide-with-milky-way\">The Guardian (2019) Nearby Galaxy set to collide with Milky Way, say scientists<\/a><\/li>\r\n \t<li>Article to Read: <a href=\"https:\/\/phys.org\/news\/2019-02-universe-mass.html\">Where is the universe hiding its missing mass?<\/a><\/li>\r\n \t<li>Article to Read: <a href=\"https:\/\/www.sciencedaily.com\/releases\/2019\/04\/190416132155.htm\">GFZ GeoForschungsZentrum Potsdam (2019) What Earth\u2019s gravity reveals about climate change<\/a><\/li>\r\n \t<li>Article to Read: <a href=\"https:\/\/phys.org\/news\/2019-06-mass-anomaly-moon-largest-crater.html\">Mass anomaly detected under the moon\u2019s largest crater<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\nEquations Introduced or Used for this Section:\r\n<p style=\"text-align: center;\">[latex]F_{g}=\\dfrac{Gm_1m_2}{{d}^2}[\/latex]<\/p>\r\n\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Solar System Data<\/caption>\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 25%;\" scope=\"col\">Body<\/th>\r\n<th style=\"width: 25%;\" scope=\"col\">Mass (kg)<\/th>\r\n<th style=\"width: 25%;\" scope=\"col\">Size (radius ... m)<\/th>\r\n<th style=\"width: 25%;\" scope=\"col\">Orbit (radius ... m)<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">Sun<\/td>\r\n<td style=\"width: 25%;\">1.9891 \u00d7 10<sup>30<\/sup><\/td>\r\n<td style=\"width: 25%;\">6.980 \u00d7 10<sup>8<\/sup><\/td>\r\n<td style=\"width: 25%;\">n\/a<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">Mercury<\/td>\r\n<td style=\"width: 25%;\">3.302 \u00d7 10<sup>23<\/sup><\/td>\r\n<td style=\"width: 25%;\">2.439 \u00d7 10<sup>6<\/sup><\/td>\r\n<td style=\"width: 25%;\">5.791 \u00d7 10<sup>10<\/sup><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">Venus<\/td>\r\n<td style=\"width: 25%;\">4.869 \u00d7 10<sup>24<\/sup><\/td>\r\n<td style=\"width: 25%;\">6.052 \u00d7 10<sup>6<\/sup><\/td>\r\n<td style=\"width: 25%;\">1.082 \u00d7 10<sup>11<\/sup><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">Earth<\/td>\r\n<td style=\"width: 25%;\">5.974 \u00d7 10<sup>24<\/sup><\/td>\r\n<td style=\"width: 25%;\">6.371 \u00d7 10<sup>6<\/sup><\/td>\r\n<td style=\"width: 25%;\">1.496 \u00d7 10<sup>11<\/sup><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">Mars<\/td>\r\n<td style=\"width: 25%;\">6.419 \u00d7 10<sup>23<\/sup><\/td>\r\n<td style=\"width: 25%;\">3.390 \u00d7 10<sup>6<\/sup><\/td>\r\n<td style=\"width: 25%;\">2.279 \u00d7 10<sup>11<\/sup><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">Jupiter<\/td>\r\n<td style=\"width: 25%;\">1.899 \u00d7 10<sup>27<\/sup><\/td>\r\n<td style=\"width: 25%;\">6.991 \u00d7 10<sup>7<\/sup><\/td>\r\n<td style=\"width: 25%;\">7.783 \u00d7 10<sup>11<\/sup><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">Saturn<\/td>\r\n<td style=\"width: 25%;\">5.685 \u00d7 10<sup>26<\/sup><\/td>\r\n<td style=\"width: 25%;\">5.832 \u00d7 10<sup>7<\/sup><\/td>\r\n<td style=\"width: 25%;\">1.429 \u00d7 10<sup>12<\/sup><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">Uranus<\/td>\r\n<td style=\"width: 25%;\">8.685 \u00d7 10<sup>25<\/sup><\/td>\r\n<td style=\"width: 25%;\">2.536 \u00d7 10<sup>7<\/sup><\/td>\r\n<td style=\"width: 25%;\">2.871 \u00d7 10<sup>12<\/sup><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">Neptune<\/td>\r\n<td style=\"width: 25%;\">1.028 \u00d7 10<sup>26<\/sup><\/td>\r\n<td style=\"width: 25%;\">2.462 \u00d7 10<sup>7<\/sup><\/td>\r\n<td style=\"width: 25%;\">4.504 \u00d7 10<sup>12<\/sup><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 25%;\">Moon<\/td>\r\n<td style=\"width: 25%;\">7.349 \u00d7 10<sup>22<\/sup><\/td>\r\n<td style=\"width: 25%;\">1.737 \u00d7 10<sup>6<\/sup><\/td>\r\n<td style=\"width: 25%;\">*3.844 \u00d7 10<sup>8<\/sup><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Conceptions of Gravity and Astronomical Motion<\/h2>\r\nRecorded concepts of gravity in Antiquity centre around the Greek philosopher Aristotle (384-322 BCE). Aristotle\u2019s concept of gravity emerged from his understanding of natural motion in that all matter will rise or fall to its natural place in the universe. He contrasted this with forced or violent motion, where an object could be made to move until the agent causing it to move stopped.\u00a0 In Aristotle\u2019s paradigm, gravity was a manifestation of objects moving to simply reach their natural place in the universe.\r\n\r\nAryabhata (476-550) held the view that the Earth rotates daily on its axis, but held onto the heliocentric model of the universe rotating around the Earth, as taught by the earlier pre-Ptolomaic Greek astronomers. While a great deal of Aryabhata\u2019s work is lost, remnants indicate that he might have believed the planets move in elliptical orbits, a view that took nearly a thousand years to resurface in the works of Nicolaus Copernicus (1473-1543). There is also evidence that Aryabhata considered gravity to be an attractive force that allowed people to stand upright no matter where they were positioned on the Earth.\r\n\r\nGalileo\u2019s (1564-1642) was the first to record counter arguments to Aristotelian views on the movement of bodies to their natural place in the universe, through his experiments with the acceleration of falling bodies. Galileo\u2019s concepts of acceleration were later confirmed by the works of Francisco Maria Grimaldi (1618-1663) and Giovanni Battista Riccioli (1589-1671). It was Riccioli\u00a0 who, through his analysis of free-fall motion in his attempts to find errors in Galileo\u2019s theories, ended up finding the first measure of the acceleration of gravity. He estimated this to be 9.36 \u00b1 0.22 m\/s<sup>2<\/sup>, which is within a 5% error of today\u2019s accepted value of 9.8 m\/s<sup>2<\/sup>.\r\n\r\nIn working to quantify Robert Hooke\u2019s\u00a0 (1635-1703) suggestion that gravitational force was an inverse square law relation , Isaac Newton (1643-1727) came up with the proportionality relation:\r\n<p style=\"text-align: center;\">[latex]{F}_g\u221d \\dfrac{m_1{m}_2}{{d}^2}[\/latex]<\/p>\r\nThe first measured value of the constant G needed for Newton\u2019s equation came from the 1797 experiments by Henry Cavendish (1731-1810) . This was accomplished by Cavendish\u2019s measure of the density of the Earth, which he measured to be 5.48 times that of water. Cavendish's work led other natural philosophers to estimate a value for G to be 6.754 \u00d7 10<sup>\u221211<\/sup> N-m<sup>2<\/sup>\/kg<sup>2<\/sup> close to the current accepted value of 6.67428 \u00d7 10<sup>\u221211<\/sup> N-m<sup>2<\/sup>\/kg<sup>2<\/sup>. The Cavendish experiments are considered to be highlights of accuracy in laboratory science; his work is taught as representing a gold standard in experimental research.\r\n\r\nShown below is a sketch of the torsion apparatus that Cavendish used to measure the gravitational constant G.\r\n\r\n<img class=\"aligncenter wp-image-615\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Cavendish_Experiment.png\" alt=\"\" width=\"500\" height=\"375\" \/>\r\n<div class=\"textbox\">\r\n<ul>\r\n \t<li>Article to Read:\u00a0<a href=\"https:\/\/physicstoday.scitation.org\/doi\/full\/10.1063\/PT.3.1716\">Anatomy of a fall: Giovanni Battista Riccioli\u00a0 and the story of g<\/a><\/li>\r\n \t<li>Hooke\u2019s personality - Hooke is described as irascible, proud and prone to taking umbrage with intellectual competitors in his later years. Of note are his arguments with Isaac Newton on who should take credit over the discoveries in gravitation.<\/li>\r\n \t<li>Extra Help:\u00a0<a href=\"https:\/\/www.physicsclassroom.com\/class\/circles\/Lesson-3\/The-Apple,-the-Moon,-and-the-Inverse-Square-Law\">The Apple, the Moon and the Inverse Square Law<\/a><\/li>\r\n \t<li>Extra Help:\u00a0<a href=\"https:\/\/www.physicsclassroom.com\/class\/circles\/Lesson-3\/Cavendish-and-the-Value-of-G\">Cavendish and the Value of G<\/a><\/li>\r\n \t<li>Article to Read: <a href=\"https:\/\/sciencedemonstrations.fas.harvard.edu\/presentations\/cavendish-experiment\">Cavendish Experiment<\/a><\/li>\r\n \t<li>Article to Read: <a href=\"https:\/\/www.quora.com\/Is-string-theory-still-in-the-running-for-a-theory-of-Everything\">Is string theory still in the running for a theory of everything?<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 17.1.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat is the gravitational force of attraction between a 70 kg person resting on the Earth's surface and the Earth?\r\n\r\n<strong>Solution<\/strong>\r\n\r\nData:\r\n<ul>\r\n \t<li>[latex]F_g[\/latex] = Find<\/li>\r\n \t<li>[latex]G[\/latex] = 6.67 \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup><\/li>\r\n \t<li>[latex]m_1[\/latex] = 5.974 \u00d7 10<sup>24<\/sup> kg<\/li>\r\n \t<li>[latex]m_2[\/latex] = 70 kg<\/li>\r\n \t<li>[latex]d[\/latex] = 6.371 \u00d7 10<sup>6<\/sup> m<\/li>\r\n<\/ul>\r\nSolution:\r\n<ul>\r\n \t<li>[latex]{F}_{g}=\\dfrac{Gm_1{m}_2}{{d}^2}[\/latex]<\/li>\r\n \t<li>[latex]{F}_{g}=\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(5.974\\times10^{24}\\text{ kg})(70\\text{ kg})}{(6.371\\times10^6\\text{ m})^2}[\/latex]<\/li>\r\n \t<li>[latex]F_g[\/latex] = 690 N<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 17.1.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat is the gravitational force of attraction between the planet Mercury and the Sun?\r\n\r\n<strong>Solution<\/strong>\r\n\r\nData:\r\n<ul>\r\n \t<li>[latex]F_g[\/latex] = Find<\/li>\r\n \t<li>[latex]G[\/latex] = 6.67 \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup><\/li>\r\n \t<li>[latex]m_1[\/latex] = 1.981 \u00d7 10<sup>31<\/sup> kg<\/li>\r\n \t<li>[latex]m_2[\/latex] = 3.302 \u00d7 10<sup>23<\/sup> kg<\/li>\r\n \t<li>[latex]d[\/latex] = 5.791 \u00d7 10<sup>10<\/sup> m<\/li>\r\n<\/ul>\r\nSolution:\r\n<ul>\r\n \t<li>[latex]{F}_{g}=\\dfrac{Gm_1{m}_2}{{d}^2}[\/latex]<\/li>\r\n \t<li>[latex]{F}_{g}=\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(1.981\\times10^{31}\\text{ kg})(3.302\\times10^{23}\\text{ kg})}{(5.791\\times10^{10}\\text{ m})^2}[\/latex]<\/li>\r\n \t<li>[latex]F_g[\/latex] = 1.31 \u00d7 10<sup>22<\/sup> N<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 17.1.3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat is the gravitational force of attraction between the electron and an alpha particle separated by 18 nm? (One nanometre (nm) = 1 \u00d7 10<sup>-9<\/sup> m)\r\n\r\n<strong>Solution<\/strong>\r\n\r\nData:\r\n<ul>\r\n \t<li>[latex]F_g[\/latex] = Find<\/li>\r\n \t<li>[latex]G[\/latex] = 6.67 \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup><\/li>\r\n \t<li>[latex]m_1[\/latex] = 6.64 \u00d7 10<sup>\u221227<\/sup> kg<\/li>\r\n \t<li>[latex]m_2[\/latex] = 9.11 \u00d7 10<sup>\u221231<\/sup> kg<\/li>\r\n \t<li>[latex]d[\/latex] = 18 \u00d7 10<sup>\u22129<\/sup>\u00a0m<\/li>\r\n<\/ul>\r\nSolution:\r\n<ul>\r\n \t<li>[latex]{F}_{g}=\\dfrac{Gm_1{m}_2}{{d}^2}[\/latex]<\/li>\r\n \t<li>[latex]{F}_\\text{g}=\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(6.64\\times10^{-27}\\text{ kg})(9.11\\times10^{-31}\\text{ kg})}{(18\\times10^{-9}\\text{ m})^2}[\/latex]<\/li>\r\n \t<li>[latex]F_g[\/latex] = 1.24 \u00d7 10<sup>\u221251<\/sup> N<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 17.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>What is the gravitational force of attraction between a mass of 1.00 kg resting on the Earth's surface and the Earth?<\/li>\r\n \t<li>What is the gravitational force of attraction between 2 apples that have a mass of 500 g and 400 g separated by 12 cm on a tree branch?<\/li>\r\n \t<li>Calculate the distance separating two ships that have a mass of 750 tonnes and 500 tonnes respectively, if the gravitational force of attraction between them is 10<sup>\u22126<\/sup> newtons.<\/li>\r\n \t<li>What is the gravitational force of attraction between the electron and proton in a hydrogen atom? (The distance separating these two particles is called the Bohr radius.)<\/li>\r\n \t<li>What is the gravitational force of attraction between the planet Venus and the Sun?<\/li>\r\n \t<li>What is the maximum gravitational force of attraction between Jupiter and the Earth?<\/li>\r\n \t<li>One problem that plagues many entry level students is the comparison of the gravitational force of attraction that exists between the Earth and the Moon, and that between the Sun and the Moon. Find these two different forces of attraction and try to find an explanation why the Moon remains orbiting the Earth.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox\">Research News: <a href=\"https:\/\/www.nature.com\/magazine-assets\/d41586-018-07591-8\/d41586-018-07591-8.pdf\">Those dusty young stars are changing the rules of planet building<\/a><\/div>\r\n<h1>17.2 Gravitational Field Strength<\/h1>\r\n<div class=\"textbox\">Extra Help: <a href=\"https:\/\/www.physicsclassroom.com\/class\/circles\/Lesson-3\/The-Value-of-g\">The Value of g<\/a><\/div>\r\nEquations Introduced or Used for this Section:\r\n<p style=\"text-align: center;\">[latex]a_g \\text{ or }g=\\dfrac{Gm}{d^2}[\/latex]<\/p>\r\nIn Newtonian mechanics or Classical Mechanics (non-relativistic) an equation quantifying gravitational strength of any body can be found by equating Newton\u2019s Second Law and Newton\u2019s Law of Gravitational Force. This can be shown as follows:\r\n<p style=\"text-align: center;\">[latex]F_{\\text{net}}= ma\\text{ or }F_g = mg\\text{ is equated to }F_{g}=\\dfrac{Gm_1{m}_2}{{d}^2}[\/latex]<\/p>\r\nThis means that ...\r\n<p style=\"text-align: center;\">[latex]mg=\\dfrac{Gm_1{m}_2}{{d}^2}[\/latex]<\/p>\r\ncancelling out the common mass leaves us with ...\r\n<p style=\"text-align: center;\">[latex]g=\\dfrac{Gm_1}{{d}^2}[\/latex]<\/p>\r\nApplying the conventional negative direction to this derivation we are left with\r\n<p style=\"text-align: center;\">[latex]g=-\\dfrac{Gm_1}{{d}^2}[\/latex]<\/p>\r\nIn most classical mechanics questions this negative value is left out and the strength of gravity is just given as a scalar value.\r\n\r\nThe units of gravitational field strength can be either m\/s<sup>2<\/sup> or N\/kg.\u00a0 \u00a0This equation relates to the gravity at any distance away from a point mass and is an inverse square law relationship.\r\n<p style=\"text-align: center;\">[latex]g \u221d \\dfrac{1}{{d}^2}[\/latex]<\/p>\r\nWhen we look at the gravitational field strength as we move towards the centre of the mass, the gravitational field strength drops inversely to zero. Beneath the planet\u2019s surface gravity is an inverse linear relationship.\r\n<p style=\"text-align: center;\">[latex]g \u221d \\dfrac{1}{{d}}[\/latex]<\/p>\r\n<img class=\"aligncenter wp-image-616 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/17-g-field-e1679437176317.jpg\" alt=\"\" width=\"656\" height=\"439\" \/>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 17.2.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat should be the surface gravity of the planet Venus?\r\n\r\n<strong>Solution<\/strong>\r\n\r\nData:\r\n<ul>\r\n \t<li>[latex]a_g[\/latex] or [latex]g[\/latex] = Find<\/li>\r\n \t<li>[latex]G[\/latex] = 6.67 \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup><\/li>\r\n \t<li>[latex]m[\/latex] = 4.869 \u00d7 10<sup>24<\/sup>\u00a0kg<\/li>\r\n \t<li>[latex]d[\/latex] = 6.052 \u00d7 10<sup>6<\/sup>\u00a0m<\/li>\r\n<\/ul>\r\nSolution:\r\n<ul>\r\n \t<li>[latex]a_g \\text{ or }g = \\dfrac{Gm}{{d}^2}[\/latex]<\/li>\r\n \t<li>[latex]g=\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(4.869\\times10^{24}\\text{ kg})}{(6.052\\times10^6\\text{ m})^2}[\/latex]<\/li>\r\n \t<li>[latex]g[\/latex] = 8.87 m\/s<sup>2<\/sup>... (N\/kg)<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 17.2.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat is the acceleration due to gravity at an orbit of 280 km above the Earth's surface[footnote]The Earth is losing 50,000 tonnes of mass every single year.\r\n\r\n<strong>Solution<\/strong>\r\n\r\nThe Earth gains 40<span style=\"margin-left: 0.25em;\">000<\/span> tonnes of space dust and asteroid.\r\nThe Earth loses 16 tonnes due to the fission in the Earth\u2019s core.\r\n\r\nThe Earth loses 95<span style=\"margin-left: 0.25em;\">000<\/span> tonnes of escaped hydrogen.\r\n\r\nThe Earth loses 1600 tonnes of escaped helium.\r\n\r\nThe net loss of Earth Mass is about 0.000000000000001% each year.\r\n\r\nSource - <a href=\"https:\/\/scitechdaily.com\/earth-loses-50000-tonnes-of-mass-every-year\">Earth loses 50,000 Tonnes of Mass Every Year<\/a>[\/footnote]?\r\n\r\nNow... [latex]d[\/latex] = radius of the Earth + distance to orbit\r\n<ul>\r\n \t<li>[latex]d[\/latex] = 280 km + 6.371 \u00d7 10<sup>6<\/sup> m or 6.651 \u00d7 10<sup>6<\/sup> m<\/li>\r\n<\/ul>\r\n<img class=\"alignright wp-image-617\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex17.2.2-e1679438124436.jpg\" alt=\"\" width=\"300\" height=\"226\" \/>Finally ...\r\n<ul>\r\n \t<li>[latex]a_g[\/latex] or [latex]g[\/latex] = Find<\/li>\r\n \t<li>[latex]G[\/latex] = 6.67 \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup><\/li>\r\n \t<li>[latex]m[\/latex] = 5.974 \u00d7 10<sup>24<\/sup>\u00a0kg<\/li>\r\n \t<li>[latex]d[\/latex] = 6.651 \u00d7 10<sup>6<\/sup>\u00a0m<\/li>\r\n<\/ul>\r\nSolution:\r\n<ul>\r\n \t<li>[latex]a_g \\text{ or }g = \\dfrac{Gm}{{d}^2}[\/latex]<\/li>\r\n \t<li>[latex]g=\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(5.974\\times10^{24}\\text{ kg})}{(6.651\\times10^6\\text{ m})^2}[\/latex]<\/li>\r\n \t<li>[latex]g[\/latex] = 9.01 m\/s<sup>2<\/sup> ... (N\/kg)<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 17.2.3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIf a rocket is orbiting at a distance from Earth where the gravity is 0.90 gs, how much further away must it move away from the Earth to experience a gravity of 0.42 gs?\r\n\r\n<strong>Solution<\/strong>\r\n\r\nFirst: d for 0.90 g\u2019s\r\n<ul>\r\n \t<li>[latex]a_g[\/latex] or [latex]g[\/latex] = 0.90 g\u2019s<\/li>\r\n \t<li>[latex]G[\/latex] = 6.67 \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup><\/li>\r\n \t<li>[latex]m[\/latex] = 5.974 \u00d7 10<sup>24<\/sup>\u00a0kg<\/li>\r\n \t<li>[latex]d[\/latex] = Find<\/li>\r\n<\/ul>\r\nSolution:\r\n<ul>\r\n \t<li>[latex]a_g \\text{ or }g = \\dfrac{Gm}{{d}^2}[\/latex]<\/li>\r\n \t<li>[latex](0.90)(9.80\\text{ m\/s}^2)=\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(5.974\\times10^{24}\\text{ kg})}{d^2}[\/latex]<\/li>\r\n \t<li>[latex]\\text{d}^2=\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(5.974\\times10^{24}\\text{ kg})}{(0.90)(9.80\\text{ m\/s}^2)}[\/latex]<\/li>\r\n \t<li>[latex]d^2[\/latex]\u00a0 = 4.52 \u00d7 10<sup>13<\/sup> m<sup>2<\/sup><\/li>\r\n \t<li>[latex]d[\/latex] = 6.72 \u00d7 10<sup>6<\/sup> m<\/li>\r\n<\/ul>\r\nSecond: d for 0.42 g\u2019s\r\n<ul>\r\n \t<li>[latex]a_g[\/latex] or [latex]g[\/latex] = 0.42 g\u2019s<\/li>\r\n \t<li>[latex]G[\/latex] = 6.67 \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup><\/li>\r\n \t<li>[latex]m[\/latex] = 5.974 \u00d7 10<sup>24<\/sup>\u00a0kg<\/li>\r\n \t<li>[latex]d[\/latex] = Find<\/li>\r\n<\/ul>\r\nSolution:\r\n<ul>\r\n \t<li>[latex]a_g \\text{ or }g = \\dfrac{Gm}{{d}^2}[\/latex]<\/li>\r\n \t<li>[latex](0.42)(9.80\\text{ m\/s}^2)=\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(5.974\\times10^{24}\\text{ kg})}{d^2}[\/latex]<\/li>\r\n \t<li>[latex]{d}^2=\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(5.974\\times10^{24}\\text{ kg})}{(0.42)(9.80\\text{ m\/s}^2)}[\/latex]<\/li>\r\n \t<li>[latex]d^2[\/latex]\u00a0 = 9.69 \u00d7 10<sup>13<\/sup> m<sup>2<\/sup><\/li>\r\n \t<li>[latex]d[\/latex] = 9.84 \u00d7 10<sup>6<\/sup> m<\/li>\r\n<\/ul>\r\nFinally...\r\n<ul>\r\n \t<li>[latex]\\Delta d[\/latex] = d<sub>90<\/sub> \u2212 d<sub>42<\/sub><\/li>\r\n \t<li>[latex]\\Delta d[\/latex] = 9.84 \u00d7 10<sup>6<\/sup> m \u2212 6.72 \u00d7 10<sup>6<\/sup> m<\/li>\r\n \t<li>[latex]\\Delta d[\/latex] = 3.12 \u00d7 10<sup>6<\/sup> m<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 17.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>What is the surface gravity of a 100 kg pumpkin of radius 0.75 m?<\/li>\r\n \t<li>What should be the surface gravity of the planet Uranus?<\/li>\r\n \t<li>What would be the surface gravity at the outer edge of the Sun?<\/li>\r\n \t<li>What is the acceleration due to gravity at an orbit of 300 km above the Earth's surface? This distance is termed the Low Earth Orbital (LEO). Ships orbiting at this distance need to be boosted higher in orbit at regular intervals due to air resistance from the very weak atmosphere that exists at this distance.\u00a0 (Atmospheric pressure at this distance averages 10<sup>\u22128<\/sup> Pa, compared to 1.013 \u00d7 10<sup>5<\/sup> Pa at the Earth\u2019s surface.)<\/li>\r\n \t<li>What acceleration does the Moon experience due to the Earth's gravity, in its orbit around the Earth? What acceleration does the Earth experience due to the Moon's gravity?<\/li>\r\n \t<li>At what distance away from the Earth has the net acceleration due to gravity fallen to \u00bd g?<\/li>\r\n \t<li>What is the acceleration due to gravity experienced by an Earth orbiting satellite at a distance of 1 Earth radius above the surface?<\/li>\r\n \t<li>What distance above the Earth\u2019s surface must one travel to experience a reduced Earth gravity of 0.7 g\u2019s?<\/li>\r\n \t<li>If a shuttle is orbiting at a distance from Earth where the gravity is 0.8 gs, how much farther away must it move from the Earth to experience a gravity of 0.6 gs?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h1>17.3 Gravitational Potential<\/h1>\r\n<div class=\"textbox\">\r\n\r\nArticle to Read: <a href=\"https:\/\/www.scientificamerican.com\/article\/gravitational-waves-reveal-the-hearts-of-neutron-stars1\/\">Gravitational Waves Reveal the Hearts of Neutron Stars<\/a>\r\n\r\n<\/div>\r\nEquations Introduced or Used for this Section:\r\n<p style=\"text-align: center;\">[latex]\u0394V =-\\dfrac{Gm_1}{d}[\/latex]<\/p>\r\n\r\n\r\n[caption id=\"attachment_618\" align=\"aligncenter\" width=\"400\"]<img class=\"wp-image-618\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/GravityPotential.jpg\" alt=\"\" width=\"400\" height=\"232\" \/> Plot of a two-dimensional slice of the gravitational potential in and around a uniform spherical body. The inflection points of the cross-section are at the surface of the body.[\/caption]\r\n\r\n<div class=\"textbox textbox--sidebar\">Video to Watch: <a href=\"https:\/\/www.youtube.com\/watch?v=dngQetgjXBk\">Gravitational Potential and Gravitational Potential\u00a0 Energy, explained<\/a><\/div>\r\nIn classical mechanics, the gravitational potential at any location equals the work per unit of mass needed and moves the object from a fixed reference position to a final position within or outside a gravitational field. Typically, the gravitational potential refers to some final position infinitely far away from the gravitational source. As a result, at an infinite distance away the potential energy of any object in relation to its source position is zero. (Epf = 0). Also note that the movement that changes an object\u2019s potential energy must be either toward or away from the gravitational source.\r\n\r\nIf we look at the work needed to move an object away from a gravitational source, we start by using the conservation of mechanical energy ... Specifically:\r\n<p style=\"text-align: center;\">[latex]W = \u2206\\text{Energy}[\/latex]... Since the [latex]\u2206 E[\/latex] is a change in potential energy... [latex]W = E_{pf} \u2212E_{pi}[\/latex]<\/p>\r\nThis can be simplified for cases where the object is moved to an infinite distance away to\r\n<p style=\"text-align: center;\">[latex]W = - E_{pi}\\text{ since }E_{pf} = 0[\/latex]<\/p>\r\n<img class=\"aligncenter size-full wp-image-619\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/17.3-graph.jpg\" alt=\"\" width=\"504\" height=\"268\" \/>\r\n\r\nWe can expand [latex]W = - E_{pi}[\/latex] to [latex]W = - m g h_i[\/latex] and replace gravity with [latex]g= - \\dfrac{Gm_1}{d}[\/latex]\r\n\r\nThis yields: [latex]W= - m\\dfrac{Gm_1}{d}{h}_{i}[\/latex]... And since h<sub>i<\/sub> and d<sub>2<\/sub> are measuring the same distance, we can cancel common terms.\r\n\r\nThis leaves us with [latex]W=-\\dfrac{Gm_1{m}_2}{d}[\/latex]\r\n\r\nThe derivation of gravitational potential is almost complete. What remains is to isolate the second mass m<sub>2<\/sub>.\r\n\r\nIsolating m<sub>2<\/sub> from the main equation leaves us with: [latex]W=-\\dfrac{Gm_1}{d}({m}_2)[\/latex]\r\n\r\nThe symbol we will use for gravitational potential is the same as the one we use for electric potential. Gravitational potential (\u2206V) where [latex]\\Delta V=-\\dfrac{Gm_1}{d}[\/latex]\r\n\r\nIn this case, the work done (joules) to move an object infinitely away from a measurable gravitational field is found by the equation:\r\n<p style=\"text-align: center;\">[latex]W=-\\dfrac{Gm_1}{d}({m}_2)[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 or\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]V=\u2212\\Delta V (m_2)[\/latex]<\/p>\r\nLikewise the gravitational potential energy of an object removed from the influence of a planetary or sun\u2019s gravitational field is also found by:\r\n<p style=\"text-align: center;\">[latex]\\Delta E_p=-\\dfrac{Gm_1}{d}({m}_2)[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 or\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\Delta E_p=-\\Delta V (m_2)[\/latex]<\/p>\r\nHow do we understand the negative value that shows up in these equations?\r\n<ul>\r\n \t<li>As you move away from a gravitational field, you will need energy to escape. You can consider this energy to be lost.<\/li>\r\n \t<li>As you move into a gravitational field, you will be gaining energy, which is added to your mass.<\/li>\r\n<\/ul>\r\nEquation Summary: To move any mass away from a gravitational source, use:\r\n<p style=\"text-align: center;\">[latex]W=-\\dfrac{Gm_1{m}_2}{d}[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 or\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 [latex]\\Delta E_p=-\\dfrac{Gm_1{m}_2}{d}[\/latex]<\/p>\r\nUsing \u2206V = [latex]-\\dfrac{Gm_1}{d}[\/latex] the above equations shorten to:\r\n<p style=\"text-align: center;\">[latex]W=-\\Delta Vm_2[\/latex]<sub>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/sub>or\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 [latex]\\Delta E_p=-\\Delta Vm_2[\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 17.3.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat is the gravitational potential at an orbital distance of 280 km above the Earth\u2019s surface?\r\n\r\n<strong>Solution<\/strong>\r\n\r\nData:\r\n<ul>\r\n \t<li>[latex]\\Delta V[\/latex] = Find<\/li>\r\n \t<li>[latex]d[\/latex] = 280 km + 6.371 \u00d7 10<sup>6<\/sup> m or 6.651 \u00d7 10<sup>6<\/sup> m<\/li>\r\n \t<li>[latex]m[\/latex] = 5.974 \u00d7 10<sup>24<\/sup>\u00a0kg<\/li>\r\n \t<li>[latex]G[\/latex] = 6.67 \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup><\/li>\r\n<\/ul>\r\nSolution:\r\n<ul>\r\n \t<li>[latex]\\Delta V=-\\dfrac{Gm_1}{d}[\/latex]<\/li>\r\n \t<li>[latex]\\Delta V=\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(5.974\\times10^{24}\\text{ kg})}{6.651\\times10^6\\text{ m}}[\/latex]<\/li>\r\n \t<li>[latex]\\Delta V[\/latex]= \u22125.99 \u00d7 10<sup style=\"word-spacing: normal;\">7<\/sup><span style=\"font-size: 0.9em; word-spacing: normal;\"> J\/kg<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 17.3.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nHow much work must be done for the space shuttle (mass = 2030 t) orbiting 280 km above the Earth\u2019s surface to permanently escape the Earth?\r\n\r\n<strong>Solution<\/strong>\r\n\r\nData:\r\n<ul>\r\n \t<li>[latex]W[\/latex] = Find<\/li>\r\n \t<li>[latex]d[\/latex] = 280 km + 6.371 \u00d7 10<sup>6<\/sup> m or 6.651 \u00d7 10<sup>6<\/sup> m<\/li>\r\n \t<li>[latex]m_1[\/latex] = 6.419\u00a0\u00d7 10<sup>23<\/sup>\u00a0kg<\/li>\r\n \t<li>[latex]m_2[\/latex] = 2030 \u00d7 10<sup>3<\/sup>\u00a0kg<\/li>\r\n \t<li>[latex]G[\/latex] = 6.67 \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup><\/li>\r\n<\/ul>\r\nSolution:\r\n<ul>\r\n \t<li>[latex]W[\/latex] = [latex]-\\dfrac{Gm_1{m}_2}{d}[\/latex]<\/li>\r\n \t<li>[latex]W[\/latex] = [latex]-\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(5.974\\times10^{24}\\text{ kg})(2030\\times10^3\\text{ kg})}{6.651\\times10^6\\text{ m}}[\/latex]<\/li>\r\n \t<li>[latex]W[\/latex] = 1.22 \u00d7 10<sup>14<\/sup> J<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 17.3.3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat is the gravitational potential of a rocket ship sitting on the surface of the planet Mars?\r\n\r\n<strong>Solution<\/strong>\r\n\r\nData:\r\n<ul>\r\n \t<li>[latex]\\Delta V[\/latex] = Find<\/li>\r\n \t<li>[latex]m_1[\/latex] = 6.419\u00a0\u00d7 10<sup>23<\/sup>\u00a0kg<\/li>\r\n \t<li>[latex]d[\/latex] = 3.390\u00a0\u00d7 10<sup>6<\/sup> m<\/li>\r\n \t<li>[latex]G[\/latex] = 6.67 \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup><\/li>\r\n<\/ul>\r\nSolution:\r\n<ul>\r\n \t<li>[latex]\\Delta V=-\\dfrac{Gm_1}{d}[\/latex]<\/li>\r\n \t<li>[latex]\\Delta V=\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(6.419\\times10^{23}\\text{ kg})}{3.390\\times10^6\\text{ m}}[\/latex]<\/li>\r\n \t<li>[latex]\\Delta V[\/latex] = \u22121.26 \u00d7 10<sup>7<\/sup> J\/kg<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 17.3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>What is the gravitational potential at an orbital distance of 300 km above the Earth\u2019s surface?<\/li>\r\n \t<li>How much work must be done for Elon Musk\u2019s 2008 Tesla Roadster (mass = 1300 kg) at an orbital distance of 300 km above the Earth\u2019s surface to permanently escape the Earth?<\/li>\r\n \t<li>What is the gravitational potential of a rocket ship sitting on the moon\u2019s surface?<\/li>\r\n \t<li>What is the gravitational potential for a rocket launched from the Earth\u2019s orbit around the Sun to escape the Sun\u2019s gravitational field?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h1>17.4.1 Earth-Moon Gravity Neutral Point<\/h1>\r\nBetween the Earth and the Moon, there exists a \u201cfluctuating\u201d position where the gravitational fields of the Earth and Moon balance each other out. This position is defined as the Earth-Moon Gravity Neutral Point.\u00a0 Described by an equation, this position can be found from\r\n<p style=\"text-align: center;\">[latex]g_{\\text{ Earth}}= g_{\\text{ Moon}}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">where [latex]g=\\dfrac{GM}{{d}^2}[\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--sidebar\">Article to Read: <a href=\"https:\/\/www.madsci.org\/posts\/archives\/1998-08\/899933769.As.r.html\">Re: What is the Earth Moon neutral point?<\/a> by David Ellis<\/div>\r\nTechnically, this position is where the gravitational fields from both the Moon and the Earth balance each other out, meaning that any object in this position having no velocity relative to the Earth and Moon should stay in that position without being pulled to the Earth or the Moon. Should any spaceship reach this position headed to Earth or the Moon, it would then be able to fall to either side of this position without using any more fuel for thrust. This can be looked at as a moving vehicle at the top of a hill ... If the vehicle is moving in any direction down the hill when at the top, then the vehicle will be able to roll down hill without using any fuel. If the vehicle is stopped at the top of the hill, then it should be able to remain stationary unless some external unbalanced force acts upon it, which would then act to nudge the vehicle downhill in one of the two directions. This position is termed fluctuating because of a number of factors which are described by NASA\u2019s David Ellis.\r\n\r\nThe gravity neutral position is visually shown by mapping the variable strength of the Earth\u2019s gravitational field as one heads to the Moon, overlaid by the variable strength of the Moon\u2019s gravitational field as one heads back to the Earth. The intersection of these two gravitational fields\u2019 lines marks the approximate position. Due to fluctuations in the location of this position, any spaceship parked there would need corrections by booster rockets to remain in the region of neutral gravity.\r\n\r\n<img class=\"wp-image-1220  aligncenter\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Earth-moon-gravitational-potential.svg_.png\" alt=\"\" width=\"754\" height=\"566\" \/>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 17.4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>Using your equations for the gravitational fields of the Earth and the Moon, calculate the distance from the Earth that this point exists.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h1>Exercise Answers<\/h1>\r\n<h2>17.1 Gravitational Force<\/h2>\r\n<ol class=\"twocolumn\">\r\n \t<li>\u00a09.82 N<\/li>\r\n \t<li>9.3 \u00d7 10<sup>\u221210<\/sup> N<\/li>\r\n \t<li>5000 m or 5.00 km<\/li>\r\n \t<li>3.63 \u00d7 10<sup>\u221247<\/sup> N<\/li>\r\n \t<li>5.52 \u00d7 10<sup>22<\/sup> N<\/li>\r\n \t<li>1.91 \u00d7 10<sup>18<\/sup> N<\/li>\r\n \t<li>Earth-Moon: 2.02 \u00d7 10<sup>20<\/sup> N,\r\nSun-Moon: 4.39 \u00d7 10<sup>20<\/sup> N<\/li>\r\n<\/ol>\r\n<h2>17.2 Gravitational Field Strength<\/h2>\r\n<ol class=\"twocolumn\">\r\n \t<li>1.2 \u00d7 10<sup>\u22128<\/sup> m\/s<sup>2 <\/sup><\/li>\r\n \t<li>9.01 m\/s<sup>2\u00a0<\/sup><\/li>\r\n \t<li>272 m\/s<sup>2 <\/sup><\/li>\r\n \t<li>8.96 m\/s<sup>2<\/sup><\/li>\r\n \t<li>Moon: 2.70 \u00d7 10<sup>\u22123<\/sup> m\/s<sup>2<\/sup>\r\nEarth: 3.32 \u00d7 10<sup>\u22125<\/sup> m\/s<sup>2<\/sup><\/li>\r\n \t<li>9.01 \u00d7 10<sup>6<\/sup> m from the Earth's centre or 2650 km above the surface<\/li>\r\n \t<li>2.45 m\/s<sup>2<\/sup> or 1\/4 g's<\/li>\r\n \t<li>1.25 \u00d7 10<sup>6<\/sup> m\u00a0 or 1250 km<\/li>\r\n \t<li>3.49 \u00d7 10<sup>5<\/sup> m<\/li>\r\n<\/ol>\r\n<h2>17.3 Gravitational Potential<\/h2>\r\n<ol class=\"twocolumn\">\r\n \t<li>[latex]\\Delta V[\/latex] = \u2212 5.97 \u00d7 10<sup>4<\/sup> J\/kg<\/li>\r\n \t<li>[latex]W[\/latex] = \u2212 7.77 \u00d7 10<sup>10<\/sup> J<\/li>\r\n \t<li>[latex]\\Delta V[\/latex] = \u2212 2.82 \u00d7 10<sup>6<\/sup> J\/kg<\/li>\r\n \t<li>[latex]\\Delta V[\/latex] = \u2212 8.87 \u00d7 10<sup>8<\/sup> J\/kg<\/li>\r\n<\/ol>\r\n<h2>17.4.1 Earth-Moon Gravity Neutral Point<\/h2>\r\n<ol>\r\n \t<li>Distance from the Earth = 3.46 \u00d7 10<sup>8<\/sup> m &amp; Distance from the Moon = 3.84 \u00d7 10<sup>7<\/sup> m<\/li>\r\n<\/ol>\r\n<h3>Media Attributions<\/h3>\r\n<ul>\r\n \t<li>\"<a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Cavendish_Experiment.png\">Cavendish Experiment<\/a>\" by Henry Cavendish is in the <a href=\"https:\/\/creativecommons.org\/share-your-work\/public-domain\/pdm\/\">public domain<\/a>.<\/li>\r\n \t<li>\"<a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:GravityPotential.jpg\">Gravity Potential<\/a>\" by AllenMcC. is licensed under a <a href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/3.0\/deed.en\">CC BY-SA 3.0<\/a> licence.<\/li>\r\n \t<li>\r\n<p id=\"firstHeading\" class=\"firstHeading mw-first-heading\">\"<a href=\"https:\/\/en.wikipedia.org\/wiki\/File:Earth-moon-gravitational-potential.svg\">Earth moon gravitational potential<\/a>\" by MikeRun is licensed under a <a href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/3.0\/deed.en\">CC BY-SA 3.0<\/a> licence.<\/p>\r\n<\/li>\r\n<\/ul>","rendered":"<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Resources<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n<li>Video to Watch: <a href=\"https:\/\/www.youtube.com\/watch?v=f7MTOb8GUwk\">Mechanical Universe &#8211; Episode 8 &#8211; Apple and the Moon<\/a><\/li>\n<li>Extra Help: <a href=\"denied:ttp:\/\/www.a-levelphysicstutor.com\/index-field.php\">A-Level Physics Tutor<\/a><\/li>\n<li>Extra Help: <a href=\"https:\/\/www.physicsclassroom.com\/class\/circles\/Lesson-3\/Gravity-is-More-Than-a-Name\">Gravity is More Than a Name<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<ul>\n<li>Article to Read: <a href=\"https:\/\/phys.org\/news\/2018-12-scientists-easier-cheaper-gravity.html\">University of Otago (2018) Scientists invent easier, cheaper way to measure gravity<\/a><\/li>\n<li>Article to Read: <a href=\"https:\/\/phys.org\/news\/2018-12-most-distant-solar.html\">Carnegie Institution for Science (2018) Outer solar system experts find \u2018far out there\u2019 dwarf planet<\/a><\/li>\n<li>Article to Read: <a href=\"https:\/\/phys.org\/news\/2018-12-nasa-reveals-saturn-worst-case-scenario.html\">NASA\u2019s Goddard Space Flight Center (2018) NASA research reveals Saturn is losing its rings at \u2018worst-case-scenario\u2019 rate<\/a><\/li>\n<\/ul>\n<\/div>\n<p>Equations Introduced and Used in this Topic:<\/p>\n<ul class=\"threecolumn\" style=\"list-style-type: none;\">\n<li>[latex]F_g=\\dfrac{Gm_1m_2}{d^2}[\/latex]<\/li>\n<li>[latex]a_g \\text{ or }g = \\dfrac{Gm}{{d}^2}[\/latex]<\/li>\n<li>[latex]\\Delta V=-\\dfrac{Gm}{d}[\/latex]<\/li>\n<\/ul>\n<p style=\"text-align: center;\">G = 6.67408(31) \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup> (round to needed accuracy)<\/p>\n<p>Where\u2026<\/p>\n<ul>\n<li>[latex]F_g[\/latex] is the Gravitational Force of attraction, measured in newtons (N)<\/li>\n<li>[latex]G[\/latex] is Newton\u2019s Gravitational Constant, currently estimated to be 6.67408(31) \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup><\/li>\n<li>[latex]m_1[\/latex] &amp; [latex]m_2[\/latex] are the Masses of the two interacting bodies, measured in Kilograms (kg)<\/li>\n<li>[latex]a_g[\/latex] or [latex]g[\/latex] is the Gravitational Field Strength of a body, measured in metres\/second squared (m\/s2) or newtons per Kilogram (N\/kg)<\/li>\n<li>[latex]d[\/latex] is the Distance away from the Mass Centre of a Body (gravitational field) or the Distance between Mass Centres of Two Bodies (gravitational fields), measured in metres (m)<\/li>\n<li>[latex]\u2206V[\/latex] is the potential difference measured in J\/kg<\/li>\n<\/ul>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Solar System Data<\/caption>\n<tbody>\n<tr>\n<th style=\"width: 25%;\" scope=\"col\">Body<\/th>\n<th style=\"width: 25%;\" scope=\"col\">Mass (kg)<\/th>\n<th style=\"width: 25%;\" scope=\"col\">Size (radius &#8230; m)<\/th>\n<th style=\"width: 25%;\" scope=\"col\">Orbit (radius &#8230; m)<a class=\"footnote\" title=\"The radius of orbits are given around the Sun, except the moon, which is given as its orbit around the Earth.\" id=\"return-footnote-198-1\" href=\"#footnote-198-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/th>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">Sun<\/td>\n<td style=\"width: 25%;\">1.9891 \u00d7 10<sup>30<\/sup><\/td>\n<td style=\"width: 25%;\">6.980 \u00d7 10<sup>8<\/sup><\/td>\n<td style=\"width: 25%;\">n\/a<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">Mercury<\/td>\n<td style=\"width: 25%;\">3.302 \u00d7 10<sup>23<\/sup><\/td>\n<td style=\"width: 25%;\">2.439 \u00d7 10<sup>6<\/sup><\/td>\n<td style=\"width: 25%;\">5.791 \u00d7 10<sup>10<\/sup><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">Venus<\/td>\n<td style=\"width: 25%;\">4.869 \u00d7 10<sup>24<\/sup><\/td>\n<td style=\"width: 25%;\">6.052 \u00d7 10<sup>6<\/sup><\/td>\n<td style=\"width: 25%;\">1.082 \u00d7 10<sup>11<\/sup><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">Earth<\/td>\n<td style=\"width: 25%;\">5.974 \u00d7 10<sup>24<\/sup><\/td>\n<td style=\"width: 25%;\">6.371 \u00d7 10<sup>6<\/sup><\/td>\n<td style=\"width: 25%;\">1.496 \u00d7 10<sup>11<\/sup><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">Mars<\/td>\n<td style=\"width: 25%;\">6.419 \u00d7 10<sup>23<\/sup><\/td>\n<td style=\"width: 25%;\">3.390 \u00d7 10<sup>6<\/sup><\/td>\n<td style=\"width: 25%;\">2.279 \u00d7 10<sup>11<\/sup><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">Jupiter<\/td>\n<td style=\"width: 25%;\">1.899 \u00d7 10<sup>27<\/sup><\/td>\n<td style=\"width: 25%;\">6.991 \u00d7 10<sup>7<\/sup><\/td>\n<td style=\"width: 25%;\">7.783 \u00d7 10<sup>11<\/sup><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">Saturn<\/td>\n<td style=\"width: 25%;\">5.685 \u00d7 10<sup>26<\/sup><\/td>\n<td style=\"width: 25%;\">5.832 \u00d7 10<sup>7<\/sup><\/td>\n<td style=\"width: 25%;\">1.429 \u00d7 10<sup>12<\/sup><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">Uranus<\/td>\n<td style=\"width: 25%;\">8.685 \u00d7 10<sup>25<\/sup><\/td>\n<td style=\"width: 25%;\">2.536 \u00d7 10<sup>7<\/sup><\/td>\n<td style=\"width: 25%;\">2.871 \u00d7 10<sup>12<\/sup><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">Neptune<\/td>\n<td style=\"width: 25%;\">1.028 \u00d7 10<sup>26<\/sup><\/td>\n<td style=\"width: 25%;\">2.462 \u00d7 10<sup>7<\/sup><\/td>\n<td style=\"width: 25%;\">4.504 \u00d7 10<sup>12<\/sup><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">Moon<\/td>\n<td style=\"width: 25%;\">7.349 \u00d7 10<sup>22<\/sup><\/td>\n<td style=\"width: 25%;\">1.737 \u00d7 10<sup>6<\/sup><\/td>\n<td style=\"width: 25%;\">3.844 \u00d7 10<sup>8<\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ul>\n<li>Solar Mass (m<sub>s<\/sub>) is the mass of the Sun. 1 Solar mass = 1.98855 \u00b1 0.00025 \u00d7 10<sup>30<\/sup> kg<\/li>\n<li>Astronomical Unit (AU) is the distance from Earth to the Sun. 1 AU = 1.49597870700 \u00d7 10<sup>11<\/sup> m<\/li>\n<\/ul>\n<h1>17.1 Gravitational Forces<\/h1>\n<div class=\"textbox\">\n<ul>\n<li>Article to Read: <a href=\"https:\/\/phys.org\/print449228257.html\">Physicists set limits on size of neutron stars<\/a><\/li>\n<li>Article to Read: <a href=\"https:\/\/www.newsweek.com\/astronomers-detect-giant-rogue-planet-floating-near-our-solar-system-1058270?spMailingID=3887126&amp;spUserID=MTI0NzM2NzgwNjYS1&amp;spJobID=1090235283&amp;spReportId=MTA5MDIzNTI4MwS2\">Astronomers Detect Giant Rouge Planet Floating Near Our Solar System<\/a><\/li>\n<li>Article to Read: <a href=\"https:\/\/www.nature.com\/articles\/d41586-018-07591-8?utm_source=briefing-dy&amp;utm_medium=email&amp;utm_campaign=briefing&amp;utm_content=20181205\">Boyle, R. (2018) These dusty young stars are changing the rules of planet-building<\/a><\/li>\n<li>Article to Read: <a href=\"https:\/\/www.theguardian.com\/science\/2019\/jan\/04\/nearby-galaxy-large-magellanic-cloud-set-to-collide-with-milky-way\">The Guardian (2019) Nearby Galaxy set to collide with Milky Way, say scientists<\/a><\/li>\n<li>Article to Read: <a href=\"https:\/\/phys.org\/news\/2019-02-universe-mass.html\">Where is the universe hiding its missing mass?<\/a><\/li>\n<li>Article to Read: <a href=\"https:\/\/www.sciencedaily.com\/releases\/2019\/04\/190416132155.htm\">GFZ GeoForschungsZentrum Potsdam (2019) What Earth\u2019s gravity reveals about climate change<\/a><\/li>\n<li>Article to Read: <a href=\"https:\/\/phys.org\/news\/2019-06-mass-anomaly-moon-largest-crater.html\">Mass anomaly detected under the moon\u2019s largest crater<\/a><\/li>\n<\/ul>\n<\/div>\n<p>Equations Introduced or Used for this Section:<\/p>\n<p style=\"text-align: center;\">[latex]F_{g}=\\dfrac{Gm_1m_2}{{d}^2}[\/latex]<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Solar System Data<\/caption>\n<tbody>\n<tr>\n<th style=\"width: 25%;\" scope=\"col\">Body<\/th>\n<th style=\"width: 25%;\" scope=\"col\">Mass (kg)<\/th>\n<th style=\"width: 25%;\" scope=\"col\">Size (radius &#8230; m)<\/th>\n<th style=\"width: 25%;\" scope=\"col\">Orbit (radius &#8230; m)<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">Sun<\/td>\n<td style=\"width: 25%;\">1.9891 \u00d7 10<sup>30<\/sup><\/td>\n<td style=\"width: 25%;\">6.980 \u00d7 10<sup>8<\/sup><\/td>\n<td style=\"width: 25%;\">n\/a<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">Mercury<\/td>\n<td style=\"width: 25%;\">3.302 \u00d7 10<sup>23<\/sup><\/td>\n<td style=\"width: 25%;\">2.439 \u00d7 10<sup>6<\/sup><\/td>\n<td style=\"width: 25%;\">5.791 \u00d7 10<sup>10<\/sup><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">Venus<\/td>\n<td style=\"width: 25%;\">4.869 \u00d7 10<sup>24<\/sup><\/td>\n<td style=\"width: 25%;\">6.052 \u00d7 10<sup>6<\/sup><\/td>\n<td style=\"width: 25%;\">1.082 \u00d7 10<sup>11<\/sup><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">Earth<\/td>\n<td style=\"width: 25%;\">5.974 \u00d7 10<sup>24<\/sup><\/td>\n<td style=\"width: 25%;\">6.371 \u00d7 10<sup>6<\/sup><\/td>\n<td style=\"width: 25%;\">1.496 \u00d7 10<sup>11<\/sup><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">Mars<\/td>\n<td style=\"width: 25%;\">6.419 \u00d7 10<sup>23<\/sup><\/td>\n<td style=\"width: 25%;\">3.390 \u00d7 10<sup>6<\/sup><\/td>\n<td style=\"width: 25%;\">2.279 \u00d7 10<sup>11<\/sup><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">Jupiter<\/td>\n<td style=\"width: 25%;\">1.899 \u00d7 10<sup>27<\/sup><\/td>\n<td style=\"width: 25%;\">6.991 \u00d7 10<sup>7<\/sup><\/td>\n<td style=\"width: 25%;\">7.783 \u00d7 10<sup>11<\/sup><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">Saturn<\/td>\n<td style=\"width: 25%;\">5.685 \u00d7 10<sup>26<\/sup><\/td>\n<td style=\"width: 25%;\">5.832 \u00d7 10<sup>7<\/sup><\/td>\n<td style=\"width: 25%;\">1.429 \u00d7 10<sup>12<\/sup><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">Uranus<\/td>\n<td style=\"width: 25%;\">8.685 \u00d7 10<sup>25<\/sup><\/td>\n<td style=\"width: 25%;\">2.536 \u00d7 10<sup>7<\/sup><\/td>\n<td style=\"width: 25%;\">2.871 \u00d7 10<sup>12<\/sup><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">Neptune<\/td>\n<td style=\"width: 25%;\">1.028 \u00d7 10<sup>26<\/sup><\/td>\n<td style=\"width: 25%;\">2.462 \u00d7 10<sup>7<\/sup><\/td>\n<td style=\"width: 25%;\">4.504 \u00d7 10<sup>12<\/sup><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 25%;\">Moon<\/td>\n<td style=\"width: 25%;\">7.349 \u00d7 10<sup>22<\/sup><\/td>\n<td style=\"width: 25%;\">1.737 \u00d7 10<sup>6<\/sup><\/td>\n<td style=\"width: 25%;\">*3.844 \u00d7 10<sup>8<\/sup><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Conceptions of Gravity and Astronomical Motion<\/h2>\n<p>Recorded concepts of gravity in Antiquity centre around the Greek philosopher Aristotle (384-322 BCE). Aristotle\u2019s concept of gravity emerged from his understanding of natural motion in that all matter will rise or fall to its natural place in the universe. He contrasted this with forced or violent motion, where an object could be made to move until the agent causing it to move stopped.\u00a0 In Aristotle\u2019s paradigm, gravity was a manifestation of objects moving to simply reach their natural place in the universe.<\/p>\n<p>Aryabhata (476-550) held the view that the Earth rotates daily on its axis, but held onto the heliocentric model of the universe rotating around the Earth, as taught by the earlier pre-Ptolomaic Greek astronomers. While a great deal of Aryabhata\u2019s work is lost, remnants indicate that he might have believed the planets move in elliptical orbits, a view that took nearly a thousand years to resurface in the works of Nicolaus Copernicus (1473-1543). There is also evidence that Aryabhata considered gravity to be an attractive force that allowed people to stand upright no matter where they were positioned on the Earth.<\/p>\n<p>Galileo\u2019s (1564-1642) was the first to record counter arguments to Aristotelian views on the movement of bodies to their natural place in the universe, through his experiments with the acceleration of falling bodies. Galileo\u2019s concepts of acceleration were later confirmed by the works of Francisco Maria Grimaldi (1618-1663) and Giovanni Battista Riccioli (1589-1671). It was Riccioli\u00a0 who, through his analysis of free-fall motion in his attempts to find errors in Galileo\u2019s theories, ended up finding the first measure of the acceleration of gravity. He estimated this to be 9.36 \u00b1 0.22 m\/s<sup>2<\/sup>, which is within a 5% error of today\u2019s accepted value of 9.8 m\/s<sup>2<\/sup>.<\/p>\n<p>In working to quantify Robert Hooke\u2019s\u00a0 (1635-1703) suggestion that gravitational force was an inverse square law relation , Isaac Newton (1643-1727) came up with the proportionality relation:<\/p>\n<p style=\"text-align: center;\">[latex]{F}_g\u221d \\dfrac{m_1{m}_2}{{d}^2}[\/latex]<\/p>\n<p>The first measured value of the constant G needed for Newton\u2019s equation came from the 1797 experiments by Henry Cavendish (1731-1810) . This was accomplished by Cavendish\u2019s measure of the density of the Earth, which he measured to be 5.48 times that of water. Cavendish&#8217;s work led other natural philosophers to estimate a value for G to be 6.754 \u00d7 10<sup>\u221211<\/sup> N-m<sup>2<\/sup>\/kg<sup>2<\/sup> close to the current accepted value of 6.67428 \u00d7 10<sup>\u221211<\/sup> N-m<sup>2<\/sup>\/kg<sup>2<\/sup>. The Cavendish experiments are considered to be highlights of accuracy in laboratory science; his work is taught as representing a gold standard in experimental research.<\/p>\n<p>Shown below is a sketch of the torsion apparatus that Cavendish used to measure the gravitational constant G.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-615\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Cavendish_Experiment.png\" alt=\"\" width=\"500\" height=\"375\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Cavendish_Experiment.png 699w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Cavendish_Experiment-300x225.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Cavendish_Experiment-65x49.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Cavendish_Experiment-225x169.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Cavendish_Experiment-350x262.png 350w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/p>\n<div class=\"textbox\">\n<ul>\n<li>Article to Read:\u00a0<a href=\"https:\/\/physicstoday.scitation.org\/doi\/full\/10.1063\/PT.3.1716\">Anatomy of a fall: Giovanni Battista Riccioli\u00a0 and the story of g<\/a><\/li>\n<li>Hooke\u2019s personality &#8211; Hooke is described as irascible, proud and prone to taking umbrage with intellectual competitors in his later years. Of note are his arguments with Isaac Newton on who should take credit over the discoveries in gravitation.<\/li>\n<li>Extra Help:\u00a0<a href=\"https:\/\/www.physicsclassroom.com\/class\/circles\/Lesson-3\/The-Apple,-the-Moon,-and-the-Inverse-Square-Law\">The Apple, the Moon and the Inverse Square Law<\/a><\/li>\n<li>Extra Help:\u00a0<a href=\"https:\/\/www.physicsclassroom.com\/class\/circles\/Lesson-3\/Cavendish-and-the-Value-of-G\">Cavendish and the Value of G<\/a><\/li>\n<li>Article to Read: <a href=\"https:\/\/sciencedemonstrations.fas.harvard.edu\/presentations\/cavendish-experiment\">Cavendish Experiment<\/a><\/li>\n<li>Article to Read: <a href=\"https:\/\/www.quora.com\/Is-string-theory-still-in-the-running-for-a-theory-of-Everything\">Is string theory still in the running for a theory of everything?<\/a><\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 17.1.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What is the gravitational force of attraction between a 70 kg person resting on the Earth&#8217;s surface and the Earth?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Data:<\/p>\n<ul>\n<li>[latex]F_g[\/latex] = Find<\/li>\n<li>[latex]G[\/latex] = 6.67 \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup><\/li>\n<li>[latex]m_1[\/latex] = 5.974 \u00d7 10<sup>24<\/sup> kg<\/li>\n<li>[latex]m_2[\/latex] = 70 kg<\/li>\n<li>[latex]d[\/latex] = 6.371 \u00d7 10<sup>6<\/sup> m<\/li>\n<\/ul>\n<p>Solution:<\/p>\n<ul>\n<li>[latex]{F}_{g}=\\dfrac{Gm_1{m}_2}{{d}^2}[\/latex]<\/li>\n<li>[latex]{F}_{g}=\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(5.974\\times10^{24}\\text{ kg})(70\\text{ kg})}{(6.371\\times10^6\\text{ m})^2}[\/latex]<\/li>\n<li>[latex]F_g[\/latex] = 690 N<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 17.1.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What is the gravitational force of attraction between the planet Mercury and the Sun?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Data:<\/p>\n<ul>\n<li>[latex]F_g[\/latex] = Find<\/li>\n<li>[latex]G[\/latex] = 6.67 \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup><\/li>\n<li>[latex]m_1[\/latex] = 1.981 \u00d7 10<sup>31<\/sup> kg<\/li>\n<li>[latex]m_2[\/latex] = 3.302 \u00d7 10<sup>23<\/sup> kg<\/li>\n<li>[latex]d[\/latex] = 5.791 \u00d7 10<sup>10<\/sup> m<\/li>\n<\/ul>\n<p>Solution:<\/p>\n<ul>\n<li>[latex]{F}_{g}=\\dfrac{Gm_1{m}_2}{{d}^2}[\/latex]<\/li>\n<li>[latex]{F}_{g}=\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(1.981\\times10^{31}\\text{ kg})(3.302\\times10^{23}\\text{ kg})}{(5.791\\times10^{10}\\text{ m})^2}[\/latex]<\/li>\n<li>[latex]F_g[\/latex] = 1.31 \u00d7 10<sup>22<\/sup> N<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 17.1.3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What is the gravitational force of attraction between the electron and an alpha particle separated by 18 nm? (One nanometre (nm) = 1 \u00d7 10<sup>-9<\/sup> m)<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Data:<\/p>\n<ul>\n<li>[latex]F_g[\/latex] = Find<\/li>\n<li>[latex]G[\/latex] = 6.67 \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup><\/li>\n<li>[latex]m_1[\/latex] = 6.64 \u00d7 10<sup>\u221227<\/sup> kg<\/li>\n<li>[latex]m_2[\/latex] = 9.11 \u00d7 10<sup>\u221231<\/sup> kg<\/li>\n<li>[latex]d[\/latex] = 18 \u00d7 10<sup>\u22129<\/sup>\u00a0m<\/li>\n<\/ul>\n<p>Solution:<\/p>\n<ul>\n<li>[latex]{F}_{g}=\\dfrac{Gm_1{m}_2}{{d}^2}[\/latex]<\/li>\n<li>[latex]{F}_\\text{g}=\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(6.64\\times10^{-27}\\text{ kg})(9.11\\times10^{-31}\\text{ kg})}{(18\\times10^{-9}\\text{ m})^2}[\/latex]<\/li>\n<li>[latex]F_g[\/latex] = 1.24 \u00d7 10<sup>\u221251<\/sup> N<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 17.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>What is the gravitational force of attraction between a mass of 1.00 kg resting on the Earth&#8217;s surface and the Earth?<\/li>\n<li>What is the gravitational force of attraction between 2 apples that have a mass of 500 g and 400 g separated by 12 cm on a tree branch?<\/li>\n<li>Calculate the distance separating two ships that have a mass of 750 tonnes and 500 tonnes respectively, if the gravitational force of attraction between them is 10<sup>\u22126<\/sup> newtons.<\/li>\n<li>What is the gravitational force of attraction between the electron and proton in a hydrogen atom? (The distance separating these two particles is called the Bohr radius.)<\/li>\n<li>What is the gravitational force of attraction between the planet Venus and the Sun?<\/li>\n<li>What is the maximum gravitational force of attraction between Jupiter and the Earth?<\/li>\n<li>One problem that plagues many entry level students is the comparison of the gravitational force of attraction that exists between the Earth and the Moon, and that between the Sun and the Moon. Find these two different forces of attraction and try to find an explanation why the Moon remains orbiting the Earth.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox\">Research News: <a href=\"https:\/\/www.nature.com\/magazine-assets\/d41586-018-07591-8\/d41586-018-07591-8.pdf\">Those dusty young stars are changing the rules of planet building<\/a><\/div>\n<h1>17.2 Gravitational Field Strength<\/h1>\n<div class=\"textbox\">Extra Help: <a href=\"https:\/\/www.physicsclassroom.com\/class\/circles\/Lesson-3\/The-Value-of-g\">The Value of g<\/a><\/div>\n<p>Equations Introduced or Used for this Section:<\/p>\n<p style=\"text-align: center;\">[latex]a_g \\text{ or }g=\\dfrac{Gm}{d^2}[\/latex]<\/p>\n<p>In Newtonian mechanics or Classical Mechanics (non-relativistic) an equation quantifying gravitational strength of any body can be found by equating Newton\u2019s Second Law and Newton\u2019s Law of Gravitational Force. This can be shown as follows:<\/p>\n<p style=\"text-align: center;\">[latex]F_{\\text{net}}= ma\\text{ or }F_g = mg\\text{ is equated to }F_{g}=\\dfrac{Gm_1{m}_2}{{d}^2}[\/latex]<\/p>\n<p>This means that &#8230;<\/p>\n<p style=\"text-align: center;\">[latex]mg=\\dfrac{Gm_1{m}_2}{{d}^2}[\/latex]<\/p>\n<p>cancelling out the common mass leaves us with &#8230;<\/p>\n<p style=\"text-align: center;\">[latex]g=\\dfrac{Gm_1}{{d}^2}[\/latex]<\/p>\n<p>Applying the conventional negative direction to this derivation we are left with<\/p>\n<p style=\"text-align: center;\">[latex]g=-\\dfrac{Gm_1}{{d}^2}[\/latex]<\/p>\n<p>In most classical mechanics questions this negative value is left out and the strength of gravity is just given as a scalar value.<\/p>\n<p>The units of gravitational field strength can be either m\/s<sup>2<\/sup> or N\/kg.\u00a0 \u00a0This equation relates to the gravity at any distance away from a point mass and is an inverse square law relationship.<\/p>\n<p style=\"text-align: center;\">[latex]g \u221d \\dfrac{1}{{d}^2}[\/latex]<\/p>\n<p>When we look at the gravitational field strength as we move towards the centre of the mass, the gravitational field strength drops inversely to zero. Beneath the planet\u2019s surface gravity is an inverse linear relationship.<\/p>\n<p style=\"text-align: center;\">[latex]g \u221d \\dfrac{1}{{d}}[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-616 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/17-g-field-e1679437176317.jpg\" alt=\"\" width=\"656\" height=\"439\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/17-g-field-e1679437176317.jpg 656w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/17-g-field-e1679437176317-300x201.jpg 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/17-g-field-e1679437176317-65x43.jpg 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/17-g-field-e1679437176317-225x151.jpg 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/17-g-field-e1679437176317-350x234.jpg 350w\" sizes=\"auto, (max-width: 656px) 100vw, 656px\" \/><\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 17.2.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What should be the surface gravity of the planet Venus?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Data:<\/p>\n<ul>\n<li>[latex]a_g[\/latex] or [latex]g[\/latex] = Find<\/li>\n<li>[latex]G[\/latex] = 6.67 \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup><\/li>\n<li>[latex]m[\/latex] = 4.869 \u00d7 10<sup>24<\/sup>\u00a0kg<\/li>\n<li>[latex]d[\/latex] = 6.052 \u00d7 10<sup>6<\/sup>\u00a0m<\/li>\n<\/ul>\n<p>Solution:<\/p>\n<ul>\n<li>[latex]a_g \\text{ or }g = \\dfrac{Gm}{{d}^2}[\/latex]<\/li>\n<li>[latex]g=\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(4.869\\times10^{24}\\text{ kg})}{(6.052\\times10^6\\text{ m})^2}[\/latex]<\/li>\n<li>[latex]g[\/latex] = 8.87 m\/s<sup>2<\/sup>&#8230; (N\/kg)<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 17.2.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What is the acceleration due to gravity at an orbit of 280 km above the Earth&#8217;s surface<a class=\"footnote\" title=\"The Earth is losing 50,000 tonnes of mass every single year.\n\nSolution\n\nThe Earth gains 40000 tonnes of space dust and asteroid.\nThe Earth loses 16 tonnes due to the fission in the Earth\u2019s core.\n\nThe Earth loses 95000 tonnes of escaped hydrogen.\n\nThe Earth loses 1600 tonnes of escaped helium.\n\nThe net loss of Earth Mass is about 0.000000000000001% each year.\n\nSource - Earth loses 50,000 Tonnes of Mass Every Year\" id=\"return-footnote-198-2\" href=\"#footnote-198-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a>?<\/p>\n<p>Now&#8230; [latex]d[\/latex] = radius of the Earth + distance to orbit<\/p>\n<ul>\n<li>[latex]d[\/latex] = 280 km + 6.371 \u00d7 10<sup>6<\/sup> m or 6.651 \u00d7 10<sup>6<\/sup> m<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignright wp-image-617\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex17.2.2-e1679438124436.jpg\" alt=\"\" width=\"300\" height=\"226\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex17.2.2-e1679438124436.jpg 316w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex17.2.2-e1679438124436-300x226.jpg 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex17.2.2-e1679438124436-65x49.jpg 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/ex17.2.2-e1679438124436-225x169.jpg 225w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/>Finally &#8230;<\/p>\n<ul>\n<li>[latex]a_g[\/latex] or [latex]g[\/latex] = Find<\/li>\n<li>[latex]G[\/latex] = 6.67 \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup><\/li>\n<li>[latex]m[\/latex] = 5.974 \u00d7 10<sup>24<\/sup>\u00a0kg<\/li>\n<li>[latex]d[\/latex] = 6.651 \u00d7 10<sup>6<\/sup>\u00a0m<\/li>\n<\/ul>\n<p>Solution:<\/p>\n<ul>\n<li>[latex]a_g \\text{ or }g = \\dfrac{Gm}{{d}^2}[\/latex]<\/li>\n<li>[latex]g=\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(5.974\\times10^{24}\\text{ kg})}{(6.651\\times10^6\\text{ m})^2}[\/latex]<\/li>\n<li>[latex]g[\/latex] = 9.01 m\/s<sup>2<\/sup> &#8230; (N\/kg)<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 17.2.3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>If a rocket is orbiting at a distance from Earth where the gravity is 0.90 gs, how much further away must it move away from the Earth to experience a gravity of 0.42 gs?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>First: d for 0.90 g\u2019s<\/p>\n<ul>\n<li>[latex]a_g[\/latex] or [latex]g[\/latex] = 0.90 g\u2019s<\/li>\n<li>[latex]G[\/latex] = 6.67 \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup><\/li>\n<li>[latex]m[\/latex] = 5.974 \u00d7 10<sup>24<\/sup>\u00a0kg<\/li>\n<li>[latex]d[\/latex] = Find<\/li>\n<\/ul>\n<p>Solution:<\/p>\n<ul>\n<li>[latex]a_g \\text{ or }g = \\dfrac{Gm}{{d}^2}[\/latex]<\/li>\n<li>[latex](0.90)(9.80\\text{ m\/s}^2)=\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(5.974\\times10^{24}\\text{ kg})}{d^2}[\/latex]<\/li>\n<li>[latex]\\text{d}^2=\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(5.974\\times10^{24}\\text{ kg})}{(0.90)(9.80\\text{ m\/s}^2)}[\/latex]<\/li>\n<li>[latex]d^2[\/latex]\u00a0 = 4.52 \u00d7 10<sup>13<\/sup> m<sup>2<\/sup><\/li>\n<li>[latex]d[\/latex] = 6.72 \u00d7 10<sup>6<\/sup> m<\/li>\n<\/ul>\n<p>Second: d for 0.42 g\u2019s<\/p>\n<ul>\n<li>[latex]a_g[\/latex] or [latex]g[\/latex] = 0.42 g\u2019s<\/li>\n<li>[latex]G[\/latex] = 6.67 \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup><\/li>\n<li>[latex]m[\/latex] = 5.974 \u00d7 10<sup>24<\/sup>\u00a0kg<\/li>\n<li>[latex]d[\/latex] = Find<\/li>\n<\/ul>\n<p>Solution:<\/p>\n<ul>\n<li>[latex]a_g \\text{ or }g = \\dfrac{Gm}{{d}^2}[\/latex]<\/li>\n<li>[latex](0.42)(9.80\\text{ m\/s}^2)=\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(5.974\\times10^{24}\\text{ kg})}{d^2}[\/latex]<\/li>\n<li>[latex]{d}^2=\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(5.974\\times10^{24}\\text{ kg})}{(0.42)(9.80\\text{ m\/s}^2)}[\/latex]<\/li>\n<li>[latex]d^2[\/latex]\u00a0 = 9.69 \u00d7 10<sup>13<\/sup> m<sup>2<\/sup><\/li>\n<li>[latex]d[\/latex] = 9.84 \u00d7 10<sup>6<\/sup> m<\/li>\n<\/ul>\n<p>Finally&#8230;<\/p>\n<ul>\n<li>[latex]\\Delta d[\/latex] = d<sub>90<\/sub> \u2212 d<sub>42<\/sub><\/li>\n<li>[latex]\\Delta d[\/latex] = 9.84 \u00d7 10<sup>6<\/sup> m \u2212 6.72 \u00d7 10<sup>6<\/sup> m<\/li>\n<li>[latex]\\Delta d[\/latex] = 3.12 \u00d7 10<sup>6<\/sup> m<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 17.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>What is the surface gravity of a 100 kg pumpkin of radius 0.75 m?<\/li>\n<li>What should be the surface gravity of the planet Uranus?<\/li>\n<li>What would be the surface gravity at the outer edge of the Sun?<\/li>\n<li>What is the acceleration due to gravity at an orbit of 300 km above the Earth&#8217;s surface? This distance is termed the Low Earth Orbital (LEO). Ships orbiting at this distance need to be boosted higher in orbit at regular intervals due to air resistance from the very weak atmosphere that exists at this distance.\u00a0 (Atmospheric pressure at this distance averages 10<sup>\u22128<\/sup> Pa, compared to 1.013 \u00d7 10<sup>5<\/sup> Pa at the Earth\u2019s surface.)<\/li>\n<li>What acceleration does the Moon experience due to the Earth&#8217;s gravity, in its orbit around the Earth? What acceleration does the Earth experience due to the Moon&#8217;s gravity?<\/li>\n<li>At what distance away from the Earth has the net acceleration due to gravity fallen to \u00bd g?<\/li>\n<li>What is the acceleration due to gravity experienced by an Earth orbiting satellite at a distance of 1 Earth radius above the surface?<\/li>\n<li>What distance above the Earth\u2019s surface must one travel to experience a reduced Earth gravity of 0.7 g\u2019s?<\/li>\n<li>If a shuttle is orbiting at a distance from Earth where the gravity is 0.8 gs, how much farther away must it move from the Earth to experience a gravity of 0.6 gs?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h1>17.3 Gravitational Potential<\/h1>\n<div class=\"textbox\">\n<p>Article to Read: <a href=\"https:\/\/www.scientificamerican.com\/article\/gravitational-waves-reveal-the-hearts-of-neutron-stars1\/\">Gravitational Waves Reveal the Hearts of Neutron Stars<\/a><\/p>\n<\/div>\n<p>Equations Introduced or Used for this Section:<\/p>\n<p style=\"text-align: center;\">[latex]\u0394V =-\\dfrac{Gm_1}{d}[\/latex]<\/p>\n<figure id=\"attachment_618\" aria-describedby=\"caption-attachment-618\" style=\"width: 400px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-618\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/GravityPotential.jpg\" alt=\"\" width=\"400\" height=\"232\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/GravityPotential.jpg 796w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/GravityPotential-300x174.jpg 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/GravityPotential-768x446.jpg 768w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/GravityPotential-65x38.jpg 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/GravityPotential-225x131.jpg 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/GravityPotential-350x203.jpg 350w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><figcaption id=\"caption-attachment-618\" class=\"wp-caption-text\">Plot of a two-dimensional slice of the gravitational potential in and around a uniform spherical body. The inflection points of the cross-section are at the surface of the body.<\/figcaption><\/figure>\n<div class=\"textbox textbox--sidebar\">Video to Watch: <a href=\"https:\/\/www.youtube.com\/watch?v=dngQetgjXBk\">Gravitational Potential and Gravitational Potential\u00a0 Energy, explained<\/a><\/div>\n<p>In classical mechanics, the gravitational potential at any location equals the work per unit of mass needed and moves the object from a fixed reference position to a final position within or outside a gravitational field. Typically, the gravitational potential refers to some final position infinitely far away from the gravitational source. As a result, at an infinite distance away the potential energy of any object in relation to its source position is zero. (Epf = 0). Also note that the movement that changes an object\u2019s potential energy must be either toward or away from the gravitational source.<\/p>\n<p>If we look at the work needed to move an object away from a gravitational source, we start by using the conservation of mechanical energy &#8230; Specifically:<\/p>\n<p style=\"text-align: center;\">[latex]W = \u2206\\text{Energy}[\/latex]&#8230; Since the [latex]\u2206 E[\/latex] is a change in potential energy&#8230; [latex]W = E_{pf} \u2212E_{pi}[\/latex]<\/p>\n<p>This can be simplified for cases where the object is moved to an infinite distance away to<\/p>\n<p style=\"text-align: center;\">[latex]W = - E_{pi}\\text{ since }E_{pf} = 0[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-619\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/17.3-graph.jpg\" alt=\"\" width=\"504\" height=\"268\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/17.3-graph.jpg 504w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/17.3-graph-300x160.jpg 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/17.3-graph-65x35.jpg 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/17.3-graph-225x120.jpg 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/17.3-graph-350x186.jpg 350w\" sizes=\"auto, (max-width: 504px) 100vw, 504px\" \/><\/p>\n<p>We can expand [latex]W = - E_{pi}[\/latex] to [latex]W = - m g h_i[\/latex] and replace gravity with [latex]g= - \\dfrac{Gm_1}{d}[\/latex]<\/p>\n<p>This yields: [latex]W= - m\\dfrac{Gm_1}{d}{h}_{i}[\/latex]&#8230; And since h<sub>i<\/sub> and d<sub>2<\/sub> are measuring the same distance, we can cancel common terms.<\/p>\n<p>This leaves us with [latex]W=-\\dfrac{Gm_1{m}_2}{d}[\/latex]<\/p>\n<p>The derivation of gravitational potential is almost complete. What remains is to isolate the second mass m<sub>2<\/sub>.<\/p>\n<p>Isolating m<sub>2<\/sub> from the main equation leaves us with: [latex]W=-\\dfrac{Gm_1}{d}({m}_2)[\/latex]<\/p>\n<p>The symbol we will use for gravitational potential is the same as the one we use for electric potential. Gravitational potential (\u2206V) where [latex]\\Delta V=-\\dfrac{Gm_1}{d}[\/latex]<\/p>\n<p>In this case, the work done (joules) to move an object infinitely away from a measurable gravitational field is found by the equation:<\/p>\n<p style=\"text-align: center;\">[latex]W=-\\dfrac{Gm_1}{d}({m}_2)[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 or\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]V=\u2212\\Delta V (m_2)[\/latex]<\/p>\n<p>Likewise the gravitational potential energy of an object removed from the influence of a planetary or sun\u2019s gravitational field is also found by:<\/p>\n<p style=\"text-align: center;\">[latex]\\Delta E_p=-\\dfrac{Gm_1}{d}({m}_2)[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 or\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\Delta E_p=-\\Delta V (m_2)[\/latex]<\/p>\n<p>How do we understand the negative value that shows up in these equations?<\/p>\n<ul>\n<li>As you move away from a gravitational field, you will need energy to escape. You can consider this energy to be lost.<\/li>\n<li>As you move into a gravitational field, you will be gaining energy, which is added to your mass.<\/li>\n<\/ul>\n<p>Equation Summary: To move any mass away from a gravitational source, use:<\/p>\n<p style=\"text-align: center;\">[latex]W=-\\dfrac{Gm_1{m}_2}{d}[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 or\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 [latex]\\Delta E_p=-\\dfrac{Gm_1{m}_2}{d}[\/latex]<\/p>\n<p>Using \u2206V = [latex]-\\dfrac{Gm_1}{d}[\/latex] the above equations shorten to:<\/p>\n<p style=\"text-align: center;\">[latex]W=-\\Delta Vm_2[\/latex]<sub>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/sub>or\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 [latex]\\Delta E_p=-\\Delta Vm_2[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 17.3.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What is the gravitational potential at an orbital distance of 280 km above the Earth\u2019s surface?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Data:<\/p>\n<ul>\n<li>[latex]\\Delta V[\/latex] = Find<\/li>\n<li>[latex]d[\/latex] = 280 km + 6.371 \u00d7 10<sup>6<\/sup> m or 6.651 \u00d7 10<sup>6<\/sup> m<\/li>\n<li>[latex]m[\/latex] = 5.974 \u00d7 10<sup>24<\/sup>\u00a0kg<\/li>\n<li>[latex]G[\/latex] = 6.67 \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup><\/li>\n<\/ul>\n<p>Solution:<\/p>\n<ul>\n<li>[latex]\\Delta V=-\\dfrac{Gm_1}{d}[\/latex]<\/li>\n<li>[latex]\\Delta V=\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(5.974\\times10^{24}\\text{ kg})}{6.651\\times10^6\\text{ m}}[\/latex]<\/li>\n<li>[latex]\\Delta V[\/latex]= \u22125.99 \u00d7 10<sup style=\"word-spacing: normal;\">7<\/sup><span style=\"font-size: 0.9em; word-spacing: normal;\"> J\/kg<\/span><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 17.3.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>How much work must be done for the space shuttle (mass = 2030 t) orbiting 280 km above the Earth\u2019s surface to permanently escape the Earth?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Data:<\/p>\n<ul>\n<li>[latex]W[\/latex] = Find<\/li>\n<li>[latex]d[\/latex] = 280 km + 6.371 \u00d7 10<sup>6<\/sup> m or 6.651 \u00d7 10<sup>6<\/sup> m<\/li>\n<li>[latex]m_1[\/latex] = 6.419\u00a0\u00d7 10<sup>23<\/sup>\u00a0kg<\/li>\n<li>[latex]m_2[\/latex] = 2030 \u00d7 10<sup>3<\/sup>\u00a0kg<\/li>\n<li>[latex]G[\/latex] = 6.67 \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup><\/li>\n<\/ul>\n<p>Solution:<\/p>\n<ul>\n<li>[latex]W[\/latex] = [latex]-\\dfrac{Gm_1{m}_2}{d}[\/latex]<\/li>\n<li>[latex]W[\/latex] = [latex]-\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(5.974\\times10^{24}\\text{ kg})(2030\\times10^3\\text{ kg})}{6.651\\times10^6\\text{ m}}[\/latex]<\/li>\n<li>[latex]W[\/latex] = 1.22 \u00d7 10<sup>14<\/sup> J<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 17.3.3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What is the gravitational potential of a rocket ship sitting on the surface of the planet Mars?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Data:<\/p>\n<ul>\n<li>[latex]\\Delta V[\/latex] = Find<\/li>\n<li>[latex]m_1[\/latex] = 6.419\u00a0\u00d7 10<sup>23<\/sup>\u00a0kg<\/li>\n<li>[latex]d[\/latex] = 3.390\u00a0\u00d7 10<sup>6<\/sup> m<\/li>\n<li>[latex]G[\/latex] = 6.67 \u00d7 10<sup>\u221211<\/sup> Nm<sup>2<\/sup>\/kg<sup>2<\/sup><\/li>\n<\/ul>\n<p>Solution:<\/p>\n<ul>\n<li>[latex]\\Delta V=-\\dfrac{Gm_1}{d}[\/latex]<\/li>\n<li>[latex]\\Delta V=\\dfrac{(6.67\\times10^{-11}\\text{ Nm}^2\\text{\/kg}^2)(6.419\\times10^{23}\\text{ kg})}{3.390\\times10^6\\text{ m}}[\/latex]<\/li>\n<li>[latex]\\Delta V[\/latex] = \u22121.26 \u00d7 10<sup>7<\/sup> J\/kg<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 17.3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>What is the gravitational potential at an orbital distance of 300 km above the Earth\u2019s surface?<\/li>\n<li>How much work must be done for Elon Musk\u2019s 2008 Tesla Roadster (mass = 1300 kg) at an orbital distance of 300 km above the Earth\u2019s surface to permanently escape the Earth?<\/li>\n<li>What is the gravitational potential of a rocket ship sitting on the moon\u2019s surface?<\/li>\n<li>What is the gravitational potential for a rocket launched from the Earth\u2019s orbit around the Sun to escape the Sun\u2019s gravitational field?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h1>17.4.1 Earth-Moon Gravity Neutral Point<\/h1>\n<p>Between the Earth and the Moon, there exists a \u201cfluctuating\u201d position where the gravitational fields of the Earth and Moon balance each other out. This position is defined as the Earth-Moon Gravity Neutral Point.\u00a0 Described by an equation, this position can be found from<\/p>\n<p style=\"text-align: center;\">[latex]g_{\\text{ Earth}}= g_{\\text{ Moon}}[\/latex]<\/p>\n<p style=\"text-align: center;\">where [latex]g=\\dfrac{GM}{{d}^2}[\/latex]<\/p>\n<div class=\"textbox textbox--sidebar\">Article to Read: <a href=\"https:\/\/www.madsci.org\/posts\/archives\/1998-08\/899933769.As.r.html\">Re: What is the Earth Moon neutral point?<\/a> by David Ellis<\/div>\n<p>Technically, this position is where the gravitational fields from both the Moon and the Earth balance each other out, meaning that any object in this position having no velocity relative to the Earth and Moon should stay in that position without being pulled to the Earth or the Moon. Should any spaceship reach this position headed to Earth or the Moon, it would then be able to fall to either side of this position without using any more fuel for thrust. This can be looked at as a moving vehicle at the top of a hill &#8230; If the vehicle is moving in any direction down the hill when at the top, then the vehicle will be able to roll down hill without using any fuel. If the vehicle is stopped at the top of the hill, then it should be able to remain stationary unless some external unbalanced force acts upon it, which would then act to nudge the vehicle downhill in one of the two directions. This position is termed fluctuating because of a number of factors which are described by NASA\u2019s David Ellis.<\/p>\n<p>The gravity neutral position is visually shown by mapping the variable strength of the Earth\u2019s gravitational field as one heads to the Moon, overlaid by the variable strength of the Moon\u2019s gravitational field as one heads back to the Earth. The intersection of these two gravitational fields\u2019 lines marks the approximate position. Due to fluctuations in the location of this position, any spaceship parked there would need corrections by booster rockets to remain in the region of neutral gravity.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1220  aligncenter\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Earth-moon-gravitational-potential.svg_.png\" alt=\"\" width=\"754\" height=\"566\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Earth-moon-gravitational-potential.svg_.png 1280w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Earth-moon-gravitational-potential.svg_-300x225.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Earth-moon-gravitational-potential.svg_-1024x768.png 1024w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Earth-moon-gravitational-potential.svg_-768x576.png 768w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Earth-moon-gravitational-potential.svg_-65x49.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Earth-moon-gravitational-potential.svg_-225x169.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Earth-moon-gravitational-potential.svg_-350x263.png 350w\" sizes=\"auto, (max-width: 754px) 100vw, 754px\" \/><\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 17.4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>Using your equations for the gravitational fields of the Earth and the Moon, calculate the distance from the Earth that this point exists.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h1>Exercise Answers<\/h1>\n<h2>17.1 Gravitational Force<\/h2>\n<ol class=\"twocolumn\">\n<li>\u00a09.82 N<\/li>\n<li>9.3 \u00d7 10<sup>\u221210<\/sup> N<\/li>\n<li>5000 m or 5.00 km<\/li>\n<li>3.63 \u00d7 10<sup>\u221247<\/sup> N<\/li>\n<li>5.52 \u00d7 10<sup>22<\/sup> N<\/li>\n<li>1.91 \u00d7 10<sup>18<\/sup> N<\/li>\n<li>Earth-Moon: 2.02 \u00d7 10<sup>20<\/sup> N,<br \/>\nSun-Moon: 4.39 \u00d7 10<sup>20<\/sup> N<\/li>\n<\/ol>\n<h2>17.2 Gravitational Field Strength<\/h2>\n<ol class=\"twocolumn\">\n<li>1.2 \u00d7 10<sup>\u22128<\/sup> m\/s<sup>2 <\/sup><\/li>\n<li>9.01 m\/s<sup>2\u00a0<\/sup><\/li>\n<li>272 m\/s<sup>2 <\/sup><\/li>\n<li>8.96 m\/s<sup>2<\/sup><\/li>\n<li>Moon: 2.70 \u00d7 10<sup>\u22123<\/sup> m\/s<sup>2<\/sup><br \/>\nEarth: 3.32 \u00d7 10<sup>\u22125<\/sup> m\/s<sup>2<\/sup><\/li>\n<li>9.01 \u00d7 10<sup>6<\/sup> m from the Earth&#8217;s centre or 2650 km above the surface<\/li>\n<li>2.45 m\/s<sup>2<\/sup> or 1\/4 g&#8217;s<\/li>\n<li>1.25 \u00d7 10<sup>6<\/sup> m\u00a0 or 1250 km<\/li>\n<li>3.49 \u00d7 10<sup>5<\/sup> m<\/li>\n<\/ol>\n<h2>17.3 Gravitational Potential<\/h2>\n<ol class=\"twocolumn\">\n<li>[latex]\\Delta V[\/latex] = \u2212 5.97 \u00d7 10<sup>4<\/sup> J\/kg<\/li>\n<li>[latex]W[\/latex] = \u2212 7.77 \u00d7 10<sup>10<\/sup> J<\/li>\n<li>[latex]\\Delta V[\/latex] = \u2212 2.82 \u00d7 10<sup>6<\/sup> J\/kg<\/li>\n<li>[latex]\\Delta V[\/latex] = \u2212 8.87 \u00d7 10<sup>8<\/sup> J\/kg<\/li>\n<\/ol>\n<h2>17.4.1 Earth-Moon Gravity Neutral Point<\/h2>\n<ol>\n<li>Distance from the Earth = 3.46 \u00d7 10<sup>8<\/sup> m &amp; Distance from the Moon = 3.84 \u00d7 10<sup>7<\/sup> m<\/li>\n<\/ol>\n<h3>Media Attributions<\/h3>\n<ul>\n<li>&#8220;<a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Cavendish_Experiment.png\">Cavendish Experiment<\/a>&#8221; by Henry Cavendish is in the <a href=\"https:\/\/creativecommons.org\/share-your-work\/public-domain\/pdm\/\">public domain<\/a>.<\/li>\n<li>&#8220;<a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:GravityPotential.jpg\">Gravity Potential<\/a>&#8221; by AllenMcC. is licensed under a <a href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/3.0\/deed.en\">CC BY-SA 3.0<\/a> licence.<\/li>\n<li>\n<p id=\"firstHeading\" class=\"firstHeading mw-first-heading\">&#8220;<a href=\"https:\/\/en.wikipedia.org\/wiki\/File:Earth-moon-gravitational-potential.svg\">Earth moon gravitational potential<\/a>&#8221; by MikeRun is licensed under a <a href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/3.0\/deed.en\">CC BY-SA 3.0<\/a> licence.<\/p>\n<\/li>\n<\/ul>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-198-1\">The radius of orbits are given around the Sun, except the moon, which is given as its orbit around the Earth. <a href=\"#return-footnote-198-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-198-2\">The Earth is losing 50,000 tonnes of mass every single year.\r\n\r\n<strong>Solution<\/strong>\r\n\r\nThe Earth gains 40<span style=\"margin-left: 0.25em;\">000<\/span> tonnes of space dust and asteroid.\r\nThe Earth loses 16 tonnes due to the fission in the Earth\u2019s core.\r\n\r\nThe Earth loses 95<span style=\"margin-left: 0.25em;\">000<\/span> tonnes of escaped hydrogen.\r\n\r\nThe Earth loses 1600 tonnes of escaped helium.\r\n\r\nThe net loss of Earth Mass is about 0.000000000000001% each year.\r\n\r\nSource - <a href=\"https:\/\/scitechdaily.com\/earth-loses-50000-tonnes-of-mass-every-year\">Earth loses 50,000 Tonnes of Mass Every Year<\/a> <a href=\"#return-footnote-198-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":125,"menu_order":17,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-198","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/pressbooks\/v2\/chapters\/198","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/wp\/v2\/users\/125"}],"version-history":[{"count":25,"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/pressbooks\/v2\/chapters\/198\/revisions"}],"predecessor-version":[{"id":1221,"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/pressbooks\/v2\/chapters\/198\/revisions\/1221"}],"part":[{"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/pressbooks\/v2\/chapters\/198\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/wp\/v2\/media?parent=198"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/pressbooks\/v2\/chapter-type?post=198"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/wp\/v2\/contributor?post=198"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/wp\/v2\/license?post=198"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}