{"id":777,"date":"2020-10-27T19:21:39","date_gmt":"2020-10-27T19:21:39","guid":{"rendered":"https:\/\/books.compclassnotes.com\/rothphys110-2e\/?p=777"},"modified":"2020-10-27T19:21:40","modified_gmt":"2020-10-27T19:21:40","slug":"section-9-8","status":"publish","type":"post","link":"https:\/\/books.compclassnotes.com\/rothphys110-2e\/2020\/10\/27\/section-9-8\/","title":{"rendered":"Chapter 9: Force"},"content":{"rendered":"\n

9.8 Universal gravitation<\/h2>\n\n\n\n

Imagine that you launch a cannonball straight out of a cannon; it goes a bit, then lands. What happens if you shoot it with a bit more velocity? What if you shoot it really<\/em> fast? You could shoot it with such a velocity that the ground is dropping out from under it at the same rate that it is falling\u2014it would never land! The cannonball would now be in orbit, as shown below.<\/p>\n\n\n\n

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Newton\u2019s Cannonball: launch a cannonball at increasing speeds
Isaac Newton [Public domain], via Wikimedia Commons File:Principia1846-513.png;<\/figcaption><\/figure><\/div>\n\n\n\n

In the 17th century, Newton discovered that gravity is the force that keeps objects in orbit, and published it in Principia Mathematica<\/em>. The implications of this were revolutionary\u2014the same force that keeps things to the ground here governs celestial bodies! Gravity acts between any<\/em> two bodies with mass; they are attracted towards<\/em> each other. The magnitude of this force is given by:<\/p>\n\n\n\n

FG<\/sup> = G (m1<\/sub> m2<\/sub><\/sup>\u2044r2<\/sup><\/sub>) <\/em><\/p>\n\n\n\n

where m1<\/sub><\/em> and m2<\/sub><\/em> are the masses of the two objects, r<\/em> is the distance between them, and G<\/em> is the gravitational constant. Newton did not know what the value of G<\/em> was; it was determined for the first time in the 1790’s by Henry Cavendish. The value of \u201cBig G\u201d in SI units is G <\/em>= 6.67\u00d710-11<\/sup> N \u00b7 m2<\/sup> \u2044 kg2<\/sup>.<\/p>\n\n\n\n

When you are near the surface of Earth, the distance from you to the center of Earth is the planet’s radius R\u2295<\/sub><\/em>= 6.371 \u00d7 106<\/sup> m. The mass of Earth is M\u2295<\/sub><\/em>= 5.972 \u00d7 1024<\/sup> kg. If you have a mass m<\/em>, this means the force of gravity Earth exerts on you is<\/p>\n\n\n\n

FG<\/sup> = G (m M\u2295<\/sub> <\/sup>\u2044 R\u2295<\/sub>2<\/sup><\/sub>) = m<\/em> (9.81) = mg<\/em><\/p>\n\n\n\n

This is how we know the value of g, the strength of Earth\u2019s gravity. In fact, you can find out the strength of Earth\u2019s gravity even if you are not close to the surface: use R\u2295<\/sub><\/em> + h<\/em> for any height h<\/em> above the surface.<\/p>\n\n\n\n

Example<\/h4>\n\n\n\n\n