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Universal Gravitation

Universal Gravitation. Imagine an apple falling It accelerates towards the center of the earth at g = 9.8 m/s 2 Now imagine the moon falling It falls with centripetal acceleration a M What’s the ratio of these accelerations?. Universal Gravitation.

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Universal Gravitation

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  1. Universal Gravitation • Imagine an apple falling • It accelerates towards the center of the earth atg = 9.8 m/s2 • Now imagine the moon falling • It falls with centripetal acceleration aM • What’s the ratio of these accelerations?

  2. Universal Gravitation • Newton realized (1687) that the same force that explained why the apple dropped, could explain the motion of the moon in its orbit! • He stated it as the Law of Universal Gravitation: “Every particle is attracted to every other particle by a force which is proportional to the product of their masses and inversely proportional to the square of the distance between them.” • In algebraic form:

  3. The Cavendish Experiment • How does one measure the constant G? • First measured by Cavendish in 1798 • Use large masses and “torsional balance” to measure twist due to gravity • Amount of twist is amplified by bouncing light off the mirror

  4. Spheres as Points? • Moon and apple accelerate to center of Earth as if it were a point particle! • But the Earth is a sphere, not a point particle. • For apple, some parts of the Earth a much closer than others • How can we assume that the Earth is a point particle?

  5. Spheres as Points? • Consider a circular shell • Forr < R, Fg = 0 • Inside each set of opposing cones, force cancels, because included mass (one side) goes as r2 and force goes as 1/ r2

  6. Spheres as Points? • Consider a circular shell • Forr > R, masses at r1 and r2 have the same magnitude of force and the same direction, so the average is in the middle, at R

  7. The Graviational Field • Can remove one of the masses from the definition of the Gravitational force, get the Gravitational Field • This is the acceleration that would be felt by a small mass • Can think of it as the way a large mass effect the space around itself

  8. Central Force Motion • Newton’s Law of Universal Gravitation is an example of a Central Force. • For two particle case, we can simplify the analysis by choosing the CM frame

  9. Central Force Motion • Demonstration of one body equivalent problem • m is called the reduced mass

  10. Central Force Motion • Angular Momentum Conservation • Since F and r are parallel, L is conserved • Motion confined to plane!

  11. Central Force Motion • Energy Conservation • CF’s conservative: work depends only on radii • What is the energy? • Really a 1D problem Centrifugal Potential

  12. Central Force Motion • Energy Conservation • CF’s conservative: work depends only on radii • What is the energy? • Really a 1D problem Centrifugal Potential

  13. Central Force Motion • Formal Solution • From energy • From ang. Momentum (need r(t)) • Now “just” integrate!

  14. Central Force Motion • The effective potential determines the stability and form of orbits • For gravity

  15. Central Force Motion U E<0 Open Orbit r Elliptical Orbit E<0 Bound Orbit Circular Orbit

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