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**Lecture 5: Gravity and Motion**

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**Describing Motion and Forces**

speed, velocity and acceleration momentum and force mass and weight Newton’s Laws of Motion conservation of momentum analogs for rotational motion

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**Torque and Angular Momentum**

A torque is a twisting force Torque = force x length of lever arm Angular momentum is torque times velocity For circular motion, L = m x v x r

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**Laws for Rotational Motion**

Analogs of all of Newton’s Laws exist for rotational motion For example, in the absence of a net torque, the total angular momentum of a system remains constant There is also a Law of Conservation of Angular Momentum

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**Conservation of Angular Momentum during star formation**

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**Newton’s Universal Law of Gravitation**

Every mass attracts every other mass through a force called gravity The force is proportional to the product of the two objects’ masses The force is inversely proportional to the square of the distance between the objects’ centers

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**Universal Law of Gravitation**

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**The Gravitational Constant G**

The value of the constant G in Newton’s formula has been measured to be G = 6.67 x 10 –11 m3/(kg s2) This constant is believed to have the same value everywhere in the Universe

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**Remember Kepler’s Laws?**

Orbits of planets are ellipses, with the Sun at one focus Planets sweep out equal areas in equal amounts of time Period-distance relation: (orbital period)2 = (average distance)3

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**Kepler’s Laws are just a special case of Newton’s Laws!**

Newton explained Kepler’s Laws by solving the law of Universal Gravitation and the law of Motion Ellipses are one possible solution, but there are others (parabolas and hyperbolas)

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Conic Sections

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**Bound and Unbound Orbits**

Unbound (comet) Unbound (galaxy-galaxy) Bound (planets, binary stars)

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**Understanding Kepler’s Laws: conservation of angular momentum**

L = mv x r = constant smaller distance smaller r bigger v planet moves faster r larger distance smaller v planet moves slower

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**Understanding Kepler’s Third Law**

Newton’s generalization of Kepler’s Third Law is given by: 4p2 a3 p2 = G(M1 + M2) 4p2 a3 p2 = GMsun Mplanet << Msun, so

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**This has two amazing implications:**

The orbital period of a planet depends only on its distance from the sun, and this is true whenever M1 << M2

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**An Astronaut and the Space Shuttle have the same orbit!**

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**Second Amazing Implication:**

If we know the period p and the average distance of the orbit a, we can calculate the mass of the sun!

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**Example: How can we use this information to find the mass of the Sun?**

Io is one of the large Galilean moons orbiting Jupiter. It orbits at a distance of 421,600 km from the center of Jupiter and has an orbital period of 1.77 days. How can we use this information to find the mass of the Sun?

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Tides

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**The Moon’s Tidal Forces on the Earth**

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Tidal Friction

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Synchronous Rotation

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Galactic Tidal Forces

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