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**Scores will be adjusted up by 10%**

Test Breakdown A >= 93% (63) A- >= 90% (60) B+ >= 87% (57) B >= 83% (54) B- >= 80% (52) Average: 63.2% + 10% Std Dev: 21.5% Scores will be adjusted up by 10% C+ >= 77% (51) C >= 70% (45) D >= 60% (38) F < 60% (37 or less) 8/1/2004

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Chapter 14 Gravitation

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**One of the fundamental forces of Nature**

Gravity One of the fundamental forces of Nature Not just the reason things fall…. Why the Earth is round Why the moon goes around the earth Why the earth goes around the sun Why there are ocean tides 8/1/2004

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**Gravitational Force m1 m2**

Fg Fg is an attractive force between any two masses Fg is not a constant unless you have a small object near the surface of a big sphere G = x N m2/kg2 8/1/2004

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**Multiple Objects Obeys principle of superposition: M1 M3 r1 r3 m r2 M2**

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**Extended Objects Fg Each bit of #1 attracts each bit of #2**

Need to integrate over whole object to get Fg 8/1/2004

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**Extended Objects: Special Case**

Can treat uniform spherical shells (and thus spheres) like point masses located at geometric center No gravitational force inside uniform spherical shell (it integrates to zero) Fg = 0 8/1/2004

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**Gravitational Force Examples**

Force between earth and sun Force between two people (assumed spherical) 8/1/2004

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Example: (Problem 14.16) What will an object weigh on the Moon’s surface if it weighs 100 N on Earth’s surface? How many Earth radii must this same object be from the center of Earth if it is to weigh the same as it does on the Moon? 8/1/2004

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Example: Find the mass of the object: On the Moon: 8/1/2004

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Example: Distance from the earth with the same weight: 8/1/2004

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**Objects Near Earth’s Surface**

As long as the distance above earth’s surface is small compared to RE, the force is approximately constant 8/1/2004

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**Variation of Gravitational Force on Earth’s Surface**

Earth is not uniformly dense Variations in crust from region to region Earth is not a sphere Bulge at the equator Apparent change from earth’s rotation In this case, 8/1/2004

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**Variation of g from Rotation**

w At the earth’s pole, there is no centripetal acceleration: N Fg ac=0 At the pole, g=ag 8/1/2004

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**Variation of g from Rotation**

w At the equator: N Fg ac At the equator, g < ag! Effectively lower gravity g = m/s2 in Pittsburgh g = m/s2 in Jamaica!! 8/1/2004

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**Gravitation Inside a Sphere**

Recall: No gravitational force exerted on an object inside a spherical shell As you travel further into a sphere, the layers above can be thought of as many spherical shells, which exert no gravitational force! Only the mass of the sphere below you matters for calculating Fg! 8/1/2004

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**“Inner” Gravity Only the mass of the inward sphere contributes to Fg**

Movie claims that earth’s core is a trillion trillion tons… 1012 1012 tons = 1027 kg Mass of entire Earth: 61024 kg !!! Real inner core: m = 1.7% MEarth ginner = 0.017g They walk as if under 1 g!!! 8/1/2004

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**Gravitational Potential Energy**

Gravity is a conservative force – what is the associated potential energy? ΔU = -W and So for point masses or spheres m1 and m2 Taking U = 0 at x = 8/1/2004

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**Gravitational Potential Energy**

For point masses or spheres m1 and m2 Note: U = at r = 0 U r F = - dU/dr Always attractive 8/1/2004

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**Gravitational Potential Energy for Astronaut between Earth and Moon**

rEM x Ftot can be zero at some x… What about Utot? 8/1/2004

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**Example: Find where Fnet on the astronaut equals zero.**

rEM x Find where Fnet on the astronaut equals zero. Which solution is real? 8/1/2004

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Path Independence The amount of work done against a gravitational potential does not depend on the path taken (conservative force) x y A B 1 2 3 8/1/2004

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Escape Velocity What speed does an object need to escape the Earth’s gravity? It needs just enough KE to get to r and stop Escape velocity from Earth is: 8/1/2004

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Example: (Problem 14.29) The mean diameters of Mars and Earth are 6.9x103 km and 1.3x104 km, respectively. The mass of Mars is 0.11 times Earth’s mass. What is the ratio of the mean density of Mars to that of Earth? What is the value of the gravitational acceleration on Mars? What is the escape speed on Mars? 8/1/2004

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Kepler’s 1st Law All planets move in elliptical orbits, with the Sun at one focus a F F’ ea q r Eccentricities, e, of planets are small (close to circular) 8/1/2004

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Kepler’s 2nd Law The rate at which a planet sweeps out an area A is constant. (Constant areal velocity) F F’ A 8/1/2004

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Kepler’s 3rd Law The square of the period of any planet is proportional to the cube of the semi-major axis of its orbit 8/1/2004

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Example: (Problem 14.42) Determine the mass of Earth from the period T (27.3 days) and the radius r (3.82x105 km) of the Moon’s orbit about Earth. Assume that the Moon orbits the center of Earth rather than the center of mass of the Earth-Moon system. 8/1/2004

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Example: Kepler’s 3rd law: 8/1/2004

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Circular Orbits r Fg v Simplest case: 8/1/2004

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**Geosynchronous Orbit r**

A satellite can stay over one location on earth. Period = 1 day 8/1/2004

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**Why is there free fall on the orbiting space shuttle?**

R=Rorbit < 2Re so gravitational force is not negligible a = GME/R2 Both shuttle and occupants accelerating toward center of earth with same acceleration 8/1/2004

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**Energy in a Circular Orbit**

M m For an elliptical orbit, substitute a for r 8/1/2004

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Ocean Tides Caused by difference in gravitational force across the extent of the earth. r Water closest to Moon pulled “upward” Water farthest from Moon pulled less and thus bulges outward Smaller effect from the sun even though it has stronger gravitational pull. Why? 8/1/2004

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**or against each other to form a neap tide**

Tides The effects from the sun and moon can work together to form a spring tide… or against each other to form a neap tide 8/1/2004

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