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

Chapter 14 Gravitation

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

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 = 6.672 x 10-11 N m2/kg2 8/1/2004

Multiple Objects Obeys principle of superposition: M1 M3 r1 r3 m r2 M2 8/1/2004

Extended Objects Fg Each bit of #1 attracts each bit of #2 Need to integrate over whole object to get Fg 8/1/2004

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

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

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

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

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

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

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

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

Variation of g from Rotation w At the equator: N Fg ac At the equator, g < ag! Effectively lower gravity g = 9.801 m/s2 in Pittsburgh g = 9.786 m/s2 in Jamaica!! 8/1/2004

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

“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: 61024 kg !!! Real inner core: m = 1.7% MEarth  ginner = 0.017g They walk as if under 1 g!!! 8/1/2004

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

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

Gravitational Potential Energy for Astronaut between Earth and Moon rEM x Ftot can be zero at some x… What about Utot? 8/1/2004

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

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

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

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

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

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

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

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

Example: Kepler’s 3rd law: 8/1/2004

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

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

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

Energy in a Circular Orbit M m For an elliptical orbit, substitute a for r 8/1/2004

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

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