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Chapter 12 Gravitation

Main Points of Chapter 12 Early observations of planetary motion Kepler’s laws Universal gravitation Potential energy Planets and satellites Escape speed Orbits Gravitation and extended objects

Main Points of Chapter 12 Tides Equality of inertial and gravitational mass Einstein’s theory of gravitation Equivalence principle

12-1 Early Observations of Planetary Motion Just looking at the sky, one would assume that the earth is the center of the universe, and that everything we see is in orbit around us. Early models of the solar system were geocentric, but some of the planets were observed to exhibit retrograde motion:

Models of Retrograde motion Ptolemaic Explanation( A.D.) The motions of the Sun, Moon, and stars were based on perfect circles with the earth at the center. The planets moved around small circular paths that in turn moved around larger circular orbits around the Earth. Copernican Explanation (heliocentric) The planets further from the sun are moving more slowly in their orbits than those closer to the sun. The retrograde motion of Mars occurs when the Earth passes by the slower moving Mars. l

12-1 Early Observations of Planetary Motion The geocentric model was made more elaborate, with epicycles upon epicycles, but ultimately was unable to explain retrograde motion.

12-1 Early Observations of Planetary Motion Kepler’s laws: 1. Planets move in planar elliptical paths with the Sun at one focus of the ellipse.

12-1 Early Observations of Planetary Motion Kepler’s laws: 2. During equal time intervals the radius vector from the Sun to a planet sweeps out equal areas.

12-1 Early Observations of Planetary Motion Kepler’s laws: where C is a constant whose value is the same for all planets. (12-1) 3. If T is the time that it takes for a planet to make one full revolution around the Sun, and if R is half the major axis of the ellipse (R reduces to the radius of the planet’s orbit if that orbit is circular), then:

12-2 Newton’s Inverse-Square Law Newton wanted to explain Kepler’s laws; found that: Force must be central Inverse-square law Possible paths must be conic sections:

Newton’s explanation of Kepler’s 3 rd law The elliptical paths described by Kepler could be explained by the inverse square law k/r 2 Where k is a constant F = ma k/r 2 = mv 2 / R using T = 2pieR/v and squaring T 2 = 4 pie 2 R 2 / v 2 T 2 = 4 pie 2 R 2 / k/mR = (4 pie 2 m/k) R 3 This is Kepler’s 3 rd law with C= (4 pie 2 m/k)

12-2 Newton’s Inverse-Square Law Law of universal gravitation includes all requirements: (12-4) G is a constant that can be measured using known masses; find: (12-7)

C= (4 pie 2 m/k) = (4 pie 2 m/GmM) T 2 = (4 pie 2 R 3 /GM) Return to Kepler’s 3 rd law

Potential energy can be derived from force: (12-8) 12-2 Newton’s Inverse-Square Law

12-3 Planets and Satellites Newton realized that falling with a sufficiently large initial horizontal velocity is orbiting – that is, the same force that causes the apple to fall from the tree also keeps the Moon in its orbit.

12-3 Planets and Satellites Escape speed is outward speed needed to escape from Earth’s gravitational potential well (that is, to make total energy nonnegative): (12-10)

Escape velocity of the earth The total energy necessary for a rocket to escape the earths gravitational attraction is 0 ½ mv esc 2 – Gm e m/R e =0=E V esc = 11.2 km/s The escape velocity is indepent of direction

12-3 Planets and Satellites Types of Orbits (12-9) Total energy: If E is negative, orbits are closed – elliptical or circular If E is positive, orbits are open – hyperbola

12-3 Planets and Satellites Some facts and definitions: Orbits are all conical sections Ellipse has semimajor and semiminor axes Ellipse equation: (12-11) Closest approach to sun is called perihelion; farthest is aphelion

12-3 Planets and Satellites Semimajor axis: Eccentricity: (12-12) (12-13)

12-3 Planets and Satellites Types of Orbits: circle, ellipses, parabola, hyperbola

12-4 Gravitation and Extended Objects The Gravitational Force Due to a Spherically Symmetric Object Within a hollow shell, the gravitational force is zero – forces from opposite sides cancel

12-4 Gravitation and Extended Objects The Gravitational Force Due to a Spherically Symmetric Object Within the object, force at r is due to mass inside r:

12-4 Gravitation and Extended Objects Dark Matter: Discovered within and around galaxies and galaxy clusters Within galaxies: needed to explain rotational speed of outer areas Within galaxy clusters: needed to make clusters gravitationally bound No matter of any kind is detectable where it is needed – scientists still figuring it out

12-4 Gravitation and Extended Objects Acceleration of gravity, g, varies with altitude above Earth’s surface, due to changing distance from Earth’s center: (12-19) For h/Re << 1 we can use 1/(1+x) 2 = 1-2x g(h) = GM/(Re+h) 2 = GM/Re(1+h/Re) 2

12-4 Gravitation and Extended Objects Acceleration of gravity, g, varies with altitude above Earth’s surface, due to changing distance from Earth’s center: (12-19) This is a very small difference, even at the top of Mt. Everest!

12-4 Gravitation and Extended Objects Tidal Forces Tidal forces occur because extended objects are not points. Near side of Earth is closer to Moon than far side, so the magnitude of the gravitational force is not the same on both sides. This difference in gravitational forces on different parts of an extended object is the tidal force.

12-4 Gravitation and Extended Objects Tidal Forces Different accelerations at A and B:

12-4 Gravitation and Extended Objects Tidal Forces The friction of the water provides the earth a negative angular acceleration. Conservation of momentum causes an increase in distance from the earth,.5m a year. Ultimately the earth will slow its rotation until one face will always the moon.

12-5 A Closer Look at Gravitation Two massive bodies will exert equal and opposite forces on each other – does that mean they orbit around each other? Yes – actually, they both orbit around their common center of mass. For a small planet and a large star, the difference between this and orbiting the star’s center is tiny. For a large planet or a double star, it can be a large effect.

12-5 A Closer Look at Gravitation If several objects are exerting gravitational forces on each other, net force is vector sum of forces, and net potential energy is sum of potential energies.

12-5 A Closer Look at Gravitation Equality of inertial and gravitational masses: The m in F = ma is the inertial mass, a measure of how resistant an object is to an external force. The m in the gravitational force equation is the gravitational mass, a measure of the gravitational force the object exerts on another mass. Apparently the inertial and gravitational masses of an object are equal, but why?

12-6 Einstein’s Theory of Gravitation Equivalence of gravitational and inertial mass led Einstein to formulate the equivalence principle: No experiment can distinguish between the following two situations: (1) a physical system at rest that is subject to a uniform gravitational force; and (2) a physical system that is uniformly accelerating in the absence of gravity.

12-6 Einstein’s Theory of Gravitation Predictions of equivalence principle: Light falls under the influence of gravity Gravitational lenses Black holes Precession of planetary orbits

Summary of Chapter 12 Kepler’s three laws: 1. Planets move in planar elliptical paths with the Sun at one focus of the ellipse. 2. During equal time intervals, the radius vector from the Sun to a planet sweeps out equal areas. 3. If T is the time that it takes for a planet to make one full revolution around the Sun, and if R is half the major axis of the ellipse (R reduces to the radius of the orbit of the planet if that orbit is circular), then (12-1) where C is a constant whose value is the same for all planets.

Summary of Chapter 12, cont. Newton showed that Kepler’s laws result from universal law of gravitation: (12-4) Not all masses are pointlike; if spherically symmetric, can treat all mass within radius of measurement to be concentrated at center Differences in gravitational force across an extended object are tidal forces

Summary of Chapter 12, cont. Einstein’s general theory of relativity gives a much more complete description of gravity; starts with equivalence principle for inertial and gravitational mass