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Chapter 8 Gravity

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Newton’s law of gravitation Any two (or more) massive bodies attract each other Gravitational force (Newton's law of gravitation) Gravitational constant G = 6.67*10 –11 N*m 2 /kg 2 = 6.67*10 –11 m 3 /(kg*s 2 ) – universal constant

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Chapter 8 Problem 15 Two identical lead spheres with their centers 14 cm apart attract each other with a 0.25-µN force. Find their mass.

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Gravitation and the superposition principle For a group of interacting particles, the net gravitational force on one of the particles is For a particle interacting with a continuous arrangement of masses (a massive finite object) the sum is replaced with an integral

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Shell theorem For a particle interacting with a uniform spherical shell of matter Result of integration: a uniform spherical shell of matter attracts a particle that is outside the shell as if all the shell's mass were concentrated at its center

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Shell theorem For a particle inside a uniform spherical shell of matter Result of integration: a uniform spherical shell of matter exerts no net gravitational force on a particle located inside it

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Gravity force near the surface of Earth Earth can be though of as a nest of shells, one within another and each attracting a particle outside the Earth’s surface Thus Earth behaves like a particle located at the center of Earth with a mass equal to that of Earth g = 9.8 m/s 2 This formula is derived for stationary Earth of ideal spherical shape and uniform density

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Gravity force near the surface of Earth In reality g is not a constant because: Earth is rotating, Earth is approximately an ellipsoid with a non-uniform density

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Gravitational field A gravitational field exists at every point in space When a particle is placed at a point where there is gravitational field, the particle experiences a force The field exerts a force on the particle The gravitational field is defined as: The gravitational field is the gravitational force experienced by a test particle placed at that point divided by the mass of the test particle

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Gravitational field The presence of the test particle is not necessary for the field to exist The source particle creates the field The gravitational field vectors point in the direction of the acceleration a particle would experience if placed in that field The magnitude is that of the freefall acceleration at that location

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Gravitational potential energy Gravitation is a conservative force (work done by it is path-independent) For conservative forces (Ch. 7):

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Gravitational potential energy To remove a particle from initial position to infinity Assuming U ∞ = 0

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Gravitational potential energy

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Orbits Accounting for the shape of Earth, projectile motion (Ch. 3) has to be modified:

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Orbits The total mechanical energy E = K + U determines the type of orbit an object follows: E < 0: The object is in a bound, elliptical orbit

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Orbits The total mechanical energy E = K + U determines the type of orbit an object follows: Special cases include circular orbits and the straight-line paths of falling objects

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Orbits The total mechanical energy E = K + U determines the type of orbit an object follows: E > 0: The orbit is unbound and hyperbolic

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Orbits The total mechanical energy E = K + U determines the type of orbit an object follows: E = 0: The borderline case gives a parabolic orbit

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Orbits Elliptical orbits of planets are described by a semimajor axis a and an eccentricity e For most planets, the eccentricities are very small (Earth's e is 0.00167)

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Orbits The “parabolic” trajectories of projectiles near Earth’s surface are actually sections of elliptical orbits that intersect Earth

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Orbits The trajectories are parabolic only in the approximation that we can neglect Earth’s curvature and the variation in gravity with distance from Earth’s center

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Orbits For a circular orbit and the Newton’s Second law Kinetic energy of a satellite Total mechanical energy of a satellite

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Orbits For an elliptic orbit it can be shown Orbits with different e but the same a have the same total mechanical energy

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Chapter 8 Problem 40 A white dwarf is a collapsed star with roughly the Sun’s mass compressed into the size of Earth. What would be (a) the orbital speed and (b) the orbital period for a spaceship in orbit just above the surface of a white dwarf?

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Escape speed Escape speed: speed required for a particle to escape from the planet into infinity (and stop there)

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Escape speed If for some astronomical object Nothing (even light) can escape from the surface of this object – a black hole

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Chapter 8 Problem 54 A projectile is launched vertically upward from a planet of mass M and radius R; its initial speed is twice the escape speed. Derive an expression for its speed as a function of the distance r from the planet’s center.

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Kepler’s laws Three Kepler’s laws 1. The law of orbits: All planets move in elliptical orbits, with the Sun at one focus 2. The law of areas: A line that connects the planet to the Sun sweeps out equal areas in the plane of the planet’s orbit in equal time intervals 3. The law of periods: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit Johannes Kepler (1571-1630) Tycho Brahe/ Tyge Ottesen Brahe de Knudstrup (1546-1601)

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Third Kepler’s law For a circular orbit and the Newton’s Second law From the definition of a period For elliptic orbits

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Chapter 8 Problem 23 The Mars Reconnaissance Orbiter circles the red planet with a 112-min period. What’s the spacecraft’s altitude?

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Questions?

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Answers to the even-numbered problems Chapter 8 Problem 16 4.62 × 10 −8 N

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Answers to the even-numbered problems Chapter 8 Problem 20 1.87 y

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Answers to the even-numbered problems Chapter 8 Problem 26 1.67 × 10 6 m

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