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Chapter 12 Gravity DEHS Physics 1

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Newton’s Law of Universal Gravity The force of gravity between any two point objects of mass m 1 and m 2 is attractive and of magnitude Here, r is the distance between the masses G is called the universal gravitation constant and it is equal to

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Inverse Square Dependence & Superposition Newton’s Law of Universal Gravitation (LUG) is an example of an inverse square dependence – The strength of gravity falls off rapidly with distance, but it never becomes zero – Gravity has an infinite range The net gravitational force like other field forces can be found using the principle of superposition, it is the vector sum of each other the forces individually

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Example 12-1 An astronaut is a distance d from his spaceship when he experiences a gravitational pull of 80 N from his ship. He drifts out to a distance of 4d. Now what gravitational force does the ship exert on the astronaut?

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Example 12-2 The picture shows an arrangement of three particles, particle 1 of mass m 1 = 6.0 kg, particles 2 and 3 of mass m 2 = m 3 = 4.0 kg, and a distance a = 2.0 cm. What is the net gravitational force on particle 1 due to the other particles?

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Example 12-3 The picture shows an arrangement of five particles, with masses m 1 = 8.0 kg, m 2 = m 3 = m 4 = m 5 = 2.0 kg, and with a = 2.0 cm and θ = 30°

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Gravity due to a Sphere The net force exerted by a sphere whose mass is SPHERICALLY distributed (does not have to by uniformly distributed) throughout an object is the same as if all of the sphere’s mass were concentrated at its center

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The Cavendish Experiment – “Weighing the Earth”

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“Weighing the Earth” and consequences Cavendish didn’t actually weigh the Earth, he accurately measured the value for G Prior to his experiment, the quantities g and R E were known from direct measurement Calculate the mass of Earth:

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“Weighing the Earth” and consequences Calculate the volume of the Earth: Calculate the average density of the Earth: Typical rocks found near the surface of the Earth, have a density of about 3.00 g/cm 3. What can you conclude?

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Acceleration due to gravity Acceleration due to gravity (g) varies by location g is determined by the distance from the center to the planet it is near and the mass of the planet

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g as a function of h

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Example 12-4 A hypothetical planet has a mass of 2.5 times that of Earth, but the same surface gravity as Earth. What is this planet’s radius (in Earth radii)?

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Example 12-5 You are explaining to friends why astronauts feel weightless orbiting in the space shuttle, and they respond that they thought gravity was just a lot weaker up there. Convince them by calculating how much weaker gravity is 300 km above Earth’s surface.

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Example 12-6 Certain neutron stars (extremely dense stars) are believed to be rotating at about 1 rev/s. If such a star has a radius of 20 km, what must be its minimum mass so that material on its surface remains in place during the rapid rotation?

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Some important Astronomy Terms Astronomical Unit (AU): is the mean distance from the Earth to the Sun (1 AU= 1.5×10 11 m) Periapsis: The point of closest approach of an object to the body being orbited – perihelion: closest distance from the Sun – perigee: closest distance from the Earth Apoapsis: The point of furthest excision of an object from the body being orbited – aphelion: furthest distance from the Sun – apogee: furthest distance from the Earth

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Kepler’s 1 st Law of Orbit – Law of Elliptical Orbits Planets follow elliptical orbits, with the Sun at one focus of the ellipse The semi-major axis (a) is also equal to the average distance from a focus to points on the ellipse

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Orbital Distance at Apoapsis and Periapsis The eccentricity (e) is defined so that ea is the distance from the center of the ellipse to the either focus In circular orbits the two foci merge to one focus and e = 0

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Orbital Shape depends on eccentricity! All orbits take the shape of a conic section, and the shape depends on the eccentricity Closed orbits includes circular orbits (e = 0) and elliptical orbits (0 < e < 1) Open trajectories include parabolas and hyperbolas where an object approach the body and will never return (an escape trajectory) – Parabolic trajectory (e = 1), object will reach infinity with zero KE (E = 0) – Hyperbolic trajectory (e > 1) object will reach infinity with excess KE (E > 0)

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Kepler’s 2 nd Law of Orbit – Law of Equal Areas As a planet moves in its orbit, it sweeps out an equal amount of area in an equal amount of time This means that they move faster when it is closer to the Sun This is a consequence of the conservation of angular momentum

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Example 12-7 The Earth’s orbit around the Sun is in the shape of an ellipse. Its aphelion is 152,000,000 km. Its perihelion, is 147,000,000 km. The Earth’s speed is 30,300 at perihelion. What is the Earth’s orbital speed at aphelion?

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Kepler’s 3 rd Law of Orbit – Law of Harmonics The period, T, of a planet is proportional to its mean distance from the Sun raised to the 3/2 power or

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Derive Kepler’s 3 rd Law

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Example 12-8 Comet Halley orbits the Sun with a with a period of 76 years, and in 1986, had a distance of closest approach to the Sun, its perihelion distance R P, of 8.9×10 10 m. (a) What is Halley’s average distance from the Sun? (b) What is the eccentricity of comet Halley? (c) What is the comet’s farthest distance from the Sun, its aphelion distance?

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Example 12-9 We SEE nothing at the exact center of our galaxy, the Milky Way. However, we know that something has to be there, because all of the stars in our galaxy orbit this point. We can observe a star, called S2 as it moves around this mysterious object called Sagittarius A*. S2 orbits Sagittarius A* with a period of 15.2 years and with a semimajor axis of 5.5 light days (= 1.42×10 14 m). What is the mass of Sagittarius A*? What is Sagittarius A*?

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Kepler’s 3 rd Law (alternate form) Because the ratio T 2 /a 3 is a constant for any objects in orbit around the same mass, we can say When comparing to Earth, remember a = 1 AU and T = 1 yr or

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Example (a)Uranus has a semi-major axis of AU. Calculate Uranus’ orbital period? (b)It takes Mercury only 88 days to complete one orbit around the Sun. Calculate Mercury’s mean distance from the Sun.

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Satellite Orbits A satellite is said to be in geosynchronous orbit if it has a period of 1 day – Such satellites are used in communications and for weather forecasting For a satellite in a circular orbit:

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Example (a)Calculate the orbital speed of a satellite in circular orbit at an altitude of 1,000 km above the surface of Earth. (b)Calculate this satellite’s orbital period.

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Gravitational Potential Energy at any r Note that U = 0 at infinity, this is the common convention for problems of an astronomical scale – Doesn’t have to be, because only differences in U matter

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U = mgh when h << R

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Energy Conservation The mechanical energy E of an object of mass m, moving at speed v, at a distance from a planet of mass M is given by For a satellite in Kepler orbit, the total energy can be shown to be:

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Orbital Speed

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Example An asteroid, headed directly toward Earth, has a speed of 12 km/s relative to Earth when the asteroid is 10 Earth radii from Earth’s center. Neglecting the effects of Earth’s atmosphere on the asteroid, find the asteroid’s speed when it reaches Earth’s surface.

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Example Venus orbits the Sun with a semi-major axis of 1.082×10 11 m and an eccentricity of Calculate Venus’ orbital speed at periheion and aphelion.

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Orbital Maneuvers The radius at which a satellite follows a circular orbital path is directly related to its speed Moving to lower orbits requires you to use your decelerating rockets – Slows satellite causing it to orbit in an elliptical orbit, a second rocket burn is required to turn it into a circular orbit Moving to higher orbits requires you to use your accelerating rockets

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The Hohmann Transfer to a lower orbit

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The Hohmann Transfer to a higher orbit

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Escape Speed Escape speed is the minimum speed required to leave a planet and to never return – To do this, K = 0 when r = ∞ (where U = 0)

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Example Suppose that you fired a cannonball straight upward with an initial speed equal to one-half the escape speed. How far from the center of the Earth does this rocket travel before momentarily coming to rest? (Ignore air resistance in the Earth’s atmosphere)

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Example What multiple of the energy needed to escape from Earth gives the energy to escape from (a) the Moon, and (b) Jupiter? (the moon’s radius is Earths and the moon’s mass is Earths, while Jupiter’s radius is Earths and its mass is Earths)

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Kepler Orbits Orbital ShapeEccentricity (e)Semi-Major Axis (a) Total Energy (E) Circlee = 0a > 0E < 0 Ellipse0 < e < 1a > 0E < 0 Parabolae = 1noneE = 0 Hyperbolae > 1a < 0E > 0

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Black Holes & the Schwarzchild Radius Looking at the escape speed equation: you can seed that escape speed increases with increasing mass and decreasing radius If an object were to be compressed to a size small enough that the escape speed necessary were the speed of light, c then we have a black hole

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Example Mr. Bailey has a mass of about 110 kg. How small would you have to compress him to turn him into a black hole?

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Evidence for Black Holes Gravitational Lensing Motion of stars near it

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