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chap8

# chap8 - Gravity Motion GravitationalFields Newtons Law of...

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Gravity  Newton’s Law of Gravitation Kepler’s Laws of Planetary  Motion Gravitational Fields

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Newton’s Law of Gravitation m 1 m 2 r There is a force of gravity between any pair of objects anywhere. The force is proportional to each mass and inversely proportional to the square of the distance between the two objects. Its equation is: F G = G m 1 m 2 r 2 The constant of proportionality is G, the universal gravitation constant. G = 6.67 · 10 -11 N·m 2 / kg 2 . Note how the units of G all cancel out except for the Newtons, which is the unit needed on the left side of the equation.
Gravity Example F G = G m 1 m 2 r 2 How hard do two planets pull on each other if their masses are 1.23 × 10 26 kg and 5.21 × 10 22 kg and they 230 million kilometers apart? This is the force each planet exerts on the other. Note the denominator is has a factor of 10 3 to convert to meters and a factor of 10 6 to account for the million. It doesn’t matter which way or how fast the planets are moving. (6.67 · 10 -11 N·m 2 / kg 2 ) (1.23 · 10 26 kg) (5.21 · 10 22 kg) = (230 · 10 3 · 10 6 m) 2 = 8.08 · 10 15 N

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3rd Law: Action-Reaction In the last example the force on each planet is the same. This is due to to Newton’s third law of motion: the force on Planet 1 due to Planet 2 is just as strong but in the opposite direction as the force on Planet 2 due to Planet 1. The effects of these forces are not the same, however, since the planets have different masses. For the big planet: a = (8.08 · 10 15 N) / (1.23 · 10 26 kg) = 6.57 · 10 -11 m/s 2 . For the little planet: a = (8.08 · 10 15 N) / (5.21 · 10 22 kg) = 1.55 · 10 -7 m/s 2 . 5.21 · 10 22 kg 1.23 · 10 26 kg 8.08 · 10 15 N 8.08 · 10 15 N
Inverse Square Law F G = G m 1 m 2 r 2 The law of gravitation is called an inverse square law because the magnitude of the force is inversely proportional to the square of the separation. If the masses are moved twice as far apart, the force of gravity between is cut by a factor of four. Triple the separation and the force is nine times weaker. What if each mass and the separation were all quadrupled? answer: no change in the force

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Calculating the Gravitational Constant In 1798 Sir Henry Cavendish suspended a rod with two small masses (red) from a thin wire. Two larger mass (green) attract the small masses and cause the wire to twist slightly, since each force of attraction produces a torque in the same direction. By varying the masses and measuring the separations and the amount of twist, Cavendish was the first to calculate G . Since G is only 6.67 · 10 -11 N·m 2 / kg 2 , the measurements had to be very precise.
Calculating the mass of the Earth Knowing G, we can now actually calculate the mass of the Earth. All we do is write the weight of any object in two different ways and equate them. Its weight is the force of gravity between it and the Earth, which is F G in the equation below. M E is the mass of the Earth, R E is the radius of the Earth, and m is the mass of the object.

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