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Chandra X-ray image of two black holes (blue) in NGC 6240
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In this equation, G is Newton s universal gravitational constant, M is the mass of the black hole, and c is the speed of light The edge of the sphere defined by the Schwarzschild radius is called the event horizon At the event horizon, the escape velocity equals the speed of light Because nothing travels faster than the speed of light, objects that cross the event horizon can never escape
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Solve The escape velocity of an object leaving the event horizon can be represented by the following equation: v
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Indirect and Direct Evidence Black holes have three physical properties that can theoretically be measured mass, angular momentum, and electric charge A black hole s
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In this equation, G is Newton s universal gravitational constant, M is the mass of the black hole, and Rs is the radius of the black hole Show that the escape velocity equals the speed of light
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71 Planetary Motion and Gravitation
Vocabulary
Kepler s first law (p 172) Kepler s second law (p 173) Kepler s third law (p 173) gravitational force (p 175) law of universal gravitation (p 175)
Key Concepts
Kepler s first law states that planets move in elliptical orbits, with the Sun at one focus Kepler s second law states that an imaginary line from the Sun to a planet sweeps out equal areas in equal times Kepler s third law states that the square of the ratio of the periods of any two planets is equal to the cube of the ratio of their distances from the Sun
TA 2 TB rA 3 rB
Newton s law of universal gravitation states that the gravitational force between any two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers The force is attractive and along a line connecting their centers F G
m1m2 r2
Newton s law of universal gravitation can be used to rewrite Kepler s third law to relate the radius and period of a planet to the mass of the Sun T2
4 2 3 r GmS
72 Using the Law of Universal Gravitation
Vocabulary
gravitational field (p 183) inertial mass (p 183) gravitational mass (p 184)
Key Concepts
The speed of an object in circular orbit is given by the following expression v
GmE r
The period of a satellite in a circular orbit is given by the following expression T 2
r3 GmE
All objects have gravitational fields surrounding them g
Gm r2
Gravitational mass and inertial mass are two essentially different concepts The gravitational and inertial masses of an object, however, are numerically equal m inertial
Fnet a
mgrav
r2Fgrav Gm
Einstein s general theory of relativity describes gravitational attraction as a property of space itself
physicsppcom/vocabulary_puzzlemaker
Concept Mapping
22 Create a concept map using these terms: planets,
stars, Newton s law of universal gravitation, Kepler s first law, Kepler s second law, Kepler s third law, Einstein s general theory of relativity
35 Show that the dimensions of g in the equation
g F/m are in m/s2 (72)
36 If Earth were twice as massive but remained the
same size, what would happen to the value of g (72)
Mastering Concepts
23 In 1609, Galileo looked through his telescope at
Jupiter and saw four moons The name of one of the moons that he saw is Io Restate Kepler s first law for Io and Jupiter (71)
Applying Concepts
37 Golf Ball The force of gravity acting on an object
near Earth s surface is proportional to the mass of the object Figure 7-18 shows a tennis ball and golf ball in free fall Why does a tennis ball not fall faster than a golf ball
24 Earth moves more slowly in its orbit during summer
in the northern hemisphere than it does during winter Is it closer to the Sun in summer or in winter (71)
25 Is the area swept out per unit of time by Earth moving
around the Sun equal to the area swept out per unit of time by Mars moving around the Sun (71)
26 Why did Newton think that a force must act on the
Moon (71)
27 How did Cavendish demonstrate that a
gravitational force of attraction exists between two small objects (71)
28 What happens to the gravitational force between
two masses when the distance between the masses is doubled (71)
Figure 7-18
29 According to Newton s version of Kepler s third law,
how would the ratio T 2/r 3 change if the mass of the Sun were doubled (71)
38 What information do you need to find the mass of
Jupiter using Newton s version of Kepler s third law
39 The mass of Pluto was not known until a satellite of
the planet was discovered Why
30 How do you answer the question, What keeps a
satellite up (72)
40 Decide whether each of the orbits shown in
Figure 7-19 is a possible orbit for a planet
c c Sun Sun
31 A satellite is orbiting Earth On which of the
following does its speed depend (72) a mass of the satellite b distance from Earth c mass of Earth
32 What provides the force that causes the centripetal
acceleration of a satellite in orbit (72)
33 During space flight, astronauts often refer to forces
as multiples of the force of gravity on Earth s surface What does a force of 5g mean to an astronaut (72)
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