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**Ch13 – Fr 2 Ch14 – Fr 28 Ch15 – Fr 54 Ch 16 – Fr 82**

Chapters Universal Gravitation 14 Satellite motion 15 Special Relativity 16 Relativity – Momentum, etc. Ch13 – Fr 2 Ch14 – Fr 28 Ch15 – Fr 54 Ch 16 – Fr 82

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**Everything pulls on everything else.**

Newton reasoned that the moon is falling toward Earth for the same reason an apple falls from a tree—they are both pulled by Earth’s gravity. Newton understood the concept of inertia developed earlier by Galileo. He knew that without an outside force, moving objects continue to move at constant speed in a straight line. He knew that if an object undergoes a change in speed or direction, then a force is responsible.

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13.1 The Falling Apple Newton saw the apple fall, or maybe even felt it fall on his head. Perhaps he looked up through the apple tree branches and noticed the moon. He may have been puzzled by the fact that the moon does not follow a straight-line path, but instead circles about Earth. He knew that circular motion is accelerated motion, which requires a force. Newton had the insight to see that the moon is falling toward Earth, just as the apple is.

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13.2 The Falling Moon The moon is actually falling toward Earth but has great enough tangential velocity to avoid hitting Earth. Newton realized that if the moon did not fall, it would move off in a straight line and leave its orbit. His idea was that the moon must be falling around Earth. Thus the moon falls in the sense that it falls beneath the straight line it would follow if no force acted on it. He hypothesized that the moon was simply a projectile circling Earth under the attraction of gravity.

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13.2 The Falling Moon This original drawing by Isaac Newton shows how a projectile fired fast enough would fall around Earth and become an Earth satellite.

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13.2 The Falling Moon Newton calculated how far the circle of the moon’s orbit lies below the straight-line distance the moon would otherwise travel in one second. His value turned out to be about the 1.4-mm distance accepted today. He was unsure of the exact Earth-moon distance and whether the correct distance to use was the distance between their centers.

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13.2 The Falling Moon It wasn’t until after Newton invented a new branch of mathematics, calculus, to prove his center-of-gravity hypothesis, that he published the law of universal gravitation. Newton generalized his moon finding to all objects, and stated that all objects in the universe attract each other.

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13.3 The Falling Earth Newton’s theory of gravity confirmed the Copernican theory of the solar system. No longer was Earth considered to be the center of the universe. It became clear that the planets orbit the sun in the same way that the moon orbits Earth. The planets continually “fall” around the sun in closed paths.

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**13.4 Newton’s Law of Universal Gravitation**

Newton discovered that gravity is universal. Everything pulls on everything else in a way that involves only mass and distance.

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**13.4 Newton’s Law of Universal Gravitation**

Your weight is less at the top of a mountain because you are farther from the center of Earth.

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**G = 6.67 × 10−11 N·m2/kg2) 13.4 Newton’s Law of Universal Gravitation**

The Universal Gravitational Constant, G The law of universal gravitation can be expressed as an exact equation when a proportionality constant is introduced. The universal gravitational constant, G, in the equation for universal gravitation describes the strength of gravity. G = 6.67 × 10−11 N·m2/kg2)

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**A simpler method was developed by Philipp von Jolly.**

13.4 Newton’s Law of Universal Gravitation G was first measured 150 years after Newton’s discovery of universal gravitation by an English physicist, Henry Cavendish. A simpler method was developed by Philipp von Jolly.

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**13.4 Newton’s Law of Universal Gravitation**

Cavendish’s first measure of G was called the “Weighing the Earth” experiment. Once the value of G was known, the mass of Earth was easily calculated. The force that Earth exerts on a mass of 1 kilogram at its surface is 10 newtons. The distance between the 1-kilogram mass and the center of mass of Earth is Earth’s radius, 6.4 × 106 meters m1 = Fd2 Gm2 from which the mass of Earth m1 = 6 × 1024 kilograms.

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**13.5 Gravity and Distance: The Inverse-Square Law**

Gravity decreases according to the inverse-square law. The force of gravity weakens as the square of distance. Butter spray travels outward from the nozzle in straight lines. Like gravity, the spray obeys an inverse-square law.

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**Gravitational force is plotted versus distance from Earth’s center.**

It also applies to all cases where the effect from a localized source spreads evenly throughout the surrounding space. Examples are light, radiation, and sound.

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**13.5 Gravity and Distance: The Inverse-Square Law**

think! Suppose that an apple at the top of a tree is pulled by Earth’s gravity with a force of 1 N. If the tree were twice as tall, would the force of gravity on the apple be only 1/4 as strong? Explain your answer. Answer: No, the twice-as-tall apple tree is not twice as far from Earth’s center. The taller tree would have to have a height equal to the radius of Earth (6370 km) before the weight of the apple would reduce to 1/4 N.

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13.6 Gravitational Field Earth can be thought of as being surrounded by a gravitational field that interacts with objects and causes them to experience gravitational forces. We can regard the moon as in contact with the gravitational field of Earth. A gravitational field occupies the space surrounding a massive body and is considered a force field.

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13.6 Gravitational Field Field lines can also represent the pattern of Earth’s gravitational field. *The field lines are closer together where the gravitational field is stronger. *Any mass in the vicinity of Earth will be accelerated in the direction of the field lines at that location. *Earth’s gravitational field follows the inverse-square law. *Earth’s gravitational field is strongest near Earth’s surface and weaker at greater distances from Earth.

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**13.7 Gravitational Field Inside a Planet**

The gravitational field of Earth at its center is zero! In a cavity at the center of Earth, your weight would be zero, because you would be pulled equally by gravity in all directions.

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**13.8 Weight and Weightlessness**

Pressure against Earth is the sensation we interpret as weight. The force of gravity, like any force, causes acceleration. Objects under the influence of gravity are pulled toward each other and accelerate. We are almost always in contact with Earth, so we think of gravity as something that presses us against Earth rather than as something that accelerates us.

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**13.8 Weight and Weightlessness**

Repeat this weighing procedure in a moving elevator and you would find your weight reading would vary during accelerated motion. When the elevator accelerates upward, the bathroom scale and floor push harder against your feet. The scale would show an increase in your weight. The sensation of weight is equal to the force that you exert against the support- ing floor.

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**13.8 Weight and Weightlessness**

Both people at right are without a support force and therefore experience weightlessness. Rather than define weight as the force of gravity that acts on you, it is more practical to define weight as the force you exert against a supporting floor. According to this definition, you are as heavy as you feel. The condition of weightlessness is not the absence of gravity, but the absence of a support force.

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13.9 Ocean Tides Newton showed that the ocean tides are caused by differences in the gravitational pull of the moon on opposite sides of Earth.

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13.9 Ocean Tides A spring tide is a high or low tide that occurs when the sun, Earth, and moon are all lined up. The tides due to the sun and the moon coincide, making the high tides higher than average and the low tides lower than average. Spring tides occur at the times of a new or full moon.

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13.9 Ocean Tides A neap tide occurs when the moon is halfway between a new moon and a full moon, in either direction. The pulls of the moon and sun are perpendicular to each other. The solar and lunar tides do not overlap, so the high tides are not as high and low tides are not as low.

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13.9 Ocean Tides When the attractions of the sun and the moon are at right angles to each other (at the time of a half moon), neap tides occur.

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**13.11 Universal Gravitation**

Newton’s Impact on Science Few theories have affected science and civilization as much as Newton’s theory of gravity. Newton demonstrated that by observation and reason, people could uncover the workings of the physical universe.

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Chapter 14 Satellites The path of an Earth satellite follows the curvature of the Earth.

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14.1 Earth Satellites If you drop a stone, it will fall in a straight-line path to the ground below. If you move your hand, the stone will land farther away. What would happen if the curvature of the path matched the curvature of Earth?

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14.1 Earth Satellites Throw a stone at any speed and one second later it will have fallen 5 m below where it would have been without gravity.

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14.1 Earth Satellites In the curvature of Earth, the surface drops a vertical distance of nearly 5 meters for every 8000 meters tangent to its surface.

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14.1 Earth Satellites The orbital speed for close orbit about Earth is 8 km/s. That is an impressive 29,000 km/h (or 18,000 mi/h). At that speed, atmospheric friction would burn an object to a crisp. A satellite must stay 150 kilometers or more above Earth’s surface—to keep from burning due to the friction.

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14.2 Circular Orbits A satellite in circular orbit around Earth is always moving perpendicular to gravity and parallel to Earth’s surface at constant speed.

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14.2 Circular Orbits The speeds of the bowling ball and the satellite are not affected by the force of gravity because there is no horizontal component of gravitational force.

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14.2 Circular Orbits The satellite is always moving at a right angle (perpendicular) to the force of gravity. It doesn’t move in the direction of gravity, which would increase its speed. It doesn’t move in a direction against gravity, which would decrease its speed. No change in speed occurs—only a change in direction. For a satellite close to Earth, the time for a complete orbit around Earth, its period, is about 90 minutes. For higher altitudes, the orbital speed is less and the period is longer.

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14.2 Circular Orbits Communications satellites are located in orbit 6.5 Earth radii from Earth’s center, so that their period is 24 hours (geosynchronous). This period matches Earth’s daily rotation. They orbit in the plane of Earth’s equator and they are always above the same place. The moon is farther away, and has a 27.3-day period.

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14.2 Circular Orbits The International Space Station (ISS) orbits at 360 kilometers above Earth’s surface. Acceleration toward Earth is somewhat less than 1 g because of altitude. This acceleration, however, is not sensed by the astronauts; relative to the station, they experience zero g.

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14.2 Circular Orbits think! Satellites in close circular orbit fall about 5 m during each second of orbit. How can this be if the satellite does not get closer to Earth? Answer: In each second, the satellite falls about 5 m below the straight-line tangent it would have taken if there were no gravity. Earth’s surface curves 5 m below an 8-km straight-line tangent. Since the satellite moves at 8 km/s, it “falls” at the same rate Earth “curves.”

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14.3 Elliptical Orbits A satellite in orbit around Earth traces an oval-shaped path called an ellipse.

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14.3 Elliptical Orbits An ellipse is the closed path taken by a point that moves in such a way that the sum of its distances from two fixed points is constant. The two fixed points in an ellipse are called foci. For a satellite orbiting a planet, the center of the planet is at one focus and the other focus could be inside or outside the planet.

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**14.5 Kepler’s Laws of Planetary Motion**

Kepler’s first law states that the path of each planet around the sun is an ellipse with the sun at one focus. Kepler’s second law states that each planet moves so that an imaginary line drawn from the sun to any planet sweeps out equal areas of space in equal time intervals. Kepler’s third law states that the square of the orbital period of a planet is directly proportional to the cube of the average distance of the planet from the sun.

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**14.5 Kepler’s Laws of Planetary Motion**

Kepler’s First Law Kepler’s expectation that the planets would move in perfect circles around the sun was shattered after years of effort. He found the paths to be ellipses.

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**14.5 Kepler’s Laws of Planetary Motion**

Kepler’s Second Law Kepler also found that the planets do not go around the sun at a uniform speed but move faster when they are nearer the sun and more slowly when they are farther from the sun. An imaginary line or spoke joining the sun and the planet sweeps out equal areas of space in equal times.

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**14.5 Kepler’s Laws of Planetary Motion**

Kepler was the first to coin the word satellite. He had no clear idea why the planets moved as he discovered. He lacked a conceptual model.

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**14.5 Kepler’s Laws of Planetary Motion**

Kepler’s Third Law After ten years of searching for a connection between the time it takes a planet to orbit the sun and its distance from the sun, Kepler discovered a third law. Kepler found that the square of any planet’s period (T) is directly proportional to the cube of its average orbital radius (r).

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**14.5 Kepler’s Laws of Planetary Motion**

This means that the ratio is the same for all planets. If a planet’s period is known, its average orbital radial distance is easily calculated. Kepler’s laws apply not only to planets but also to moons or any satellite in orbit around any body.

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**14.5 Kepler’s Laws of Planetary Motion**

Kepler was familiar with Galileo’s concepts of inertia and accelerated motion, but he failed to apply them to his own work. Like Aristotle, he thought that the force on a moving body would be in the same direction as the body’s motion. Kepler never appreciated the concept of inertia. Galileo, on the other hand, never appreciated Kepler’s work and held to his conviction that the planets move in circles.

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14.6 Escape Speed When a payload is put into Earth-orbit by a rocket, the speed and direction of the rocket are very important. If the rocket were launched vertically and quickly achieved a speed of 8 km/s, it would soon come crashing back at 8 km/s. To achieve orbit, the payload must be launched horizontally at 8 km/s once above air resistance.

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14.6 Escape Speed The initial thrust of the rocket lifts it vertically. Another thrust tips it from its vertical course. When it is moving horizontally, it is boosted to the required speed for orbit.

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14.6 Escape Speed Earth Neglecting air resistance, fire anything at any speed greater than 11.2 km/s, and it will leave Earth, going more and more slowly, but never stopping. How much work is required to move a payload against the force of Earth’s gravity to a distance very, very far (“infinitely far”) away? Gravity diminishes rapidly with distance due to the inverse-square law. Most of the work done in launching a rocket occurs near Earth.

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14.6 Escape Speed The value of PE for a 1-kilogram mass infinitely far away is 62 million joules (MJ). To put a payload infinitely far from Earth’s surface requires at least 62 MJ of energy per kilogram of load. A KE per unit mass of 62 MJ/kg corresponds to a speed of 11.2 km/s. The escape speed is the minimum speed necessary for an object to escape permanently from a gravitational field.

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14.6 Escape Speed

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14.6 Escape Speed The escape speeds refer to the initial speed given by a brief thrust, after which there is no force to assist motion. But we could escape Earth at any sustained speed greater than zero, given enough time.

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Chapter 15 Motion through space is related to motion in time. Motion through space is related to motion in time. The first person to understand the relationship between space and time was Albert Einstein. Einstein stated in 1905 that in moving through space we also change our rate of proceeding into the future—time itself is altered. His theories changed the way scientists view the workings of the universe.

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15.1 Space-Time From the viewpoint of special relativity, you travel through a combination of space and time. You travel through space-time. Newton and other investigators before Einstein thought of space as an infinite expanse in which all things exist. Einstein theorized both space and time exist only within the universe. There is no time or space “outside.” Einstein reasoned that space and time are two parts of one whole called space-time. Einstein’s special theory of relativity describes how time is affected by motion in space at constant velocity, and how mass and energy are related.

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15.1 Space-Time The universe does not exist in a certain part of infinite space, nor does it exist during a certain era in time. Space and time exist within the universe.

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15.1 Space-Time You are moving through time at the rate of 24 hours per day. This is only half the story. To get the other half, convert your thinking from “moving through time” to “moving through space-time.” When you stand still, all your traveling is through time. When you move a bit, then some of your travel is through space and most of it is still through time.

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15.1 Space-Time If you were able to travel at the speed of light, all your traveling would be through space, with no travel through time! Light travels through space only and is timeless. From the frame of reference of a photon traveling from one part of the universe to another, the journey takes no time at all!

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15.1 Space-Time Whenever we move through space, we, to some degree, alter our rate of moving into the future. This is known as time dilation, or the stretching of time. The special theory of relativity that Einstein developed rests on two fundamental assumptions, or postulates.

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**15.2 The First Postulate of Special Relativity**

The first postulate of special relativity states that all the laws of nature are the same in all uniformly moving frames of reference. Einstein reasoned all motion is relative and all frames of reference are arbitrary. A spaceship, for example, cannot measure its speed relative to empty space, but only relative to other objects.

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**15.2 The First Postulate of Special Relativity**

Spaceman A considers himself at rest and sees spacewoman B pass by, while spacewoman B considers herself at rest and sees spaceman A pass by. Spaceman A and spacewoman B will both observe only the relative motion.

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**15.2 The First Postulate of Special Relativity**

Einstein’s first postulate of special relativity assumes our inability to detect a state of uniform motion. Many experiments can detect accelerated motion, but none can, according to Einstein, detect the state of uniform motion.

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**15.3 The Second Postulate of Special Relativity**

The second postulate of special relativity states that the speed of light in empty space will always have the same value regardless of the motion of the source or the motion of the observer. Einstein asked: “What would a light beam look like if you traveled along beside it?” In classical physics, the beam would be at rest to such an observer. Einstein became convinced that this was impossible.

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**15.3 The Second Postulate of Special Relativity**

Einstein concluded that if an observer could travel close to the speed of light, he would measure the light as moving away at 300,000 km/s. Einstein’s second postulate of special relativity assumes that the speed of light is constant.

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**Everyone who measures the speed of light will get the same value, c.**

15.3 The Second Postulate of Special Relativity The speed of light is constant regardless of the speed of the flashlight or observer. Everyone who measures the speed of light will get the same value, c.

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15.4 Time Dilation Time dilation occurs ever so slightly for everyday speeds, but significantly for speeds approaching the speed of light. Einstein proposed that time can be stretched depending on the motion between the observer and the events being observed. The stretching of time is time dilation.

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15.4 Time Dilation Einstein showed the relation between the time t0 in the observer’s own frame of reference and the relative time t measured in another frame of reference is: where v represents the relative velocity between the observer and the observed and c is the speed of light.

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15.5 Space and Time Travel The amounts of energy required to propel spaceships to relativistic speeds are billions of times the energy used to put the space shuttles into orbit.

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15.5 Space and Time Travel Before the theory of special relativity was introduced, it was argued that humans would never be able to venture to the stars. Our life span is too short to cover such great distances. Alpha Centauri is the nearest star to Earth, after the sun, and it is 4 light-years away. A round trip even at the speed of light would require 8 years. The center of our galaxy is some 30,000 light-years away, so it was reasoned that a person traveling even at the speed of light would have to survive for 30,000 years to make such a voyage!

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15.5 Space and Time Travel A person’s heart beats to the rhythm of the realm of time it is in. Astronauts traveling at 99% the speed of light could go to the star Procyon (11.4 light-years distant) and back in 23.0 years in Earth time. Because of time dilation, it would seem that only 3 years had gone by for the astronauts. It would be the space officials greeting them on their return who would be 23 years older.

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15.5 Space and Time Travel At higher speeds, the results are even more impressive. At a speed of 99.99% the speed of light, travelers could travel slightly more than 70 light-years in a single year of their own time. At % the speed of light, this distance would be pushed appreciably farther than 200 years. A 5-year trip for them would take them farther than light travels in 1000 Earth-time years.

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15.6 Length Contraction When an object moves at a very high speed relative to an observer, its measured length in the direction of motion is contracted. For moving objects, space as well as time undergoes changes. The observable shortening of objects moving at speeds approaching the speed of light is length contraction. The amount of contraction is related to the amount of time dilation. For everyday speeds, the amount of contraction is much too small to be measured.

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15.6 Length Contraction For relativistic speeds, the contraction would be noticeable. At 87% of c, it would appear to you to be 0.5 meter long. At 99.5% of c, it would appear to you to be 0.1 meter long. As relative speed gets closer and closer to the speed of light, the measured lengths of objects contract closer and closer to zero. The width of a stick, perpendicular to the direction of travel, doesn’t change.

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15.6 Length Contraction A meter stick traveling at 87% the speed of light relative to an observer would be measured as only half as long.

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As relative speed increases, contraction in the direction of motion increases. Lengths in the perpendicular direction do not change. The contraction of speeding objects is the contraction of space itself. Space contracts in only one direction, the direction of motion. Lengths along the direction perpendicular to this motion are the same in the two frames of reference.

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**The contraction of speeding objects is the contraction of space itself**

The contraction of speeding objects is the contraction of space itself. Space contracts in only one direction, the direction of motion. Lengths along the direction perpendicular to this motion are the same in the two frames of reference. Relativistic length contraction is stated mathematically: v is the speed of the object relative to the observer c is the speed of light L is the length of the moving object as measured by the observer L0 is the measured length of the object at rest

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15.6 Length Contraction Suppose that an object is at rest, so that v = 0. When 0 is substituted for v in the equation, we find L = L0. When 0.87c is substituted for v in the equation, we find L = 0.5L0. Or when 0.995c is substituted for v, we find L = 0.1L0.

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**Ch16 – Relativity: Momentum, Mass, Energy & Gravity**

According to special relativity, mass and energy are equivalent. According to general relativity, gravity causes space to become curved and time to undergo changes.

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One of the most celebrated outcomes of special relativity is the discovery that mass and energy are one and the same thing—as described by E = mc2. Einstein’s general theory of relativity, developed a decade after his special theory of relativity, offers another celebrated outcome, an alternative to Newton’s theory of gravity.

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**16.1 Momentum and Inertia in Relativity**

When a beam of electrons is directed into a magnetic field, the charged particles experience a force that deflects them from their normal paths. For a particle with a small momentum, the path curves sharply. For a particle with a large momentum, the path curves only a little—its trajectory is “stiffer.” A particle moving only a little faster than another (99.9% of c instead of 99% of c) will have much greater momentum and will follow a straighter path in the magnetic field.

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**16.2 Equivalence of Mass and Energy**

A remarkable insight of Einstein’s special theory of relativity is his conclusion that mass is simply a form of energy. A piece of matter has an “energy of being” called its rest energy. Einstein concluded that it takes energy to make mass and that energy is released when mass disappears. Rest mass is, in effect, a kind of potential energy.

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**16.2 Equivalence of Mass and Energy**

Conversion of Mass to Energy The amount of rest energy E is related to the mass m by the most celebrated equation of the twentieth century: E = mc2 where c is again the speed of light. This equation gives the total energy content of a piece of stationary matter of mass m.

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**16.1 Momentum and Inertia in Relativity**

If we push an object that is free to move, it will accelerate. If we push with a greater and greater force, we expect the acceleration in turn to increase. It might seem that the speed should increase without limit, but there is a speed limit in the universe—the speed of light.

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**In nuclear reactions, rest mass decreases by about 1 part in 1000. **

16.2 Equivalence of Mass and Energy In nuclear reactions, rest mass decreases by about 1 part in 1000. The sun is so massive that in a million years only one ten-millionth of the sun’s rest mass will have been converted to radiant energy. The present stage of thermonuclear fusion in the sun has been going on for the past 5 billion years, and there is sufficient hydrogen fuel for fusion to last another 5 billion years.

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**16.2 Equivalence of Mass and Energy**

E = mc2 is not restricted to chemical and nuclear reactions. A change in energy of any object at rest is accompanied by a change in its mass. A light bulb filament has more mass when it is energized with electricity than when it is turned off. A hot cup of tea has more mass than the same cup of tea when cold. A wound-up spring clock has more mass than the same clock when unwound.

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**16.1 Momentum and Inertia in Relativity**

Momentum equals mass times velocity: p = mv (we use p for momentum) To Newton, infinite momentum would mean infinite speed. Einstein showed that a new definition of momentum is required: where v is the speed of an object and c is the speed of light.

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**16.3 The Correspondence Principle**

According to the correspondence principle, if the equations of special relativity (or any other new theory) are to be valid, they must correspond to those of Newtonian mechanics—classical mechanics—when speeds much less than the speed of light are considered.

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**16.3 The Correspondence Principle**

The relativity equations for time dilation, length contraction, and momentum are

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**16.3 The Correspondence Principle**

These equations reduce to a Newtonian value for speeds that are very small compared with c. Then, the ratio (v/c)2 is very small, and may be taken to be zero. The relativity equations become

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16.4 General Relativity The special theory of relativity is about motion observed in uniformly moving frames of reference. Einstein was convinced that the laws of nature should be expressed in the same form in every frame of reference. This motivation led him to develop the general theory of relativity—a new theory of gravitation, in which gravity causes space to become curved and time to slow down.

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16.4 General Relativity Everything inside is weightless when the spaceship isn’t accelerating. When the spaceship accelerates, an occupant inside feels “gravity.”

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16.4 General Relativity The Principle of Equivalence Einstein concluded, in what is now called the principle of equivalence, that gravity and accelerated motion through space-time are related. You cannot tell whether you are being pulled by gravity or being accelerated. The effects of gravity and acceleration are equivalent.

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16.4 General Relativity If the ship is accelerating, the floor overtakes the ball and it hits the wall below the level at which it was thrown. An observer outside the ship still sees a straight-line path. An observer in the accelerating ship sees that the path is curved. The same holds true for a beam of light. The only difference is in the amount of path curvature.

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16.4 General Relativity Using his principle of equivalence, Einstein took another giant step that led him to the general theory of relativity. He reasoned that since acceleration (a space-time effect) can mimic gravity (a force), perhaps gravity is not a separate force after all. Perhaps it is nothing but a manifestation of space-time. From this bold idea he derived the mathematics of gravity as being a result of curved space-time.

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**16.5 Gravity, Space, and a New Geometry**

Space-time has four dimensions—three space dimensions (length, width, and height) and one time dimension (past to future). Einstein perceived a gravitational field as a geometrical warping of four-dimensional space-time. Four-dimensional geometry is altogether different from the three-dimensional geometry introduced by Euclid centuries earlier. Euclidean geometry is no longer valid when applied to objects in the presence of strong gravitational fields.

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**16.5 Gravity, Space, and a New Geometry**

Four-Dimensional Geometry The rules of Euclidean geometry pertain to figures that can be drawn on a flat surface. The ratio of the circumference of a circle to its diameter is equal to . All the angles in a triangle add up to 180°. The shortest distance between two points is a straight line. The rules of Euclidean geometry are valid in flat space, but if you draw circles or triangles on a curved surface like a sphere or a saddle-shaped object the Euclidean rules no longer hold.

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**16.5 Gravity, Space, and a New Geometry**

The sum of the angles of a triangle is not always 180°. On a flat surface, the sum is 180°. On a spherical surface, the sum is greater than 180°. On a saddle-shaped surface, the sum is less than 180°.

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**16.5 Gravity, Space, and a New Geometry**

The path of a light beam follows a geodesic. Three experimenters on Earth, Venus, and Mars measure the angles of a triangle formed by light beams traveling between them. The light beams bend when passing the sun, resulting in the sum of the three angles being larger than 180°.

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**16.5 Gravity, Space, and a New Geometry**

The light rays joining the three planets form a triangle. Since the sun’s gravity bends the light rays, the sum of the angles of the resulting triangle is greater than 180°.

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**16.5 Gravity, Space, and a New Geometry**

Instead of visualizing gravitational forces between masses, we abandon altogether the idea of gravitational force and think of masses responding in their motion to the curvature or warping of the space-time they inhabit. General relativity tells us that the bumps, depressions, and warpings of geometrical space-time are gravity.

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**16.5 Gravity, Space, and a New Geometry**

Space-time near a star is curved in a way similar to the surface of a waterbed when a heavy ball rests on it.

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**16.5 Gravity, Space, and a New Geometry**

Gravitational Waves Every object has mass, and therefore makes a bump or depression in the surrounding space-time. When an object moves, the surrounding warp of space and time moves to readjust to the new position. These readjustments produce ripples in the overall geometry of space-time. The ripples that travel outward from the gravitational sources at the speed of light are gravitational waves.

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**16.5 Gravity, Space, and a New Geometry**

think! Whoa! We learned previously that the pull of gravity is an interaction between masses. And we learned that light has no mass. Now we say that light can be bent by gravity. Isn’t this a contradiction? Answer: There is no contradiction when the mass-energy equivalence is understood. It’s true that light is massless, but it is not “energyless.” The fact that gravity deflects light is evidence that gravity pulls on the energy of light. Energy indeed is equivalent to mass!

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**16.6 Tests of General Relativity**

Upon developing the general theory of relativity, Einstein predicted that the elliptical orbits of the planets precess about the sun, starlight passing close to the sun is deflected, and gravitation causes time to slow down.

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**16.6 Tests of General Relativity**

Starlight bends as it grazes the sun. Point A shows the apparent position; point B shows the true position. (The deflection is exaggerated.)

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**16.6 Tests of General Relativity**

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