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Conceptual Physics - Chapter 19: Vibrations and Waves
Transcript of Conceptual Physics - Chapter 19: Vibrations and Waves
Answer: The minimum speed for close Earth orbit is 8 km/s; the maximum speed is 1.2 km/s. Above that speed the object would escape from the Earth’s gravity. Answer: The sum of kinetic and potential energies is constant for both circular and elliptical orbits. 28. Is the sum of kinetic and potential energies a constant for satellites in circular orbits, elliptical orbits, or both? 27. With respect to the apogee and perigee of an elliptical orbit, where is the gravitational potential greatest? Least?
Answer: The gravitational potential energy is the greatest when it is the farthest away (the apogee); it is the least when it is closest (the perigee). 26. Why is kinetic energy a variable for a satellite in an elliptical orbit?
Answer: The kinetic energy changes because there is a component of force in the direction of motion, causing the velocity to change. 25. Why is kinetic energy a constant for a satellite in circular orbit?
Answer: Kinetic energy is constant because the force is perpendicular to the motion and therefore the velocity is constant. 24. After discovering his first two laws, how much more time did it take Kepler to discover his third law?
Answer: 10 years 23. In Kepler's thinking, what was the direction of force on a planet? In Newton's thinking, what was the direction of force?
Answer: Kepler thought the force acted in the direction of motion of the satellite; Newton realized that the force acted between the satellite and the Sun. 22. Did Kepler view planets as projectiles moving under the influence of the sun?
Answer: No. 21. What did Kepler discover about the speed of planets and their distance from the sun?
Answer: He discovered that the speed of the planets decreases as they get farther away from the sun. 20. Who gathered the data that showed that the planets travel in elliptical paths around the sun? Who discovered this fact? Who explained this fact?
Answer: Tycho Brahe gathered the data; Johannes Kepler analyzed the data and discovered that the planets travelled in elliptical paths. Isaac Newton explained why. 19. At what part of an elliptical orbit does a satellite have the greatest speed? The least speed?
Answer: A satellite has its greatest speed when it is closest to the Earth; it has its slowest speed when it is the farthest away. 18. Why does the force of gravity change the speed of a satellite in an elliptical orbit?
Answer: In the case of a satellite in an elliptical orbit, there is a component of gravity in the direction of motion, so as it is going away it slows down, and while it is coming toward the Earth, it will speed up. 17. For orbits of greater altitude, is the period greater or less?
Answer: For orbits of higher altitude, the orbital speed is less, the distance is greater, and therefore the period is longer. 16. How much time is taken for a complete revolution of a satellite in close orbit about the Earth?
Answer: 90 minutes. 15. Why doesn't the force of gravity change the speed of a satellite in circular orbit?
Answer: Because the force of gravity is always perpendicular to the velocity of the satellite (there is no component of the force in the direction of motion, so it has no effect on the velocity.) 14. Are the planets of the solar system simply projectiles falling around and around the sun?
Answer: Yes. 13. Why is it important that the projectile in the last question be above the Earth's atmosphere?
Answer: If the projectile is travelling through the atmosphere, it will slow down and not follow the curvature of the Earth and will eventually hit the ground. 12. Why will a projectile that moves horizontally at 8 km/s follow a curve that matches the curvature of the Earth?
Answer: The projectile will fall 5 m for every 8 km it travels sideways, and that matches the curvature of the Earth. 11. How can a projectile “fall around the Earth”?
Answer: If the projectile is launched fast enough so that it falls parallel to the curvature of the Earth. 10. A projectile is launched vertically at 100 m/s. If air resistance can be neglected, at what speed will it return to its initial level?
Answer: It will return at the same speed, 100 m/s. 9. A projectile is launched upward at an angle of 75° from the horizontal and strikes the ground a certain distance down range. For what other angle of launch at the same speed would this projectile land just as far away?
Answer: The complement of the angle, or 15 degrees. 8. Does your answer to the last question depend on the angle at which the projectile is launched?
Answer: No. The distances fallen will be the same regardless of the initial launch angle. 7. A projectile falls beneath the straight-line path it would follow if there were no gravity. How many meters does it fall below this line if it has been traveling for 1 s? For 2 s?
Answer: After 1 second, it will fall 5 m below the straight-line path (5t2 with t=1); after 2 seconds, it will have fallen 20 m below that line (5t2 with t=2). 6. A rock is thrown upward at an angle. What happens to the vertical component of its velocity as it rises? As it falls?
Answer: The vertical component decreases as it rises (gravity is slowing it down) and increases as it falls (gravity causes it to speed up). 5. A rock is thrown upward at an angle. What happens to the horizontal component of its velocity as it rises? As it falls?
Answer: The remains constant throughout its motion, both as it rises and as it falls. 4. True or false: When air resistance does not affect the motion of a projectile, its horizontal and vertical components of velocity remain constant.
Answer: False. The horizontal component of the velocity remains constant, but the vertical velocity component changes because of gravity. 3. Why does the vertical component of velocity for a projectile change with time, whereas the horizontal component of velocity doesn't?
Answer: Because the force of gravity acts vertically to change the vertical component of the velocity, but there is no force acting horizontally to change the horizontal component. 2. What exactly is a projectile?
Answer: Any object that is projected by some means and continues in motion by its own inertia. Why does a horizontally moving projectile have to have a large speed to become an earth satellite?
Answer: It must be travelling fast enough so that it follows the curvature of the Earth and doesn’t hit the ground. Review Questions Ellipse The oval path followed by a satellite. The sum of the distances from any point on the path to two points called foci is a constant. When the foci are together at one point, the ellipse is a circle. As the foci get farther apart, the ellipse gets more “eccentric.”
Escape Speed The speed that a projectile, space probe, or similar object must reach to escape the gravitational influence of the Earth or celestial body to which it is attracted. Summary of Terms, cont. Projectile Any object that moves through the air or through space under the influence of gravity.
Parabola The curved path followed by a projectile near the Earth under the influence of gravity only.
Satellite A projectile or small celestial body that orbits a larger celestial body. Summary of Terms It is interesting to note that the accuracy with which an unmanned rocket reaches its destination is not accomplished by staying on a preplanned path or by getting back on that path if the rocket strays off course. No attempt is made to return the rocket to its original path. Instead, the control center in effect asks, “Where is it now and what is its velocity? What is the best way to get it to its destination, given its present situation?” With the aid of high-speed computers, the answers to these questions are used in finding a new path. Corrective thrusters put the rocket on this new path. This process is repeated over and over again all the way to the goal. Escape Speed, cont. It is important to point out that the escape speed of a body is the initial speed given by a brief thrust, after which there is no force to assist motion. One could escape the Earth at any sustained speed more than zero, given enough time. For example, suppose a rocket is launched to a destination such as the moon. If the rocket engines burn out when still close to the Earth, the rocket needs a minimum speed of 11.2 kilometers per second. But if the rocket engines can be sustained for long periods of time, the rocket could go to the moon without ever attaining 11.2 kilometers per second. Escape Speed, cont. Pioneer 10, launched from Earth in 1972, passed the outermost planet in 1984 and is now wandering in our galaxy. Figure 10-32 Pioneer 10 passed the orbit of Pluto in 1984. Unless it collides with another body, it will wander indefinitely through interstellar space. Like a bottle cast into the sea with a note inside, Pioneer 10 contains information about the Earth that might be of interest to extra-terrestrials, in hopes that it will one day wash up and be found on some distant “seashore.” Escape Speed, cont. The first probe to escape the solar system, Pioneer 10, was launched from Earth in 1972 with a speed of only 15 kilometers per second. The escape was accomplished by directing the probe into the path of oncoming Jupiter. It was whipped about by Jupiter's great gravitational field, picking up speed in the process—similar to the increase in the speed of a baseball encountering an oncoming bat. Its speed of departure from Jupiter was increased enough to exceed the escape speed from the sun at the distance of Jupiter. Escape Speed, cont. Table 10.1 Escape Speeds at the Surface of Bodies in the Solar System The escape speeds from various bodies in the solar system are shown in Table 10.1. Note that the escape speed from the surface of the sun is 620 kilometers per second. Even at a 150,000,000-kilometer distance from the sun (Earth's distance), the escape speed to break free of the sun's influence is 42.5 kilometers per second—considerably more than the escape speed from the Earth. An object projected from the Earth at a speed greater than 11.2 kilometers per second but less than 42.5 kilometers per second will escape the Earth but not the sun. Rather than recede forever, it will take up an orbit around the sun. Escape Speed, cont. If we give a payload any more energy than 62 million joules per kilogram at the surface of the Earth or, equivalently, any more speed than 11.2 kilometers per second, then, neglecting air drag, the payload will escape from the Earth never to return. As it continues outward, its PE increases and its KE decreases. Its speed becomes less and less, though it is never reduced to zero. The payload outruns the gravity of the Earth. It escapes. Escape Speed, cont. It turns out that the change of PE of a 1-kilogram body moved from the surface of the Earth to infinite distance is 62 million joules (62 MJ). So to put a payload infinitely far from the Earth's surface requires at least 62 million joules of energy per kilogram of load. We won't go through the calculation here, but 62 million joules per kilogram corresponds to a speed of 11.2 kilometers per second, whatever the total mass involved. This is the escape speed from the surface of the Earth. Escape Speed, cont. How much work would be required to lift a payload against the force of Earth gravity to a distance very very far (“infinitely far”) away? We might think that the change of PE would be infinite because the distance is infinite. But gravity diminishes with distance by the inverse-square law. The force of gravity on the payload would be strong only near the Earth. Most of the work done in launching a rocket occurs within 10,000 km or so of the Earth. Escape Speed, cont. In today's space-faring age, it is more accurate to say “What goes up may come down,” for there is a critical starting speed that lets a projectile outrun gravity and escape the Earth. This critical speed is called the escape speed or, if direction is involved, the escape velocity. From the surface of the Earth, escape speed is 11.2 kilometers per second. Launch a projectile at any speed greater than that and it will leave the Earth, traveling slower and slower, never stopping due to Earth gravity. We can understand the magnitude of this speed from an energy point of view. Escape Speed, cont. If Superman tosses a ball 8 km/s horizontally from the top of a mountain high enough to be just above air drag (a), then about 90 minutes later he can turn around and catch it (neglecting the Earth's rotation). Tossed slightly faster (b), it will take an elliptical orbit and return in a slightly longer time. Tossed at more than 11.2 km/s (c), it will escape the Earth. Tossed at more than 42.5 km/s (d), it will escape the solar system. Figure 10-31 We know that a cannonball fired horizontally at 8 kilometers per second from Newton's mountain would find itself in orbit. But what would happen if the cannonball were instead fired at the same speed vertically? It would rise to some maximum height, reverse direction, and then fall back to Earth. Then the old saying “What goes up must come down” would hold true, just as surely as a stone tossed skyward will be returned by gravity (unless, as we shall see, its speed is big enough). Escape Speed 2. Why does the force of gravity change the speed of a satellite when it is in an elliptical orbit, but not when it is in a circular orbit? Check Yourself 1. KE is maximum at the perigee A; PE is maximum at the apogee C; the total energy is the same everywhere in the orbit. Check Your Answer In elliptical orbit, a component of force exists along the direction of the satellite's motion. This component changes the speed and, thus, the KE. (The perpendicular component changes only the direction.) Figure 10-30 At all points along the elliptical orbit, except at the apogee and perigee, there is a component of gravitational force parallel to the direction of motion of the satellite. This component of force changes the speed of the satellite. Or we can say that (this component of force) × (distance moved) = ΔKE. Either way, when the satellite gains altitude and moves against this component, its speed and KE decrease. The decrease continues to the apogee. Once past the apogee, the satellite moves in the same direction as the component, and the speed and KE increase. The increase continues until the satellite whips past the perigee and repeats the cycle. Energy Conservation and Satellite Motion, cont. The sum of KE and PE for a satellite is a constant at all points along its orbit. Figure 10-29 The force of gravity on the satellite is always toward the center of the body it orbits. For a satellite in circular orbit, no component of force acts along the direction of motion. The speed and thus the KE do not change. Figure 10-28 In a circular orbit the distance between the satellite and the center of the attracting body does not change, which means the PE of the satellite is the same everywhere in orbit. Then, by the conservation of energy, the KE must also be constant. So a satellite in circular orbit coasts at an unchanging PE, KE, and speed (Figure 10.28). Energy Conservation and Satellite Motion, cont. Recall from Chapter 7 that an object in motion possesses kinetic energy (KE) due to its motion. An object above the Earth's surface possesses potential energy (PE) by virtue of its position. Everywhere in its orbit, a satellite has both KE and PE. The sum of the KE and PE is a constant all through the orbit. The simplest case occurs for a satellite in circular orbit. Energy Conservation and Satellite Motion It is interesting to note that Kepler was familiar with Galileo's ideas about 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. Further understanding of planetary motion required someone who could integrate the findings of these two great scientists. The rest is history, for this task fell to Isaac Newton. Kepler’s Laws of Planetary Motion, cont. The squares of the times of revolutions (periods) of the planets are proportional to the cubes of their average distances from the sun. (T2 ∼ R3 for all planets.) Ten years later Kepler discovered a third law. He had spent these years searching for a connection between the size of a planet's orbit and its period around the sun. From Brahe's data Kepler found that the square of a period is proportional to the cube of its average distance from the sun. This means that the fraction T2/R3 is the same for all the planets, where T is the planet's period, and R is the planet's average orbit radius. Law three is: Kepler’s Laws of Planetary Motion, cont. Kepler was the first to coin the word satellite. He had no clear idea as to why the planets moved as he discovered. He lacked a conceptual model. Kepler didn't see that a satellite is simply a projectile under the influence of a gravitational force directed toward the body that the satellite orbits. This is Kepler's second law: The line from the sun to any planet sweeps out equal areas of space in equal time intervals. Kepler’s Laws of Planetary Motion, cont. Figure 10-27 The triangular-shaped area swept out during a month when a planet is orbiting far from the sun (triangle ASB) is equal to the triangular area swept out during a month when the planet is orbiting closer to the sun (triangle CSD). Kepler’s Laws of Planetary Motion, cont. Each planet moves in an elliptical orbit with the sun at one focus of the ellipse.
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. They do this in such a way that an imaginary line or spoke joining the sun and the planet sweeps out equal areas of space in equal times. Kepler’s Laws of Planetary Motion, cont. After Brahe's death, Kepler converted Brahe's measurements to values that would be obtained by a stationary observer outside the solar system. Kepler's expectation that the planets would move on perfect circles around the sun was shattered after years of effort. He found the paths to be ellipses. This is Kepler's first law of planetary motion: Kepler’s Laws of Planetary Motion, cont. Johannes Kepler (1571–1630) Tycho Brahe (1546–1601) The satellite has its greatest speed as it whips around A and has its lowest speed at position C. After passing C, it gains speed as it falls back to A to repeat its cycle. Check Your Answer Figure 10-26 (a) The parabolic path of the cannonball is part of an ellipse that extends within the Earth. The Earth's center is the far focus. (b) All paths of the cannonball are ellipses. For less than orbital speeds, the center of the Earth is the far focus; for circular orbit, both foci are the Earth's center; for greater speeds, the near focus is the Earth's center. Figure 10-25 Elliptical orbit. An Earth satellite that has a speed somewhat greater than 8 km/s overshoots a circular orbit (a) and travels away from the Earth. Gravitation slows it to a point where it no longer moves farther from the Earth (b). It falls toward the Earth gaining the speed it lost in receding (c) and follows the same path as before in a repetitious cycle. Whereas the speed of a satellite is constant in a circular orbit, speed varies in an elliptical orbit. When the initial speed is greater than 8 kilometers per second, the satellite overshoots a circular path and moves away from the Earth, against the force of gravity. It therefore loses speed. The speed it loses in receding is regained as it falls back toward the Earth, and it finally rejoins its original path with the same speed it had initially (Figure 10.25). The procedure repeats over and over, and an ellipse is traced each cycle. Elliptical Orbits, cont. The shadows cast by the ball are all ellipses, one for each lamp in the room. The point at which the ball makes contact with the table is the common focus of all three ellipses. Figure 10.24 A simple method for constructing an ellipse. Figure 10.23 An ellipse is a specific curve: the closed path taken by a point that moves in such a way that the sum of its distances from two fixed points (called foci) is constant. For a satellite orbiting a planet, one focus is at the center of the planet; the other focus could be inside or outside of the planet. An ellipse can be easily constructed by using a pair of tacks, one at each focus, a loop of string, and a pencil (Figure 10.23). The closer the foci are to each other, the closer the ellipse is to a circle. When both foci are together, the ellipse is a circle. So we see that a circle is a special case of an ellipse. Elliptical Orbits, cont. If a projectile just above the drag of the atmosphere is given a horizontal speed somewhat greater than 8 kilometers per second, it will overshoot a circular path and trace an oval path called an ellipse. Elliptical Orbits 2. In each second, the satellite falls about 5 M below the straight-line tangent it would have taken if there were no gravity. The Earth's surface also curves 5 M beneath a straight-line 8-km tangent. The process of falling with the curvature of the Earth continues from tangent line to tangent line, so the curved path of the satellite and the curve of the Earth's surface “match” all the way around the Earth. Satellites in fact do crash to the Earth's surface from time to time when they encounter air drag in the upper atmosphere that decreases their orbital speed. Check Your Answer 2. Satellites in close circular orbit fall about 5 meters during each second of orbit. Why doesn't this distance accumulate and send satellites crashing into the Earth's surface? Check Yourself 1. False. What satellites are above is the atmosphere and air drag—not gravity! It's important to note that Earth's gravity extends throughout the universe in accord with the inverse-square law. Check Your Answer 1. True or false: The space shuttle orbits at altitudes in excess of 150 kilometers to be above both gravity and the Earth's atmosphere. Check Yourself The initial thrust of the rocket pushes it up above the atmosphere. Another thrust to a tangential speed of at least 8 km/s is required if the rocket is to fall around rather than into the Earth. Figure 10.22 Putting a payload into Earth orbit requires control over the speed and direction of the rocket that carries it above the atmosphere. A rocket initially fired vertically is intentionally tipped from the vertical course. Then, once above the drag of the atmosphere, it is aimed horizontally, whereupon the payload is given a final thrust to orbital speed. We see this in Figure 10.22, where for the sake of simplicity the payload is the entire single-stage rocket. With the proper tangential velocity it falls around the Earth, rather than into it, and becomes an Earth satellite. Circular Satellite Orbits, cont. For a satellite close to the Earth, the period (the time for a complete orbit about the Earth) is about 90 minutes. For higher altitudes, the orbital speed is less, distance is more, and the period is longer. For example, communication satellites located in orbit 5.5 Earth radii above the surface of the Earth have a period of 24 hours. This period matches the period of daily Earth rotation. For an orbit around the equator, these satellites stay above the same point on the ground. The moon is even farther away and has a period of 27.3 days. The higher the orbit of a satellite, the less its speed, the longer its path, and the longer its period. Circular Satellite Orbits, cont. Note that a satellite in circular orbit is always moving in a direction perpendicular to the force of gravity that acts on it. The satellite does not move in the direction of the force, which would increase its speed, nor does it move in a direction against the force, which would decrease its speed. Instead, the satellite moves at right angles to the gravitational force that acts on it. With no component of motion along this force, no change in speed occurs—only change in direction. So we see why a satellite in circular orbit sails parallel to the surface of the Earth at constant speed—a very special form of free fall. Circular Satellite Orbits, cont. What speed will allow the ball to clear the gap? Figure 10-21 (a) The force of gravity on the bowling ball is at 90° to its direction of motion, so it has no component of force to pull it forward or backward, and the ball rolls at constant speed. (b) The same is true even if the bowling alley is larger and remains “level” with the curvature of the Earth. Figure 10-20 Consider a bowling alley that completely surrounds the Earth, elevated high enough to be above the atmosphere and air drag. The bowling ball will roll at constant speed along the alley. If a part of the alley is cut away, the ball would roll off its edge and hit the ground below. A faster ball encountering the gap would hit the ground farther along the gap. Is there a speed whereby the ball will clear the gap (like a motorcyclist who drives off a ramp and clears a gap to meet a ramp on the other side)? The answer is yes: 8 kilometers per second will clear that gap— and any gap—even a 360° gap. It would be in circular orbit. Circular Satellite Orbits, cont. Fired fast enough, the cannonball will go into orbit. Figure 10-19 Note that in circular orbit, the speed of a satellite is not changed by gravity: only the direction changes. We can understand this by comparing a satellite in circular orbit with a bowling ball rolling along a bowling alley. Why doesn't the gravity that acts on the bowling ball change its speed? The answer is that gravity pulls straight downward with no component of force acting forward or backward. Circular Satellite Orbits, cont. An 8-kilometers-per-second cannonball fired horizontally from Newton's mountain would follow the Earth's curvature and glide in a circular path around the Earth again and again (provided the cannoneer and the cannon got out of the way). Fired slower, the cannonball would strike the Earth's surface; fired faster it would overshoot a circular orbit as we will discuss shortly. Newton calculated the speed for circular orbit, and since such a cannon-muzzle velocity was clearly impossible, he did not foresee humans launching satellites (and also because he probably didn't consider multi-stage rockets). Circular Satellite Orbits No, no, a thousand times no! If any moving object were beyond the pull of gravity, it would move in a straight line and would not curve around the Earth. Satellites remain in orbit because they are being pulled by gravity, not because they are beyond it. For the altitudes of most Earth satellites, the Earth's gravitational field is only a few percent weaker than at the Earth's surface. Check Your Answer Check Yourself One of the beauties of physics is that there are usually different ways to view and explain a given phenomenon. Is the following explanation valid? Satellites remain in orbit instead of falling to the Earth because they are beyond the main pull of Earth's gravity. Both cannonball and moon have tangential velocity (parallel to the Earth's surface) sufficient to ensure motion around the Earth rather than into it. If there is no resistance to reduce its speed, the moon or any Earth satellite “falls” around and around the Earth indefinitely. Similarly with the planets that continually fall around the sun in closed paths. Why don't the planets crash into the sun? They don't because of their tangential velocities. What would happen if their tangential velocities were reduced to zero? The answer is simple enough: their motion would be straight toward the sun and they would indeed crash into it. Any objects in the solar system without sufficient tangential velocities have long ago crashed into the sun. What remains is the harmony we observe. Fast-Moving Projectiles--Satellites, cont. If a cannonball were fired with a low horizontal speed, it would follow a curved path and soon hit the Earth below. If it were fired faster, its path would be less curved and it would hit the Earth farther away. If the cannonball were fired fast enough, Newton reasoned, the curved path would become a circle and the cannonball would circle the Earth indefinitely. It would be in orbit. Fast-Moving Projectiles--Satellites, cont. “The greater the velocity … with which (a stone) is projected, the farther it goes before it falls to the Earth. We may therefore suppose the velocity to be so increased, that it would describe an arc of 1, 2, 5, 10, 100, 1000 miles before it arrived at the Earth, till at last, exceeding the limits of the Earth, it should pass into space without touching.”—Isaac Newton, System of the World. Figure 10.18 Satellite motion was understood by Isaac Newton, who reasoned that the moon was simply a projectile circling the Earth under the attraction of gravity. This concept is illustrated in a drawing by Newton (Figure 10.18). He compared motion of the moon to a cannonball fired from the top of a high mountain. He imagined that the mountaintop was above the Earth's atmosphere, so that air drag would not impede the motion of the cannonball. Fast-Moving Projectiles--Satellites, cont. The space shuttle is a projectile in a constant state of free fall. Because of its tangential velocity, it falls around the Earth rather than vertically into it. Figure 10-17 At this speed, atmospheric friction would burn the baseball or even a piece of iron to a crisp. This is the fate of bits of rock and other meteorites that enter the Earth's atmosphere and burn up, appearing as “falling stars.” That is why satellites such as the space shuttles are launched to altitudes of 150 kilometers or more—to be above almost all of the atmosphere and nearly free of air drag. A common misconception is that satellites orbiting at high altitudes are free from gravity. Nothing could be further from the truth. The force of gravity on a satellite 200 kilometers above the Earth's surface is nearly as strong as at the surface. The high altitude is to put the satellite beyond the Earth's atmosphere, where air drag is almost totally absent, but not beyond Earth's gravity. Fast-Moving Projectiles—Satellites (cont.) This means that if you were floating in a calm ocean with your head close to the water surface, you would be able to see only the top of a 5-meter mast on a ship 8 kilometers away. So if a baseball could be thrown fast enough to travel a horizontal distance of 8 kilometers during the one second it takes to fall 5 meters, then it would follow the curvature of the Earth. This is a speed of 8 kilometers per second. If this doesn't seem fast, convert it to kilometers per hour and you get an impressive 29,000 kilometers per hour (or 18,000 miles per hour)! Fast-Moving Projectiles-Satellites, cont. Earth's curvature—not to scale! Figure 10-15 An Earth satellite is simply a projectile that falls around the Earth rather than into it. The speed of the satellite must be great enough to ensure that its falling distance matches the Earth's curvature. A geometrical fact about the curvature of our Earth is that its surface drops a vertical distance of 5 meters for every 8000 meters tangent to the surface (Figure 10.15). Fast-Moving Projectiles--Satellites, cont. Consider the baseball pitcher on the tower in Figure 10.13. If gravity did not act on the ball, the ball would follow a straight line path shown by the dashed line. But there is gravity, so the ball falls below this straight line path. In fact, as discussed above, 1 second after the ball leaves the pitcher's hand it will have fallen a vertical distance of 5 meters below the dashed line—whatever the pitching speed. It is important to understand this, for it is the crux of satellite motion. Fast-Moving Projectiles—Satellites The ball is thrown horizontally, so the pitching speed is horizontal distance divided by time. A horizontal distance of 20 M is given, but the time is not stated. However, you can find the time because you know the vertical distance the ball drops—5 m, which takes 1 s! This means it travels horizontally 20 M in 1 s. So its horizontal component of velocity must be 20 m/s. From the equation for constant speed (which applies to horizontal motion) v = d/t = (20m)/(1s) = 20 m/s. It is interesting to note that consideration of the equation for constant speed, v = d/t guides thinking about the crucial factor in this problem—the time. Check Your Answers Check Yourself The boy on the tower throws a ball 20 M downrange as shown . What is his pitching speed? Figure 10-13 Baseball games normally take place on level ground. For the short-range projectile motion on the playing field, the Earth can be considered to be flat because the flight of the baseball is not affected by the Earth's curvature. For very long-range projectiles, however, the curvature of the Earth's surface must be taken into account. We'll now see that if an object is projected fast enough, it will fall all the way around the Earth and become an Earth satellite. Upwardly Launched Projectiles, cont. Without air drag, speed lost while going up equals speed gained while coming down; time going up equals time coming down. Figure 10-12 When air drag is small enough to be negligible, a projectile will rise to its maximum height in the same time it takes to fall from that height to the ground (Figure 10.12). This is because its deceleration by gravity while going up is the same as its acceleration by gravity while coming down. The speed it loses while going up is therefore the same as the speed gained while coming down. So the projectile arrives at the ground with the same speed it had when it was initially projected. Upwardly Launched Projectiles, cont. In Chapter 3 we stated that airborne time during a jump is independent of horizontal speed. Now we see why this is so—horizontal and vertical components of motion are independent of each other. The rules of projectile motion apply to jumping. Once one's feet are off the ground, only the force of gravity acts on the jumper (neglecting air drag). Hang-time depends only on the vertical component of lift-off velocity. It turns out that jumping lift-off force can be somewhat increased by the action of running, so hang-time for a running jump usually exceeds hang-time for a standing jump. But once the feet are off the ground, only the vertical component of lift-off velocity determines hang-time. Hang Time Revisited 3. They are the same! Check Your Answers 3. Consider a batted baseball following a parabolic path on a day when the sun is directly overhead. How does the speed of the ball's shadow across the field compare with the ball's horizontal component of velocity? Check Yourself 2. At what part of its trajectory does the baseball have minimum speed? Check Yourself 1. Vertical acceleration is g because the force of gravity is vertical. Horizontal acceleration is zero because no horizontal force acts on the ball. Check Your Answer 1. A baseball is batted at an angle into the air. If we neglect air drag, what is the ball's acceleration vertically? Horizontally? Check Yourself In the presence of air drag, the trajectory of a high-speed projectile falls short of the idealized parabolic path. Figure 10-11 Aha, but launching speeds are not equal for such a projectile thrown at different angles. In throwing a javelin or putting a shot, a significant part of the launching force goes into combating gravity—the steeper the angle, the less speed it has when leaving the thrower's hand.
You can test this yourself: throw a heavy boulder horizontally, then vertically—you'll find the horizontal throw to be considerably faster than the vertical throw. So maximum range for heavy projectiles thrown by humans is attained for angles less than 45 degrees—and not because of air drag. Upwardly Launched Projectiles, cont. Without air drag, maximum range for a baseball would occur when it is batted 45 degrees above the horizontal. Because of air drag, best range occurs when the ball leaves the bat at about 43 degrees. Air drag is more significant for golf balls, where angles of about 38 degrees result in maximum range. For heavy projectiles like javelins and the shot, air drag has less effect on range. A javelin, being heavy and presenting a very small cross section to the air, follows an almost perfect parabola when thrown. So does a shot. For such a projectile, maximum range for equal launch speeds would occur for a launch angle of about 45 degrees (slightly less because the launching height is above ground level). Upwardly Launched Projectiles, cont. Ranges of a projectile shot at the same speed at different projection angles. Figure 10.9 Figure 10.9 shows the paths of several projectiles, all with the same initial speed but different launching angles. The figure neglects the effects of air drag, so the trajectories are all parabolas. Notice that these projectiles reach different altitudes, or heights above the ground. They also have different horizontal ranges, or distances traveled horizontally. The remarkable thing to note from Figure 10.9 is that the same range is obtained from two different launching angles when the angles add up to 90 degrees! An object thrown into the air at an angle of 60 degrees, for example, will have the same range as if it were thrown at the same speed at an angle of 30 degrees. For the smaller angle, of course, the object remains in the air for a shorter time. The greatest range occurs when the launching angle is 45 degrees—and when air drag is negligible. Upwardly Launched Projectiles, cont. Figure 10.8 shows the trajectory traced by a projectile launched with the same speed at a steeper angle. Notice the initial velocity vector has a greater vertical component than when the angle of launch is smaller. This greater component results in a trajectory that reaches a greater height. But the horizontal component is less, and the range is less. Upwardly Launched Projectiles, cont. The velocity of a projectile at various points along its trajectory. Note that the vertical component changes and the horizontal component is the same everywhere. Figure 10.7 In Figure 10.7 we see vectors representing both horizontal and vertical components of velocity for a projectile following a parabolic trajectory. Notice that the horizontal component is everywhere the same, and only the vertical component changes. Note also that the actual velocity is represented by the vector that forms the diagonal of the rectangle formed by the vector components. At the top of the trajectory the vertical component is zero, so the actual velocity there is only the horizontal component of velocity. Everywhere else the magnitude of velocity is greater (just as the diagonal of a rectangle is greater than either of its sides). Upwardly Launched Projectiles, cont. 2. With no air drag the cannonball will travel a horizontal distance of 100 m [d = = (20 m/s)(5 s) = 100 m]. Note that since gravity acts only vertically and there is no acceleration in the horizontal direction, the cannonball travels equal horizontal distances in equal times. This distance is simply its horizontal component of velocity multiplied by the time (and not 5t2, which applies only to vertical motion under the acceleration of gravity). Check Your Answer 2. If the horizontal component of the cannonball's velocity were 20 m/s, how far downrange would the cannonball be at the end of 5 s? Check Yourself 1. Vertical distance beneath the dashed line at the end of 5 s is 125 m [d = 5t2 = 5(5)2 = 5(25) = 125 m]. Interestingly enough, this distance doesn't depend on the angle of the cannon. If air drag is neglected, any projectile will fall 5t2 meters below where it would have reached if there were no gravity. Check Your Answer Note another thing from Figure 10.6. The cannonball moves equal horizontal distances in equal time intervals. That's because no acceleration takes place horizontally. The only acceleration is vertical, in the direction of Earth's gravity. The vertical distance it falls below the imaginary straight-line path during equal time intervals continuously increases with time.
So we see that the analysis of the motion of an upwardly launched projectile is as simple as that of a horizontally launched projectile. For both, the distance of fall below the projected straight-line motion is the same as for free fall from rest. Upwardly Launched Projectiles, cont. We can put it another way: shoot a projectile skyward at some angle and pretend there is no gravity. After so many seconds t, it should be at a certain point along a straight-line path. But because of gravity, it isn't. Where is it? The answer is that it's directly below this point. How far below? The answer in meters is 5t2 (or, more precisely, 4.9t2). How about that! Upwardly Launched Projectiles, cont. Consider a cannonball shot at an upward angle (Figure 10.6). Pretend for a moment that there is no gravity; according to the law of inertia, the cannonball would follow the straight-line path shown by the dashed line. But there is gravity, so this doesn't happen. What really happens is that the cannonball continuously falls beneath the imaginary line until it finally strikes the ground. Get this: the vertical distance it falls beneath any point on the dashed line is the same vertical distance it would fall if it were dropped from rest and had been falling for the same amount of time. This distance, as introduced in Chapter 3, is given by d = ½ gt2, where t is the elapsed time. Upwardly Launched Projectiles Vertical and horizontal components of a stone's velocity. Figure 10.5 Both bullets fall the same vertical distance with the same acceleration g due to gravity and therefore strike the ground at the same time. Can you see that this is consistent with our analysis of Figures 10.3 and 10.4? We can reason this another way by asking which bullet would strike the ground first if the rifle were pointed at an upward angle. In this case, the bullet that is simply dropped would hit the ground first. Now consider the case where the rifle is pointed downward. The fired bullet hits first. So upward, the dropped bullet hits first; downward, the fired bullet hits first. There must be some angle at which there is a dead heat—where both hit at the same time. Can you see it would be when the rifle is neither pointing upward nor downward—that is, when it is pointing horizontally? Check Your Answer At the instant a horizontally held rifle is fired over a level range, a bullet held at the side of the rifle is released and drops to the ground. Which bullet, the one fired downrange or the one dropped from rest, strikes the ground first? Check Yourself A strobe light photograph of two golf balls released simultaneously from a mechanism that allows one ball to drop freely while the other is projected horizontally. Figure 10.4 The trajectory of a projectile that accelerates only in the vertical direction while moving at a constant horizontal velocity is a parabola. When air drag is small enough to neglect, as it is for a heavy object that doesn't build up great speed, the trajectory is parabolic. Projectiles Launched Horizontally, cont. Simulated photographs of a moving ball illuminated with a strobe light. Figure 10.3 Projectile motion is nicely analyzed in Figure 10.3, which shows a simulated multiple- flash exposure of a ball rolling off the edge of a table. Investigate it carefully, for there's a lot of good physics there.
On the left we notice equally timed sequential positions of the ball without the effect of gravity. Only the effect of the ball's horizontal component of motion is shown. Next we see vertical motion without a horizontal component. The curved path in the third view is best analyzed by considering the horizontal and vertical components of motion separately. Projectiles Launched Horizontally The curved path of a projectile is a combination of horizontal and vertical motion. The horizontal component of velocity for a projectile is completely independent of the vertical component of velocity when air drag is small enough to ignore. Then the constant horizontal velocity component is not affected by the vertical force of gravity. Each component is independent of the other. Their combined effects produce the curved paths of projectiles. Projectile Motion, cont. Drop it, and it accelerates downward and covers a greater vertical distance each second. Figure 10.2b The vertical component of motion for a projectile following a curved path is just like the motion described in Chapter 3 for a freely falling object. The vertical component is exactly the same as for an object falling freely straight down, as shown in Figure 10.2b. The faster the object falls, the greater the distance covered in each successive second. Or if projected upward, the vertical distances of travel become less with time on the way up. Projectile Motion, cont. Roll a ball along a level surface, and its velocity is constant because no component of gravitational force acts horizontally. Figure 10.2a The horizontal component of velocity for a projectile is no more complex than the horizontal velocity of a bowling ball rolling freely along a level bowling alley. If the retarding effect of friction can be ignored, there is no horizontal force on the ball and its velocity is constant. It rolls of its own inertia and covers equal distances in equal intervals of time (Figure 10.2a). The horizontal component of a projectile's motion is just like the bowling ball's motion along the alley. Projectile Motion, cont. Without gravity, you could toss a rock at an angle skyward and it would follow a straight-line path. Because of gravity, however, the path curves. A tossed rock, a cannonball, or any object that is projected by some means and continues in motion by its own inertia is called a projectile. To the cannoneers of earlier centuries, the curved paths of projectiles seemed very complex. Today we see these paths are surprisingly simple when we look at the horizontal and vertical components of velocity separately. Projectile Motion If Superman threw a rock fast enough, it would orbit the Earth if there were no air drag. Figure 10.1 Then you wonder how fast Superman would have to throw the rock to clear the horizon ahead. And how fast he'd have to throw it so that its curved path matched the curve of the Earth. For if he could do that, and air drag were somehow eliminated, the rock would follow a curved path completely around the Earth and become an Earth satellite! A satellite is, after all, no more than a projectile moving fast enough to continually clear the horizon as it falls. Introduction, cont. Line up your sight so that the bottom of the middle of the straightedge just touches the juncture between sky and ocean, and you'll note a space between sky and ocean at the ends. You're seeing the Earth's curvature. Now toss a rock horizontally toward the horizon. It quickly falls several meters to the ground below in front of you. It curves as it falls. You'll note that the faster you throw the rock, the wider the curve. Introduction, cont. From the top of the mountain Mauna Kea in Hawaii (or any high vantage point where the distant ocean horizon is sharp and clear) you can see the curvature of the Earth. You have to eyeball the line where ocean and sky meet against a long straightedge in front of your eyes. Otherwise you can't be sure if your eyes are playing tricks on you. Introduction 2. At any point on its orbital path, a satellite is moving in the direction of a tangent to its path. In circular orbit the gravitational force is always perpendicular to the tangent. There is no component of gravitational force along the tangent, and only the direction of motion changes—not the speed. In elliptical orbit, however, the satellite moves in directions that are not perpendicular to the force of gravity. Then components of force do exist along the tangent, which change the speed of the satellite. A component of force tangent to the direction the satellite moves does work to change its KE. Check Your Answer 1. The orbital path of a satellite is shown in the sketch. In which marked positions A through D does the satellite have the greatest KE? Greatest PE? Greatest total energy? Check Yourself In an elliptical orbit the situation is different. Both speed and distance vary. PE is greatest when the satellite is farthest away (at the apogee) and least when the satellite is closest (at the perigee). Note that the KE will be least when the PE is most, and the KE will be most when the PE is least. At every point in the orbit, the sum of KE and PE is the same (Figure 10.29). Energy Conservation and Satellite Motion, cont. Kepler's laws apply not only to planets but also to moons or any satellite in orbit around any body. Except for Pluto (of which Kepler had no knowledge), the elliptical orbits of the planets are very nearly circular. Only the precise measurements of Brahe showed the slight differences. Kepler’s Laws of Planetary Motion, cont. You know that if you toss a rock upward, it goes slower the higher it rises because it's going against gravity. And you know that when it returns it's going with gravity and its speed increases. Kepler didn't see that a satellite behaves the same way. Going away from the sun, it slows. Going toward the sun, it speeds up. A satellite, whether a planet orbiting the sun, or one of today's satellite orbiting the Earth, moves slower against the gravitational field and faster with the field. Kepler didn't see this simplicity, and instead fabricated complex systems of geometrical figures to find sense in his discoveries. These proved futile. Kepler’s Laws of Planetary Motion, cont. Newton's law of gravitation was preceded by three important discoveries about planetary motion by the German astronomer, Johannes Kepler, who started as a junior assistant to the famed Danish astronomer, Tycho Brahe. Brahe headed the world's first great observatory in Denmark, just before the advent of the telescope. Using huge brass protractor-like instruments called quadrants, Brahe measured the positions of planets over twenty years so accurately that his measurements are still valid today. Brahe entrusted his data to Kepler. Kepler’s Laws of Planetary Motion The orbital path of a satellite is shown in the sketch. In which of the marked positions A through d does the satellite have the greatest speed? Lowest speed? Check Yourself Interestingly enough, the parabolic path of a projectile such as a tossed baseball or a cannonball is actually a tiny segment of a skinny ellipse that extends within and just beyond the center of the Earth (Figure 10.26a). In Figure 10.26b, we see several paths of cannonballs fired from Newton's mountain. All these ellipses have the center of the Earth as one focus. As muzzle velocity is increased, the ellipses are less eccentric (more nearly circular); and when muzzle velocity reaches 8 kilometers per second, the ellipse rounds into a circle and does not intercept the Earth's surface. The cannonball coasts in circular orbit. At greater muzzle velocities, the orbiting cannonball traces the familiar external ellipse. Elliptical Orbits, cont. If the speed of the stone and the curvature of its trajectory are great enough, the stone may become a satellite. Figure 10-16 Throw a stone at any speed and one second later it will have fallen 5 M below where it would have been without gravity. Figure 10-14 2. A ball's minimum speed occurs at the top of its trajectory. If it is launched vertically, its speed at the top is zero. If launched at an angle, the vertical component of velocity is zero at the top, leaving only the horizontal component. So the speed at the top is equal to the horizontal component of the ball's velocity at any point. Isn't that nice? Check Your Answers Maximum range is attained when a ball is batted at an angle of nearly 45°. Figure 10-10 Trajectory for a steeper projection angle. Figure 10.8 1. Suppose the cannonball in the figure below were fired faster. How many meters below the dashed line would it be at the end of the 5 s? Check Yourself With no gravity, the projectile would follow a straight-line path (dashed line). But because of gravity, the projectile falls beneath this line the same vertical distance it would fall if released from rest. Compare the distances fallen with those given in Table 3.3 in Chapter 3. (With g = 9.8 m/s2, these distances are more precisely 4.9 m, 19.6 m, and 44.1 m.) Figure 10.6 There are two important things to notice. The first is that the ball's horizontal component of velocity doesn't change as the falling ball moves forward. The ball travels the same horizontal distance in equal times between each flash. That's because there is no component of gravitational force acting horizontally. Gravity acts only downward, so the only acceleration of the ball is downward. The second thing to notice is that the vertical positions become farther apart with time. The vertical distances traveled are the same as if the ball were simply dropped. Note the curvature of the ball's path is the combination of horizontal motion that remains constant, and vertical motion that undergoes acceleration due to gravity. Projectiles Launched Horizontally, cont. Light and sound are both vibrations that propagate throughout space as waves. But they are two very different kinds of waves. Sound is the propagation of vibrations through a material medium--a solid, liquid, or gas. If there is no medium to vibrate, then no sound is possible. Sound cannot travel in a vacuum. The source of all waves--sound, light, or whatever --is something that is vibrating. We shall begin our study of vibrations and waves by considering the motion of a simple pendulum. Surprisingly, the time of a to-and-fro swing, called the period, does not depend on the mass of the pendulum or on the size of the arc through which it swings. Figure 19.1
Drop two balls of different mass and they accelerate at g. Let them slide without friction down the same incline and they slide together at the same fraction of g. Tie them to strings of the same length so they are pendulums, and they swing to and fro in unison. In all cases, the motions are independent of mass. A long pendulum has a longer period than a short pendulum; that is, it swings to and fro less frequently than a short pendulum. A grandfather's clock pendulum with a length of about 1 m, for example, swings with a leisurely period of 2 s, while the much shorter pendulum of a cuckoo clock swings with a period that is less than a second. In addition to length, the period of a pendulum depends on the acceleration of gravity. Oil and mineral prospectors use very sensitive pendulums to detect slight differences in this acceleration. The acceleration due to gravity varies due to the variety of underlying formations. Figure 19.2
Frank Oppenheimer at the San Francisco Exploratorium demonstrates (a) a straight line traced by a swinging pendulum bob that leaks sand on the stationary conveyor belt. (b) When the conveyor belt is uniformly moving, a sine curve is traced. A sine curve can also be traced by a bob attached to a spring undergoing vertical simple harmonic motion (Figure 19.3). A sine curve is a pictorial representation of a wave. Just as with a water wave, the high points of a sine wave are called crests, and the low points are called troughs. The straight dashed line in the figure represents the “home” position, or midpoint of the vibration. The term amplitude refers to the distance from the midpoint to the crest (or trough) of the wave. So the amplitude equals the maximum displacement from equilibrium. Figure 19.3
When the bob vibrates up and down, a marking pen traces out a sine curve on paper that is moved horizontally at constant speed. The wavelength of a wave is the distance from the top of one crest to the top of the next one. Or equivalently, the wavelength is the distance between any successive identical parts of the wave. The wavelengths of waves at the beach are measured in meters, the wavelengths of ripples in a pond in centimeters, and the wavelengths of light in billionths of a meter (nanometers). How frequently a vibration occurs is described by its frequency. The frequency of a vibrating pendulum, or object on a spring, specifies the number of to-and-fro vibrations it makes in a given time (usually one second). A complete to-and-fro oscillation is one vibration. If it occurs in one second, the frequency is one vibration per second. If two vibrations occur in one second, the frequency is two vibrations per second. The unit of frequency is called the hertz (Hz), after Heinrich Hertz, who demonstrated radio waves in 1886. One vibration per second is 1 hertz; two vibrations per second is 2 hertz, and so on. Higher frequencies are measured in kilohertz (kHz, thousands of hertz), and still higher frequencies in megahertz (MHz, millions of hertz) or gigahertz (GHz, billions of hertz). AM radio waves are measured in kilohertz, while FM radio waves are measured in megahertz; radar and microwave ovens operate at gigahertz frequencies. A station at 960 kHz on the AM radio dial, for example, broadcasts radio waves that have a frequency of 960,000 vibrations per second. A station at 101.7 MHz on the FM dial broadcasts radio waves with a frequency of 101,700,000 hertz. These radio-wave frequencies are the frequencies at which electrons are forced to vibrate in the antenna of a radio station's transmitting tower. The source of all waves is something that vibrates. The frequency of the vibrating source and the frequency of the wave it produces are the same. But light is different. It can travel through a vacuum. As we shall learn in later chapters, light is a vibration of electric and magnetic fields--a vibration of pure energy. Light can pass through many materials, but it needs none. This is evident when light from the sun travels through the vacuum of space to reach us on Earth. Figure 19.4
Electrons in the transmitting antenna vibrate 940,000 times each second and produce 940-kHz radio waves. If an object's frequency is known, its period can be calculated, and vice versa. Suppose, for example, that a pendulum makes two vibrations in one second. Its frequency is 2 Hz. The time needed to complete one vibration--that is, the period of vibration--is ½ second. Or if the vibration frequency is 3 Hz, then the period is 1/3 second. The frequency and period are the inverse of each other:
or vice versa: 1. What is the frequency in vibrations per second of a 60-Hz wave? What is its period? Check Yourself A 60-Hz wave vibrates 60 times per second and has a period of 1/60 second. 2. Gusts of wind make the Sears Building in Chicago sway back and forth at a vibration frequency of about 0.1 Hz. What is its period of vibration? The period is 1/frequency = 1/(0.1 Hz) = 1/(0.1 vibration/s) = 10 s. Each vibration therefore takes 10 seconds. Wave motion can be most easily understood by first considering the simple case of a horizontally stretched rope. If one end of such a rope is shaken up and down, a rhythmic disturbance travels along the rope. Each particle of the rope moves up and down, while at the same time the disturbance moves along the length of the rope. The medium, rope or whatever, returns to its initial condition after the disturbance has passed. What is propagated is the disturbance, not the medium itself. Perhaps a more familiar example of wave motion is provided by a water wave. If a stone is dropped into a quiet pond, waves will travel outward in expanding circles, the centers of which are at the source of the disturbance. In this case we might think that water is being transported with the waves, since water is splashed onto previously dry ground when the waves meet the shore. We should realize, however, that barring obstacles the water will run back into the pond, and things will be much as they were in the beginning: The surface of the water will have been disturbed, but the water itself will have gone nowhere. A leaf on the surface will bob up and down as the waves pass, but will end up where it started. Again, the medium returns to its initial condition after the disturbance has passed. Let us consider another example of a wave to illustrate that what is transported from one place to another is a disturbance in a medium, not the medium itself. If you view a field of tall grass from an elevated position on a gusty day, you will see waves travel across the grass. The individual stems of grass do not leave their places; instead, they swing to and fro. Furthermore, if you stand in a narrow footpath, the grass that blows over the edge of the path, brushing against your legs, is very much like the water that doused the shore in our earlier example. While wave motion continues, the tall grass swings back and forth, vibrating between definite limits but going nowhere. When the wave motion stops, the grass returns to its initial position. Wave Speed The speed of periodic wave motion is related to the frequency and wavelength of the waves. We can understand this by considering the simple case of water waves (Figures 19.5 and 19.6). Figure 19.5 - Water waves Figure 19.6 - A top view of water waves. Imagine that we fix our eyes on a stationary point on the surface of water and observe the waves passing by this point. We can measure how much time passes between the arrival of one crest and the arrival of the next one (the period), and also observe the distance between crests (the wavelength). We know that speed is defined as distance divided by time. In this case, the distance is one wavelength and the time is one period, so wave speed 5 wavelength/period. For example, if the wavelength is 10 meters and the time between crests at a point on the surface is 0.5 second, the wave moves 10 meters in 0.5 seconds and its speed is 10 meters divided by 0.5 seconds, or 20 meters per second.
Since period is equal to the inverse of frequency, the formula wave speed = wavelength/period can also be written
This relationship holds true for all kinds of waves, whether they are water waves, sound waves, or light waves. Check Yourself 1. If a train of freight cars, each 10 m long, rolls by you at the rate of three cars each second, what is the speed of the train? 30 m/s. We can see this in two ways. (a) According to the speed definition from Chapter 2, v = d/t = (3 × 10 m)/1 s = 30 m/s, since 30 m of train passes you in 1 s. (b) If we compare our train to wave motion, where wavelength corresponds to 10 m and frequency is 3 Hz, then Speed = wavelength × frequency = 10 m × 3 Hz = 10 m × 3/s = 30 m/s. 2. If a water wave oscillates up and down three times each second and the distance between wave crests is 2 m, what is its frequency? Its wavelength? Its wave speed? The frequency of the wave is 3 Hz, its wavelength is 2 m, and its wave speed = wavelength × frequency = 2 m × 3/s = 6 m/s. It is customary to express this as the equation v = f where v is wave speed, (the Greek letter lambda) is wavelength, and f is wave frequency. Figure 19.8 - A transverse wave. Waves in the stretched strings of musical instruments and upon the surfaces of liquids are transverse. We will see later that electromagnetic waves, which make up radio waves and light, are also transverse. Longitudinal Waves Not all waves are transverse. Sometimes parts that make up a medium move to and fro in the same direction in which the wave travels. Motion is along the direction of the wave rather than at right angles to it. This produces a longitudinal wave. Both a transverse and a longitudinal wave can be demonstrated with a spring or a Slinky stretched out on the floor, as shown in Figure 19.9. A transverse wave is demonstrated by shaking the end of a Slinky from side to side. A longitudinal wave is demonstrated by rapidly pulling and pushing the end of the Slinky toward and away from you. In this case we see that the medium vibrates parallel to the direction of energy transfer. Part of the Slinky is compressed, and a wave of compression travels along the spring. In between successive compressions is a stretched region, called a rarefaction. Both compressions and rarefactions travel in the same direction along the Slinky. Sound waves are longitudinal waves. Figure 19.9
Both waves transfer energy from left to right. When the end of the Slinky is shaken from side to side, a transverse wave is produced. When it's pushed and pulled rapidly along its length, a longitudinal wave is produced. Figure 19.10
Waves generated by an earthquake. P waves are longitudinal and travel through both molten and solid materials. S waves are transverse and travel only through solid materials. Reflections and refractions of the waves provide information about the Earth's interior. Waves that travel in the ground generated by earthquakes are of two main types: longitudinal P waves, and transverse S waves. (Geology students often remember P waves as “push-pull” waves, and S waves as “side-to-side” waves.) S waves cannot travel through liquid matter, while P waves can travel through both molten and solid parts of the Earth's interior. Study of these waves reveals much about the Earth's interior. The wavelength of a longitudinal wave is the distance between successive compressions or equivalently, the distance between successive rarefactions. The most common example of longitudinal waves is sound in air. Elements of air vibrate to and fro about some equilibrium position as the waves move by. We will treat sound waves in detail in the next chapter. When more than one wave occupies the same space at the same time, the displacements add at every point. This is the superposition principle. So when the crest of one wave overlaps the crest of another, their individual effects add together to produce a wave of increased amplitude. This is called constructive interference (Figure 19.11). When the crest of one wave overlaps the trough of another, their individual effects are reduced. The high part of one wave simply fills in the low part of another. This is called destructive interference. Figure 19.11
Constructive and destructive interference in a transverse wave. Wave interference is easiest to see in water. In Figure 19.12 we see the interference pattern made when two vibrating objects touch the surface of water. We can see the regions where a crest of one wave overlaps the trough of another to produce regions of zero amplitude. At points along these regions, the waves arrive out of step. We say they are out of phase with each other. Figure 19.12
Two sets of overlapping water waves produce an interference pattern. The left diagram is an idealized drawing of the expanding waves from the two sources. The right diagram is a photograph of an actual interference pattern. Interference is characteristic of all wave motion, whether the waves are water waves, sound waves, or light waves. We will treat the interference of sound in the next chapter and the interference of light in Chapter 28. Standing Waves If we tie a rope to a wall and shake the free end up and down, we produce a train of waves in the rope. The wall is too rigid to shake, so the waves are reflected back along the rope. By shaking the rope just right, we can cause the incident and reflected waves to form a standing wave, where parts of the rope, called the nodes, are stationary. Nodes are the regions of minimal or zero displacement, with minimal or zero energy. Antinodes (not labeled in Figure 19.13), on the other hand, are the regions of maximum displacement and maximum energy. You can hold your fingers just over and under the nodes and the rope doesn't touch them. Other parts of the rope, especially the antinodes, would make contact with your fingers. Antinodes occur halfway between nodes. Figure 19.13
The incident and reflected waves interfere to produce a standing wave. Standing waves are the result of interference (and as we will see in the next chapter, resonance). When two sets of waves of equal amplitude and wavelength pass through each other in opposite directions, the waves are steadily in and out of phase with each other. This occurs for a wave that reflects upon itself. Stable regions of constructive and destructive interference are produced It is easy to make standing waves yourself. Tie a rope, or better, a rubber tube between two firm supports. Shake the tube from side to side with your hand near one of the supports. If you shake the tube with the right frequency, you will set up a standing wave as shown in Figure 19.14a. Shake the tube with twice the frequency, and a standing wave of half the previous wavelength, having two loops, will result. (The distance between successive nodes is a half wavelength; two loops make up a full wavelength.) Triple the frequency, and a standing wave with one-third the original wavelength, having three loops, results, and so forth. Figure 19.14
(a) Shake the rope until you set up a standing wave of one segment (1/2 wavelength). (b) Shake with twice the frequency and produce a wave with two segments (1 wavelength). (c) Shake with three times the frequency and produce three segments (1½ wavelengths). Standing waves are set up in the strings of musical instruments when plucked, bowed, or struck. They are set up in the air in an organ pipe, a trumpet, or a clarinet, and the air of a soda-pop bottle when air is blown over the top. Standing waves can be set up in a tub of water or a cup of coffee by sloshing it back and forth with the right frequency. Standing waves can be produced with either transverse or longitudinal vibrations. 1. Is it possible for one wave to cancel another wave so that no amplitude remains? Check Yourself Yes. This is called destructive interference. In a standing wave in a rope, for example, parts of the rope have no amplitude--the nodes. 2. Suppose you set up a standing wave of three segments, as shown below. If you shake with twice as much frequency, how many wave segments will occur in your new standing wave? How many wavelengths? If you impart twice the frequency to the rope, you'll produce a standing wave with twice as many segments. You'll have six segments. Since a full wavelength has two segments, you'll have three complete wavelengths in your standing wave. Doppler Effect A pattern of water waves produced by a bug jiggling its legs and bobbing up and down in the middle of a quiet puddle is shown in Figure 19.15. The bug is not going anywhere but is merely treading water in a fixed position. The waves it makes are concentric circles, because wave speed is the same in all directions. If the bug bobs in the water at a constant frequency, the distance between wave crests (the wavelength) is the same in all directions. Waves encounter point A as frequently as they encounter point B. This means that the frequency of wave motion is the same at points A and B, or anywhere in the vicinity of the bug. This wave frequency is the same as the bobbing frequency of the bug. Figure 19.15
Top view of water waves made by a stationary bug jiggling in still water. Suppose the jiggling bug moves across the water at a speed less than the wave speed. In effect, the bug chases part of the waves it has produced. The wave pattern is distorted and is no longer made of concentric circles (Figure 19.16). The center of the outer wave was made when the bug was at the center of that circle. The center of the next smaller wave was made when the bug was at the center of that circle, and so forth. The centers of the circular waves move in the direction of the swimming bug. Although the bug maintains the same bobbing frequency as before, an observer at B would see the waves coming more often. The observer would measure a higher frequency. This is because each successive wave has a shorter distance to travel and therefore arrives at B more frequently than if the bug weren't moving toward B. An observer at A, on the other hand, measures a lower frequency because of the longer time between wave-crest arrivals. This is because to reach A, each crest has to travel farther than the one ahead of it due to the bug's motion. This change in frequency due to the motion of the source (or receiver) is called the Doppler effect (after the Austrian scientist Christian Doppler, 1803-1853). Figure 19.16
Water waves made by a bug swimming in still water toward point B. Water waves spread over the flat surface of the water. Sound and light waves, on the other hand, travel in three-dimensional space in all directions like an expanding balloon. Just as circular waves are closer together in front of the swimming bug, spherical sound or light waves ahead of a moving source are closer together and reach a receiver more frequently. The Doppler effect is evident when you hear the changing pitch of a car horn as the car passes you. When the car approaches, the pitch is higher than normal (higher like a higher note on a musical scale). This is because the crests of the sound waves are hitting your ear more frequently. And when the car passes and moves away, you hear a drop in pitch because the crests of the waves are hitting your ear less frequently. Figure 19.17
The pitch (frequency) of sound increases when a source moves toward you and decreases when the source moves away. The Doppler effect also occurs for light. When a light source approaches, there is an increase in its measured frequency; and when it recedes, there is a decrease in its frequency. An increase in frequency is called a blue shift, because the increase is toward the high frequency, or blue end of the color spectrum. A decrease in frequency is called a red shift, referring to a shift toward the lower-frequency, or red, end of the color spectrum. Distant galaxies, for example, show a red shift in the light they emit. A measurement of this shift permits a calculation of their speeds of recession. A rapidly spinning star shows a red shift on the side turning away from us and a blue shift on the side turning toward us. This enables a calculation of the star's spin rate. Check Yourself When a sound source moves toward you, do you measure an increase or decrease in wave speed? Neither! It is the frequency of a wave that undergoes a change where there is motion of the source, not the wave speed. Be clear about the distinction between frequency and speed. How frequently a wave vibrates is altogether different from how fast the disturbance moves from one place to another. Bow Waves When the speed of a source is as great as the speed of the waves it produces, something interesting happens. The waves pile up in front of the source. Consider the bug in our previous example when it swims as fast as the wave speed. Can you see that the bug will keep up with the waves it produces? Instead of the waves moving ahead of the bug, they superimpose and hump up on one another directly in front of the bug (Figure 19.18). The bug moves right along with the leading edge of the waves it is producing. Figure 19.18
Wave pattern made by a bug swimming at wave speed. A similar thing happens when an aircraft travels at the speed of sound. In the early days of jet aircraft, it was believed that this pile-up of sound waves in front of the airplane imposed a “sound barrier” and that to go faster than the speed of sound, the plane would have to “break the sound barrier.” What actually happens is that the overlapping wave crests disrupt the flow of air over the wings, making it more difficult to control the craft. But the barrier is not real. Just as a boat can easily travel faster than the waves it produces, with sufficient power an aircraft easily travels faster than the speed of sound. Then we say that it is supersonic. A supersonic airplane flies into smooth, undisturbed air because no sound wave can propagate out in front of it. Similarly, a bug swimming faster than the speed of water waves finds itself always entering into water with a smooth, unrippled surface. When the bug swims faster than wave speed, ideally it produces a wave pattern as shown in Figure 19.19. It outruns the waves it produces. The waves overlap at the edges, and the pattern made by these overlapping waves is a V shape, called a bow wave, which appears to be dragging behind the bug. The familiar bow wave generated by a speedboat knifing through the water is not a typical oscillatory wave. It is a disturbance produced by the overlapping of many circular waves. Figure 19.19
A bow wave, the pattern made by a bug swimming faster than wave speed. The points at which adjacent waves overlap (x) produce the V shape. Some wave patterns made by sources moving at various speeds are shown in Figure 19.20. Note that after the speed of the source exceeds wave speed, increased speed of the source produces a narrower V shape. Figure 19.20
Patterns made by a bug swimming at successively greater speeds. Overlapping at the edges occurs only when the bug swims faster than wave speed. Shock Waves A speedboat knifing through the water generates a two-dimensional bow wave. A supersonic aircraft similarly generates a three-dimensional shock wave. Just as a bow wave is produced by overlapping circles that form a V, a shock wave is produced by overlapping spheres that form a cone. And just as the bow wave of a speedboat spreads until it reaches the shore of a lake, the conical wake generated by a supersonic craft spreads until it reaches the ground. The bow wave of a speedboat that passes by can splash and douse you if you are at the water's edge. In a sense, you can say that you are hit by a “water boom.” In the same way, when the conical shell of compressed air that sweeps behind a supersonic aircraft reaches listeners on the ground below, the sharp crack they hear is described as a sonic boom. Figure 19.21
This aircraft has just cracked the wave barrier. The cloud is water vapor that has just condensed out of the rapidly expanding air in the rarefied region behind the wall of compressed air. We don't hear a sonic boom from slower-than-sound, or subsonic, aircraft because the sound waves reaching our ears are perceived as one continuous tone. Only when the craft moves faster than sound do the waves overlap to reach the listener in a single burst. The sudden increase in pressure is much the same in effect as the sudden expansion of air produced by an explosion. Both processes direct a burst of high pressure air to the listener. The ear is hard pressed to distinguish between the high pressure from an explosion and the high pressure from many overlapping waves. A water skier is familiar with the fact that next to the high hump of the V-shaped bow wave is a V-shaped depression. The same is true of a shock wave, which usually consists of two cones: a high-pressure cone generated at the bow of the supersonic aircraft and a low-pressure cone that follows at the tail of the craft. The edges of these cones are visible in the photograph of the supersonic bullet below. Figure 19.22
Shock wave of a bullet piercing a sheet of Plexiglas. Light deflecting as it passes through the compressed air makes the shock visible. Look carefully and see the second shock wave originating at the tail of the bullet. Between these two cones the air pressure rises sharply to above atmospheric pressure, then falls below atmospheric pressure before sharply returning to normal beyond the inner tail cone (Figure 19.24). This overpressure suddenly followed by underpressure intensifies the sonic boom. Figure 19.23 A shock wave. Figure 19.24
The shock wave is actually made up of two cones--a high-pressure cone with the apex at the bow of the aircraft and a low-pressure cone with the apex at the tail. A graph of the air pressure at ground level between the cones takes the shape of the letter N. A common misconception is that sonic booms are produced when an aircraft flies through the “sound barrier”--that is, just as the aircraft surpasses the speed of sound. This is the same as saying that a boat produces a bow wave when it first overtakes its own waves. This is not so. The fact is that a shock wave and its resulting sonic boom are swept continuously behind and below an aircraft traveling faster than sound, just as a bow wave is swept continuously behind a speedboat. In Figure 19.25, listener B is in the process of hearing a sonic boom. Listener C has already heard it, and listener A will hear it shortly. The aircraft that generated this shock wave may have broken through the sound barrier hours ago! Figure 19.25
The shock wave has not yet reached listener A, but is now reaching listener B and has already reached listener C. It is not necessary that the moving source be “noisy” to produce a shock wave. Once an object is moving faster than the speed of sound, it will make sound. A supersonic bullet passing overhead produces a crack, which is a small sonic boom. If the bullet were larger and disturbed more air in its path, the crack would be more boomlike.
When a lion tamer cracks a circus whip, the cracking sound is actually a sonic boom produced by the tip of the whip when it travels faster than the speed of sound. Both the bullet and the whip are not vibrating so they are not sound sources. But when traveling at supersonic speeds they produce their own sound as they generate shock waves. Summary of Terms Sine curve A wave form traced by simple harmonic motion, which can be made visible on a moving conveyor belt by a pendulum swinging at right angles above the moving belt.
Amplitude For a wave or vibration, the maximum displacement on either side of the equilibrium (midpoint) position. Wavelength The distance between successive crests, troughs, or identical parts of a wave.
Frequency For a vibrating body or medium, the number of vibrations per unit time. For a wave, the number of crests that pass a particular point per unit time. Hertz The SI unit of frequency. One hertz (symbol Hz) equals one vibration per second.
Period The time in which a vibration is completed. The period of a wave equals the period of the source, and is equal to 1/frequency. Wave speed The speed with which waves pass a particular point: Transverse wave A wave in which the medium vibrates in a direction perpendicular (transverse) to the direction in which the wave travels. Light waves and water waves are transverse. Longitudinal wave A wave in which the medium vibrates in a direction parallel (longitudinal) to the direction in which the wave travels. Sound waves are longitudinal. Interference pattern The pattern formed by superposition of different sets of waves that produces reinforcement in some places and cancellation in others.
Standing wave A stationary wave pattern formed in a medium when two sets of identical waves pass through the medium in opposite directions.
Doppler effect The shift in received frequency due to motion of a vibrating source toward or away from a receiver. Bow wave The V-shaped disturbance made by an object moving across a liquid surface at a speed greater than the wave speed.
Shock wave The cone-shaped disturbance made by an object moving at supersonic speed through a fluid.
Sonic boom The loud sound resulting from the incidence of a shock wave. Review Questions 1. What is a wiggle in time called? A wiggle in space and time? A wiggle in time is called a vibration; a wiggle in space and time is called a wave. 2. Distinguish between the propagation of sound waves and light waves. Sound is the propagation of vibrations through a material medium--a solid, liquid, or gas. Light is a vibration of electric and magnetic fields. 3. What is the source of all waves? The source of all waves is something that is vibrating. 4. What feature about a pendulum makes it useful in a grandfather clock? The period of a pendulum does not depend on the mass of the pendulum or on the size of the arc through which it swings. 5. What is meant by the period of a pendulum? The period of a pendulum is the time it takes to swing back and forth. 6. Which has the longer period, a short or a long pendulum? The longer the length of a pendulum, the longer its period. 7. How is a sine curve related to a wave? A sine curve is a pictorial representation of a wave. 8. Distinguish between these different parts of a wave: period, amplitude, wavelength, and frequency. Period is the time for one complete vibration; Amplitude is the distance from the midpoint to the crest (or trough) of the wave; the wavelength is the distance from the top of one crest to the top of the next one; the frequency is the number of complete vibrations in a given time (usually one second). 9. How many vibrations per second are represented in a radio wave of 101.7 MHz? The number of vibrations per second is equal to the frequency--so there are 101.7 million vibrations per second. 10. How do frequency and period relate to each other? Frequency and period are reciprocals of each other. 11. In one word, what is it that moves from source to receiver in wave motion? Energy No. A vibrating rope transfers energy along the rope while the rope moves back and forth; a water wave transfers energy from the ocean to the shore but the water moves up and down. 12. Does the medium in which a wave travels move with the wave? Give examples to support your answer. 13. What is the relationship among frequency, wavelength, and wave speed? Speed =
frequency x wavelength 14. In what direction are the vibrations relative to the direction of wave travel in a transverse wave? The vibrations are perpendicular to the direction of wave travel. 15. In what direction are the vibrations relative to the direction of wave travel in a longitudinal wave? In a longitudinal wave, the vibrations are in the same direction (parallel) to the travel direction of the wave. 16. The wavelength of a transverse wave is the distance between successive crests (or troughs). What is the wavelength of a longitudinal wave? The distance between the high pressure (compressions) or low pressure (rarefactions) regions in the wave. 17. What is meant by the superposition principle? When more than one wave occupies the same space at the same time, the displacements add at every point. 18. Distinguish between constructive interference and destructive interference. In constructive interference, the crest of one wave adds to the crest of another, and the amplitude increases. In destructive interference, the crest of one wave adds to the trough of another and the amplitude decreases. 19. What kinds of waves can show interference? Interference is characteristic of all wave motion. 20. What causes a standing wave? A standing waves results when an incident wave and a reflective wave interfere and cause parts of the rope to remain stationary. 21. What is a node? What is an antinode? A node is a region of minimal or zero displacement, with minimal or zero energy. An antinode is a region of maximum displacement and maximum energy. 22. In the Doppler effect, does frequency change? Does wavelength change? Does wave speed change? In the Doppler effect, the frequency of the wave changes and the wavelength changes, but the wave velocity does not change. 23. Can the Doppler effect be observed with longitudinal waves, transverse waves, or both? The Doppler effect can be observed with both types of waves. 24. What is meant by a blue shift and a red shift for light? A red shift is a decrease in frequency of a light wave (toward the red end of the spectrum) and a blue shift is an increase in frequency (toward the blue end of the spectrum). 25. How fast must a bug swim to keep up with the waves it produces? How fast must it move to produce a bow wave? It will keep up with the waves it produces if it swims at the same speed as the wave travels. If it swims at a higher speed, it will produce a bow wave. 26. How fast does a supersonic aircraft fly compared with the speed of sound? Faster than the speed of sound. 27. How does the V shape of a bow wave depend on the speed of the source? As the speed of the source increases past the wave speed, the source produces a narrower V-shape. 28. A bow wave on the surface of water is two-dimensional. How about a shock wave in air? A shock wave in air is three-dimensional, producing overlapping spheres that form a cone. 29. True or false: A sonic boom occurs only when an aircraft is breaking through the sound barrier. Defend your answer. False. A sonic boom is created continuously as long as an aircraft is flying faster than the speed of sound. It is only heard when the bow wave created by the plane reaches the ear of a listener. 30. True or false: In order for an object to produce a sonic boom, it must be “noisy.” Give two examples to support your answer. False. Once an object is moving faster than the speed of sound, it will make sound. A bullet and a whip are two examples.