Chapter 3 Gravity and Motion

Transcription

1 Chapter 3 Gravity and Motion 3.1Inertia Central to Galileo's laws of motion is the concept of inertia. Inertia is the tendency of a body at rest to remain at rest or of a body in motion to keep moving in a straight line at a constant speed. Aristotle noted that bodies at rest resist being moved, but he failed to link this property to the tendency of objects to keep moving once they are set in motion. Aristotle pictured that some force must be steadily applied to an object to ensure that it would keep moving. Kepler recognized inertia's importance and in fact was first to use that term. However, Galileo not only proposed this property of matter but also demonstrated it by real experiment. In one such experiment, Galileo rolled a ball down a sloping board over and over again and noticed that it always sped up as it rolled down the slope (fig. 3.1). He next rolled the ball up a sloping board and noticed that it always slowed down as it approached the top. He hypothesized that if a ball rolled on a flat surface and no forces such asfriction acted on it, its speed would neither increase nor decrease but remain constant. That is, in the absence of forces, inertia keeps an object already in motion moving at a fixed speed. Inertia is familiar to us all even in everyday life. Apply the brakes of your car suddenly, and the inertia of the bag of groceries beside you keeps the bag moving forward at its previous speed until it hits the dashboard or spills onto the floor. FIGURE 3.1 A ball rolling down a slope speeds up. A ball rolling up a slope slows down. A ball rolling on a flat surface rolls at a constant speed if no forces (such as friction) act on it. Newton recognized the special importance of inertia. He described it in what is now called Newton's first law of motion (sometimes referred to simply as the law of inertia). The law can be stated as follows: A body continues in a state of rest or of motion in a straight line at a constant speed unless made to change that state by forces acting on it. In applying Newton's first law, we should note two important points. First, we have not defined force yet but have relied on our intuitive feeling that a force is anything that pushes or pulls. Second, we need to note that when we use the term force, we are talking about net force; that is, the total of all forces acting on a body. For example, if a box at rest is pushed equally by two opposing forces, the forces are balanced. Therefore, the box experiences no net force and accordingly does not move (fig. 3.2). Newton's first law may not sound impressive at first, but it carries an idea that is crucial in astronomy: that if a body is not moving in a straight line at a constant speed, some net force must be acting on it.

2 FIGURE 3.2 Balanced forces lead to no acceleration. Actually, Newton was preceded in stating this law by the seventeenth-century Dutch scientist Christiaan Huygens. However, Newton went on to develop additional physical laws and more important for astronomy showed how to apply them to the Universe. For example, let us look at what happens if we swing a mass tied to a string in a circle. Newton's law tells us that the mass's inertia will carry it in a straight line if no forces act. What force, then, is acting on the circling mass? The force is the one exerted by the string, preventing the mass from moving in a straight line and keeping it moving in a circle. We can feel that force as an outward pull on the string as the mass swings around. Page 75 We can see the importance of the force if the string suddenly breaks. Many people expect the mass to retain some memory of its former circular motion and travel along the path labeled B in Figure 3.3 at least a little while longer. Or sometimes they think it should move outward along the path labeled C because of the sudden loss of the inward force. However, with the force of the string no longer acting on it, the mass flies off in a straight line in the same direction it was moving at the moment the string broke, as illustrated by the arrow labeled A in figure 3.3, demonstrating Newton's first law. FIGURE 3.3 For a mass on a string to travel in a circle, a force (green arrow) must act along the string to overcome inertia. Without that force, inertia makes the mass move in a straight line.

3 We can translate this example to an astronomical setting and apply it to the orbit of the Moon around the Earth, or of the Earth around the Sun, or to the Sun in orbit around the Milky Way. Each of these bodies follows a curved path. Therefore, each must have a force acting on it, the origin and nature of which we will now describe. 3.2Orbital Motion and Gravity Newton was not the first person to attempt to discover and define the force that holds planets in orbit around the Sun. Nearly 100 years before Newton, Kepler recognized that some force must hold the planets in their orbits and proposed that something similar to magnetism might be responsible. Newton was not even the first person to suggest that gravity was responsible. That honor belongs to Robert Hooke, another Englishman, who noted gravity's role in celestial motions several years before Newton published his law of gravity in Newton's contribution is nevertheless special because he demonstrated the properties that gravity must have if it is to control planetary motion. Moreover, Newton went on to derive equations that describe not only gravity but also its effects on motion. The solution of these equations allowed astronomers to predict the position and motion of the planets and other astronomical bodies. Page 76 According to legend, Newton realized gravity's role when he saw an apple falling from a tree. The falling apple drawn downward to the Earth's surface presumably made him speculate whether Earth's gravity might extend to the Moon. Influenced by an apple or not, Newton correctly deduced that Earth's gravity, if weakened by distance, could explain the Moon's motion. Newton's Cannon A NI MATION Newton's cannon Most of Newton's work is highly mathematical, but as part of his discussion of orbital motion, he described a thought experiment to demonstrate how an object can move in orbit. Thought experiments are not actually performed; rather, they serve as a way to think about problems. In Newton's thought experiment, we imagine a cannon on a mountain peak firing a projectile (fig. 3.4A). From our everyday experience, we know that whenever an object is thrown horizontally, gravity pulls it downward so that its path is an arc. Moreover, the faster we throw, the farther the object travels before striking the ground. FIGURE 3.4

4 (A) A cannon on a mountain peak fires a projectile. If the projectile is fired faster, it travels farther before hitting the ground. (B) At a sufficiently high speed, the projectile travels so far that the Earth's surface curves out from under it, and the projectile is in orbit. Q. Why are space vehicles normally launched from regions near the equator? Why are space vehicles normally launched to the east? answer We now imagine increasing the projectile's speed more and more, allowing the projectile to travel ever farther. As the distance traveled becomes very large, we see that the Earth's surface curves away below the projectile (fig. 3.4B). Therefore, if the projectile moves sufficiently fast, the Earth's surface may curve away from it in such a way that the projectile will never hit the ground. Such is the nature of orbital motion and how the Moon orbits the Earth. The balance between inertia and the force of gravity maintains the orbit. We can analyze this thought experiment more specifically with Newton's first law of motion. According to that law, in the absence of forces the projectile will travel in a straight line at constant speed. But because a force, gravity, is acting on the projectile, its path is not straight but curved. Moreover, the law helps us understand that the projectile does not stop, but continues moving, because its inertia carries it forward. If we apply the same reasoning to the Moon's motion, we conclude that. the Moon moves along its orbit because its inertia carries it forward, and its path is curved (not a straight line) because the force of gravity deflects it. Notice that in the above discussion we used no formulas. All we needed was Newton's first law and the idea that gravity supplies the deflecting force. However, if we are to understand the particulars of orbital motion, we require additional laws. For example, to determine how rapidly the projectile must move to be in orbit, we need laws that have a mathematical formulation. 3.3Newton's Second Law of Motion We stated that an object's inertia causes it to move at a constant speed in a straight line in the absence of forces. However, suppose forces do act on the object. How much deviation from straight-line motion will such forces produce? To answer that question, we need to define more carefully what we mean by motion. Motion of an object is a change in its position, which we can characterize in two ways: by the direction the object is moving and by the object's speed. For example, suppose a car is moving east at 30 miles per hour. If the car's speed and direction remain constant, we say it is in uniform motion (fig. 3.5A). If the car changes either its speed or its direction, it is no longer moving uniformly, as depicted in figures 3.5B and C. Such nonuniform motion is defined as acceleration.

5 FIGURE 3.5 Views of a car in uniform motion and accelerating. (A) Uniform motion implies no change in speed or direction. The car moves in a straight line at a constant speed. If an object's (B)speed or (C) direction changes, the object undergoes an acceleration. Q. Is it possible for an object to travel with constant speed and still accelerate? answer Acceleration We are all familiar with acceleration as a change in speed. For example, when we step on the accelerator in a car and it speeds up from 30 to 40 miles per hour, we say the car is accelerating. Its speed has changed, and its motion is therefore nonuniform. Although in everyday usage acceleration implies an increase in speed, scientifically any change in speed is an acceleration. Thus, technically, a car accelerates when we apply the brakes and it slows down. Strictly speaking, the wordspeed indicates the rate of motion, irrespective of its direction. The term velocityrefers to a speed in a given direction. Thus, a body's velocity changes if either its speed or its direction changes. With velocity so defined, acceleration is simply the change in velocity over some interval of time. In the example above, we produced an acceleration by changing an object's speed. We can also produce an acceleration by changing an object's direction of motion. For example, suppose we drive a car around a circular track at a steady speed of 30 miles per hour. At each moment, the car's direction of travel is changing, and therefore it is not in uniform motion. Similarly, a mass swung on a string or a planet orbiting the Sun is experiencing nonuniform motion and is therefore accelerating. In fact, a body moving in a circular orbit constantly accelerates, even though its speed is not changing. How do we produce an acceleration? Newton realized that for an object to accelerate, a force must act on it. For example, to accelerate change the direction of the mass whirling on a string, we must constantly exert a pull on the string. Similarly, to accelerate a shopping cart, we must exert a force on it. In addition, experiments show that the acceleration we get is proportional to the force we apply. That is, a larger force produces a larger acceleration. For example, if we push a shopping cart gently, its acceleration is slight. If we push harder, its acceleration is greater. But experience shows us that more than just force is at work here. For a given push, the amount of acceleration also depends on how full the cart is. A lightly loaded cart may scoot away under a slight push, but a heavily loaded cart hardly budges, as illustrated in figure 3.6. Thus, the acceleration produced by a given force also depends on the amount of matter being accelerated.

6 FIGURE 3.6 A loaded cart will not accelerate as easily as an empty cart. Page 78 Mass The amount of matter an object contains is determined by a quantity that scientists call mass. Technically, mass measures an object's inertia. The more inertia, the more mass, and vice versa. Scientists generally measure the mass of ordinary objects in kilograms (abbreviated kg). One kilogram equals 1000 grams or about 2.2 pounds of mass. For example, under normal conditions, a liter of water (roughly a quart) has a mass of 1 kilogram, but it is important to remember that mass is not the same as weight. Because an object's mass describes the amount of matter in it, its mass in kilograms is a fixed quantity. An object's weight, however, measures the force of gravity on it, a point we will explore more insection 3.7. Thus, although an object's mass is fixed, its weight changes if the local gravity changes. For example, on Earth we have one weight but on the Moon, where gravity is less, we have a lesser weight, but no matter where we are, we have the same mass. F = Any force m = Mass of the body being accelerated a = Amount of acceleration Mass is the final quantity needed to understand Newton's second law of motion. This law states that when a force, F, acts on an object whose mass is m, it produces an acceleration, a, according to the equation F = ma. However, the second law's meaning is clearer if we write: or, stated in words: The acceleration of a body is proportional to the net force exerted on it, but inversely proportional to the mass of the body. This astonishingly simple equation allows scientists to predict virtually all features of a body's motion. With a = F / m and with knowledge of the masses and the forces in action, engineers and scientists can,

7 for example, drop a spacecraft safely between Saturn and its rings or use a computer to design an airplane that will fly successfully the first time it is tested. FIGURE 3.7 Skateboarders illustrate Newton's third law of motion. (A) When X pushes on Y, an equal push is given to X by Y. Because both are the same mass, they both accelerate to the same speed. (B) When X pushes on Z, who is more massive, the resulting acceleration is smaller for Z in proportion to how much more massive he is. 3.4Newton's Third Law of Motion Newton's studies of motion and gravity led him to yet another critical law, which relates the forces that bodies exert on each other. This additional relation, Newton's third law of motion, is sometimes called the law of action reaction. This law states: When two objects interact, they create equal and opposite forces on each other. Two skateboarders side by side may serve as a simple example of the third law (fig. 3.7). When one skateboarder pushes on another, bothmove. According to Newton's third law, when X exerts a force on Y, Y exerts an equal force on X, so that both are accelerated. Note that although the forces between the skateboarders are equal and opposite in direction, the resulting magnitude of the acceleration is not necessarily the same. Here we have to recall Newton's second law: a = F/m. If one of the skateboarders weighs much more than the other, the acceleration of the more massive of the two will be smaller by the same factor that the mass is bigger (fig. 3.7B). The same principle explains why a spacecraft accelerates in response to firing its rockets. It is often incorrectly believed that the propellant pushes against the ground or air, but if that were true, a spacecraft would be unable to accelerate once it was in space. By Newton's third law, pushing propellant out the rocket nozzle produces an equal push in the opposite direction on the ship (fig. 3.8). This also shows how an astronaut who accidentally floats free from a spacecraft can redirect her motion back to the

8 spacecraft. By throwing an object in a direction away from the spacecraft, an equal and opposite force is exerted on the astronaut, who is thereby pushed back toward the ship. FIGURE 3.8 A rocket accelerates according to Newton's third law. Ignited propellants are pushed out the back of the rocket at high speed. They apply an equal and opposite force on the rocket, accelerating the ship forward. Nothing needs to be behind the rocket to be pushed on. FIGURE 3.9 Gravity produces a force of attraction (green arrows) between bodies. The strength of the force depends on the product of their masses, m and M, and the square of their separation, d 2. G is the universal gravitational constant 3.5The Law of Gravity Using Newton's three laws, we can determine an object's motion if we know its initial state of motion and the forces acting on it. For astronomical bodies, that force is often limited to gravity, and so to predict their motion, we need to know how to calculate gravity's force. Once again we encounter Newton's work, for it was he who first worked out the law of gravity. On the basis of his study of the Moon's motion, Newton concluded the following: Every mass exerts a force of attraction on every other mass. The strength of the force is directly proportional to the product of the masses divided by the square of their separation. We can write this extremely important result in a shorthand mathematical manner as follows: Let m and M be the masses of the two bodies (fig. 3.9) and let the distance between their centers be d. Then the strength of the force of gravity between them, F, is F = Strength of the gravitational force between two bodies M = Mass of one body m = Masws of second body d = Separation between the bodies' centers

9 G = Gravitational constant. The factor G is a constant whose value is found by measuring the force between two bodies of known mass and separation. The resulting number for G depends on the units chosen to measure M, m, d, and F, but once determined, G is the same as long as the same units are used. For example, if M and mare measured in kilograms, d in meters, and F in SI units, * then G = meters 3 /(kilogram.second 2 ) [m 3 kg 1 s 2 ]. Writing the law of gravity as an equation helps us see several important points. If either M or m increases and the other factors remain the same, the force increases. If d (the distance between two objects) increases, the force gets weaker. Moreover, the force weakens as the square of the distance. That is, if the distance between two masses is doubled, the gravitational force between them decreases by a factor of four, not two. Finally, although one object's gravitational force on another weakens with increasing distance, the gravitational force never completely disappears. Thus, the gravitational attraction of an object reaches across the entire Universe, so the Earth's gravity not only holds you on to Earth's surface but also extends to the Moon and exerts the force that holds the Moon in orbit around the Earth. A NI MATION Gravity produces a force of attraction between bodies. Gravity Produces a Force of Attraction Between Bodies The gravitational force between the Earth and the Moon provides another astronomical example of Newton's third law and takes us a step closer to understanding orbital motion. The gravitational force of the Earth on the Moon is exactly equal to the gravitational force of the Moon on the Earth. We can see this from Newton's law of gravity where the gravitational force between two objects depends on the product of their masses, so we get the same force regardless of whether we let the Earth act on the Moon or vice versa. Why, then, does the Moon orbit the Earth and not the other way around? Newton's second law supplies the answer. Written in the form a = F/m, we see that the acceleration an object feels is inversely proportional to its mass; that is, the more massive it is, the more force is required to accelerate it. Thus, even though the forces acting on the Earth and Moon are precisely equal, the Moon accelerates 81 times more because it is 81 times less massive than the Earth. As a result, because the Moon's acceleration is so much larger than the Earth's, the Moon does most of the moving (fig. 3.10A). In fact, however, the Earth does move a little bit as the Moon orbits it, much as you must move if you swing a child or a heavy object around you (fig. 3.10B).

10 FIGURE 3.10 (A) The Earth and Moon exert equally strong gravitational forces on each other, but the Moon is 81 times less massive, so that force produces an acceleration (and resulting velocity) that is 81 times larger. (B) You feel a similar reaction force as the Earth when swinging around a heavy object. 3.6Measuring an Object's Mass Using Orbital Motion Knowledge of orbital motion is important for more than just understanding the paths of astronomical objects. From the orbit's properties (such as size and orbital period), astronomers can deduce physical properties of the orbiting objects, such as their mass. Newton drew upon Kepler's third law in deriving his law of gravity. Although this is not simple to see, it turns out that the exponents in Kepler's law (the square of P and the cube of a) are set by the power (square of d) in the law of gravity. The basic method for determining an astronomical object's mass uses a modified form of Kepler's third law and was first worked out by Newton using his laws of motion and gravity. The underlying idea is very simple: The masses of the orbiting objects determine the gravitational force between them. The gravitational force in turn sets the properties of the orbit. Thus, from knowledge of the orbit, astronomers can work backward to find the masses of the objects. To see how this can be done, we consider a very simple case: orbital motion in a perfect circle with the orbiting object having a mass so small that it can be ignored compared with that of the central body. These restrictions are met to high precision in many astronomical systems, such as the Earth's motion around the Sun and the Sun's motion around the Milky Way. By assuming that the mass of the central body is large compared with the orbiting body, we can ignore the acceleration of the central body (as we just discussed) and assume it is at rest. These assumptions simplify the problem but are not essential for its solution. To work out the orbital properties of an object moving around another, we use Newton's laws of motion and his law of gravity. From the first law, we know that if an object moves along a circular path, there is a net force (an unbalanced force) acting on it because balanced forces give straight-line motion. This force (known as a centripetal force) must be applied to any object moving in a circle, whether it is a car rounding a curve, a mass swung on a string, or the Earth orbiting the Sun. Orbital Velocity INTERACTIVE

11 Orbital velocity Page 82 Using Newton's second law of motion and some algebra and geometry gives us an equation that shows that if a mass (m) moves with a velocity (V) around a circle at a distance (d) from the center, the force needed to hold it in a circular orbit is F = Force needed to hold a body in orbit m = Mass of the body d = Radius of the orbit V = Velocity of the body From this equation, we can determine what force the Sun must be exerting on a planet based on the orbital velocity and distance of the planet. Let the Sun's mass be M and the planet's mass be m, where the mass m is assumed to be much smaller than M. Then if the planet moves in a circular orbit at distance d, the gravitational force between the Sun and the planet is given by GMm/d 2, and this must match the centripetal force mv 2 /d to hold the planet in orbit. Setting these two forces equal to each other, we have: M = Mass of an orbited body d = Radius of the orbit V = Velocity of the orbit G = Gravitational constant Multiplying both sides of the equation by d 2 and dividing both by Gm, we obtain This equation can be used to determine the mass of the central object if the orbiting object's velocity and distance from it are known. The same method can be used to find the mass of any object around which another object orbits. This is illustrated in Astronomy by the Numbers: Weighing the Sun. Thus, gravity becomes a tool for determining the mass of astronomical objects, and we shall use this method many times throughout our study of the Universe. A S T R O N O M Y BY THE NUMBERS WEIGHING THE SUN To find the mass (M) of the Sun, we need d, the distance of the Earth from the Sun, and V, the speed with which the Earth is moving in its orbit.

12 We find V by dividing the circumference of Earth's orbit by the time it takes the Earth to complete one orbit, a length of time we call the orbital period, P. The circumference of a circle of radius d is 2πd (note that in this case the radius is d). Therefore, Putting this expression for V into the equation for M gives This is a modified form of Kepler's third law and it is important because we can use it to measure the mass of any object given the orbital period, P, and orbital distance, d, of a much smaller object moving around it. To calculate the Sun's mass, we need P and d for the Earth's orbit. If we measure P in seconds and d in meters, we can use the SI value of the gravitational constant G = m 3 kg 1 s 2. To find P in seconds, we remember that it takes the Earth one year to orbit the Sun. Thus, P is one year. We can express P in seconds by multiplying the number of seconds in a minute (60), times the number of minutes in an hour (60), times the number of hours in a day (24), times the number of days in a year (365.25). The result of that calculation, rounded off to three significant figures, is Similarly, we need d, the Earth Sun distance, in meters, which we can look up in the appendix and find is m. Putting these values and the value of p and G into the expression for M we find 3.7Surface Gravity One of Galileo's famous discoveries was that objects of different masses all fall at the same rate. Many people incorrectly believe that lighter objects fall slower, but that is just a result of air friction. In a vacuum all objects accelerate downward at the same rate. Astronomers call this acceleration surface gravity, which gives a measure of the gravitational attraction at a planet's or star's surface. This acceleration determines not only how fast objects fall, but since an object's weight depends on its mass and the acceleration of gravity, surface gravity also determines what a mass weighs. We can determine the strength of a planet's surface gravity as follows. The law of gravity states that a planet of mass M exerts a gravitational force F on a body of mass m at a distance d from its center given by GMm/d 2. However, at the planet's surface, d = R, the planet's radius, so the force on the body in other words its weight is

13 g = Surface gravity G = Gravitational constant M = Mass of the attracting body R = Radius of the attracting body The second equality in this equation is just Newton's second law, which tells us the acceleration a mass m experiences when the force F is applied. We can cancel out the m's to give us the resulting acceleration, in other words the surface gravity, which astronomers usually denote by the letter g: Note that the surface gravity does not depend on the mass of the object that is falling, which explains Galileo's discovery about falling objects. However, the surface gravity of other astronomical bodies is usually different from Earth's, depending on their mass and radius. For example, the Moon's surface gravity is about 6 times smaller than the Earth's, as illustrated in Figure 3.11 and calculated in Astronomy by the Numbers below. FIGURE 3.11 Apollo 16 astronaut John Young easily leaps high in the Moon's low gravity despite a massive space suit. A S T R O N O M Y BY THE NUMBERS THE SURFACE GRAVITY OF THE EARTH AND MOON To compare the surface gravity of the Moon with that of the Earth, we need to know the mass and radius of both. Those numbers are given in table 3.1. We can make this calculation two ways. The first way is to plug into the equation for g the values of M and R appropriate for the Earth and then repeat the process with values appropriate for the Moon. An easier way is to work with ratios. To do that, we write out on one

14 line the expression for g on the Earth and then write out on the line below it the value for g on the Moon. Next we draw a horizontal line between them to indicate division, which gives the following equation: Notice that by doing this, the value of G, the gravitational constant, cancels out. If we then group the terms in M and the terms in R separately, we find d This shows that the ratio of g on the Earth to g on the Moon is about 6:1, so that you weigh about 6 times more on the Earth than you would on the Moon. INTERACTIVE Gravity variations The surface gravity formula tells us that two planets with the same radius but different masses will have different surface gravities the planet with the larger mass will have the larger surface gravity. Similarly, if two planets have the same mass but different radii, the planet with the larger radius will have a smaller surface gravity. Page 84 Besides telling us how much more or less we would weigh, surface gravity affects the steepness of geological features, the degree of compression of a planet's atmosphere, and even the overall shape of an astronomical body. Small objects such as asteroids are not spherical because the gravitational pull at their surface is too weak to crush them into round shapes. 3.8Escape Velocity V esc = Escape velocity G = Gravitational constant M = Mass of the body to be escaped from R = Radius of the body to be escaped from

15 To overcome a planet's gravitational force and escape into space, a rocket must achieve a critical speed known as theescape velocity. The escape velocity, V esc, for a spherical body, such as a planet or star, can be found from the law of gravity and Newton's laws of motion and is given by the following formula: Here, M stands for the mass of the body from which we are attempting to escape, and R is its radius, as shown in figure The escape velocity is the speed an object must have to be able to move away from a body and not be drawn back by its gravity. We can understand how such a speed might exist if we think about throwing an object upward. The faster the object is tossed upward, the higher it goes and the longer it takes to fall back. Escape velocity is the speed an object needs so that it will never fall back, as depicted infigure Thus, escape velocity is of great importance in space travel if craft are to move away from one object and not be drawn back to it. FIGURE 3.12 Escape velocity is the speed an object must have to overcome the gravitational force of a body such as a planet and not fall back. Notice in the equation for V esc that if two objects of the same radius are compared, the larger mass will have the larger escape velocity. Likewise, if two objects of the same mass are compared, the one with the smaller radius will have the greater escape velocity. The escape velocity from the Moon's surface is calculated in the Astronomy by the Numbers box below. A similar calculation shows that the escape velocity from the Earth is 11 kilometers per second. Thus, it is much easier to blast a rocket off the Moon than off the Earth. Inchapter 7, we will see that this low escape velocity is partly responsible for the Moon's lack of an atmosphere, as illustrated in figure 3.13.

16 FIGURE 3.13 The Moon's escape velocity is almost five times smaller than the Earth's. A low escape velocity, in general, leads to the absence of an atmosphere on a planet or satellite A S T R O N O M Y BY THE NUMBERS THE ESCAPE VELOCITY FROM THE MOON To illustrate the use of the formula for escape velocity, we apply it to the Moon using the data in table 3.1. We insert the Moon's mass and radius in the formula and find Note that when we take the square root, we take the square root of both the number and the units. This is easy to do. Just divide the exponents by 2. Thus Carrying out the calcuation gives This escape velocity is about 5000 mph, which is about five times smaller than Earth's. Page 85 Although escape velocity is usually calculated from the surface of a body, where R is the body's radius, the escape velocity can also be found at any larger distance. For example, at Earth's distance from the Sun (1 AU), we would find that a spacecraft would need to be launched at 42 km/sec to escape the Sun's gravity. Thus a spacecraft launched just fast enough to escape the Earth would not have enough speed to leave the Solar System. INTERACTIVE Escape velocity Escape Velocity Escape velocity is particularly important in understanding the nature of black holes. In chapter 15, we will see that a black hole is an object whose escape velocity exceeds the speed of light. Thus, light cannot escape from it, thereby making it black. A black hole has such a huge escape velocity, not necessarily because it has an unusually large mass, but because it has an abnormally small radius.

17 SUMMARY A gravitational force exists between any two objects in the Universe. The strength of this force depends on the masses of the bodies and their separation. Gravitational forces hold astronomical bodies together and in orbit about one another. An object's inertia makes it remain at rest or move in a straight line at a constant speed unless the object is acted on by a net force. Thus, for a planet to orbit the Sun, the Sun's force of gravity must act on it. The law of inertia and Newton's other laws of motion, when combined with the law of gravity, allow us to relate the size and speed of orbital motion to the mass of the central object. The gravitational force exerted by a planet determines its surface gravity and escape velocity. The former determines your weight on a planet. The latter is the speed necessary to leave the surface and escape without falling back.