4-6 Newton formulated laws of motion and gravity that describe fundamental properties of physical reality

The same laws of motion that hold sway on Earth apply throughout the universe

Until the mid-seventeenth century, virtually all attempts to describe the motions of the heavens were empirical, or based directly on data and observations. From Ptolemy to Kepler, astronomers would adjust their ideas and calculations by trial and error until they ended up with answers that agreed with observation.

Figure 4-18: Isaac Newton (1642–1727) Using mathematical techniques that he devised, Isaac Newton formulated the law of universal gravitation and demonstrated that the planets orbit the Sun according to simple mechanical rules.
(Corbis Images)

Isaac Newton (Figure 4-18) introduced a new approach. He began with three quite general statements, now called Newton’s laws of motion. These laws, deduced from experimental observation, apply to all forces and all objects. Newton then showed that Kepler’s three laws follow logically from these laws of motion and from a formula for the force of gravity that he derived from observation.

In other words, Kepler’s laws are not just an empirical description of the motions of the planets, but a direct consequence of the fundamental laws of physical matter. Using this deeper insight into the nature of motions in the heavens, Newton and his successors were able to accurately describe not just the orbits of the planets but also the orbits of the Moon and comets.

 

Newton’s First Law

Newton’s laws of motion describe objects on Earth as well as in the heavens. Thus, we can understand each of these laws by considering the motions of objects around us. We begin with Newton’s first law of motion:

An object remains at rest, or moves in a straight line at a constant speed, unless acted upon by a net outside force.

By force we mean any push or pull that acts on the object. An outside force is one that is exerted on the object by something other than the object itself. The net, or total, outside force is the combined effect of all of the individual outside forces that act on the object.

Right now, you are demonstrating the first part of Newton’s first law—remaining at rest. As you sit in your chair reading this passage, there are two outside forces acting on you: The force of gravity pulls you downward, and the chair pushes up on you. These two forces are of equal strength but of opposite direction, so their effects cancel one another—there is no net outside force. Hence, your body remains at rest as stated in Newton’s first law. If you try to lift yourself out of your chair by grabbing your knees and pulling up, you will remain at rest because this force is not an outside force: It comes from your body.

CAUTION!

It is easy to confuse the net (or total) outside force on an object (central to Newton’s first law) with individual outside forces on an object. If you want to make a book move across the floor in a straight line at a constant speed, you must continually push on it. You might therefore think that your push is a net outside force. But another force also acts on the book—the force of friction as the book rubs across the floor. The force of your push and the force of friction combine to make the net outside force. If you push the book at a constant speed, then the force of your push exactly balances the force of friction, so there is no net outside force. If you stop pushing, there will be nothing to balance the effects of friction. Then the friction will be a net outside force and the book will slow to a stop.

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Newton’s first law tells us that if no net outside force acts on a moving object, it can only move in a straight line and at a constant speed. This means that a net outside force must be acting on the planets since they don’t move in straight lines but instead move around elliptical paths. Another way to see that planetary orbits require a net outside force is that a planet would fly off into space at a constant speed along a straight line if there were no net outside force acting on it. Because planets don’t fly off, Newton concluded that a force must act continuously on the planets to keep them in their elliptical orbits.

CONCEPT CHECK 4-15

Imagine a 10-kg rock speeding through empty space at 200 m/s, so far away from other objects that there is no gravitational force (or any other outside forces) exerted on the rock. Describe the rock’s motion.

Newton’s Second Law

Newton’s second law describes how the motion of an object changes if there is a net outside force acting on it. To appreciate Newton’s second law, we must first understand three quantities that describe motion—speed, velocity, and acceleration.

Speed is a measure of how fast an object is moving. Speed and direction of motion together constitute an object’s velocity. Compared with a car driving north at 100 km/h (62 mi/h), a car driving east at 100 km/h has the same speed but a different velocity. We can restate Newton’s first law to say that an object has a constant velocity (its speed and direction of motion do not change) if no net outside force acts on the object.

Acceleration is the rate at which velocity changes. Because velocity involves both speed and direction, acceleration can result from changes in either. Contrary to popular use of the term, acceleration does not simply mean speeding up. A car is accelerating if it is speeding up, and it is also accelerating if it is slowing down or turning (that is, changing the direction in which it is moving).

You can verify these statements about acceleration if you think about the sensations of riding in a car. If the car is moving with a constant velocity (in a straight line at a constant speed), you feel the same, aside from vibrations, as if the car were not moving at all. But you can feel it when the car accelerates in any way: You feel thrown back in your seat if the car speeds up, thrown forward if the car slows down, and thrown sideways if the car changes direction in a tight turn. In Box 4-3 we discuss the reasons for these sensations, along with other applications of Newton’s laws to everyday life.

An apple falling from a tree is a good example of acceleration that involves only an increase in speed. Initially, at the moment the stem breaks, the apple’s speed is zero. After 1 second, its downward speed is 9.8 meters per second, or 9.8 m/s (32 feet per second, or 32 ft/s). After 2 seconds, the apple’s speed is twice this, or 19.6 m/s. After 3 seconds, the speed is 29.4 m/s. Because the apple’s speed increases by 9.8 m/s for each second of free fall, the rate of acceleration is 9.8 meters per second per second, or 9.8 m/s2 (32 ft/s2). Thus, Earth’s gravity gives the apple a constant acceleration of 9.8 m/s2 downward, toward the center of Earth.

A planet revolving about the Sun along a perfectly circular orbit is an example of acceleration that involves change of direction only. As the planet moves along its orbit, its speed remains constant. Nevertheless, the planet is continuously being accelerated because its direction of motion is continuously changing.

Newton’s second law of motion says that in order to give an object an acceleration (that is, to change its velocity), a net outside force must act on the object. To be specific, this law says that the acceleration of an object is proportional to the net outside force acting on the object. That is, the harder you push on an object, the greater the resulting acceleration. This law can be succinctly stated as an equation. If a net outside force F acts on an object of mass m, the object will experience an acceleration a such that

Newton’s second law

F = ma

The mass of an object is a measure of the total amount of material in the object. It is usually expressed in kilograms (kg) or grams (g). For example, the mass of the Sun is 2 × 1030 kg, the mass of a hydrogen atom is 1.7 × 10-27 kg, and the mass of an average adult is 75 kg. The Sun, a hydrogen atom, and a person have these masses regardless of where they happen to be in the universe.

CAUTION!

It is important not to confuse the concepts of mass and weight. Weight is the force of gravity that acts on an object and, like any force, is usually expressed in pounds or newtons (1 newton = 0.225 pounds). For example, astronauts feel lighter on the Moon because they weigh less in the Moon’s weaker gravity, but an astronaut’s mass on the Moon is the same as her mass on Earth.

We can use Newton’s second law to relate mass and weight. We have seen that the acceleration caused by Earth’s gravity is 9.8 m/s2. When a 50-kg swimmer falls from a diving board, the only outside force acting on her as she falls is her weight. Thus, from Newton’s second law (F = ma), her weight is equal to her mass multiplied by the acceleration due to gravity:

50 kg × 9.8 m/s2 = 490 newtons = 110 pounds

Note that this answer is correct only when the swimmer is on Earth. She would weigh less on the Moon, where the pull of gravity is weaker, and more on Jupiter, where the gravitational pull is stronger. Floating deep in space, she would have no weight at all; she would be “weightless.” Nevertheless, in all these circumstances, she would always have exactly the same mass, because mass is an inherent property of matter unaffected by details of the environment. Whenever we describe the properties of planets, stars, or galaxies, we speak of their masses, never of their weights.

We have seen that a planet is continually accelerating as it orbits the Sun. From Newton’s second law, this means that there must be a net outside force that acts continually on each of the planets. As we will see in the next section, this force is the gravitational attraction of the Sun.

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Newton’s Third Law

The last of Newton’s general laws of motion is called Newton’s third law of motion:

Whenever one object exerts a force on a second object, the second object exerts an oppositely directed force of equal strength on the first object.

Newton’s third law is sometimes described in terms of actions and reactions: When two objects interact by exerting forces on each other, these action and reaction forces are equal in magnitude but opposite in direction.

For example, consider your weight; Earth pulls down on you with a gravitational force equal to your weight. But you also pull back up on Earth with a gravitational force of equal strength. In another example, consider a bat hitting a ball. Clearly, the bat exerts a large force on the ball when hitting a home run (this can be called an action force). However, the ball exerts a force of equal strength on the bat as well (this can be called a reaction force). It might seem totally counterintuitive that a small, passive ball could exert an equally large force on a bat that has someone’s full swing behind it. However, imagine a spring between the ball and bat during the hit; a spring is a very good approximation for the deformed ball. Regardless of which object moves in order to compress the spring, the spring pushes outward with an equal force on both ends (outward on the ball and bat).

CONCEPT CHECK 4-16

Two sumo wrestlers push against each other during a match. One wrestler is much larger than the other. The larger wrestler’s feet remain on the floor while the smaller wrestler’s feet slip as he is accelerated in a push right out of the ring. Compare the force that the larger wrestler exerts on the smaller wrestler to the force the smaller wrestler exerts on the larger wrestler.

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CONCEPT CHECK 4-17

In midair after stepping off a diving board, a diver is pulled down to the water by her weight. However, the diver pulls up on Earth with a force of equal strength, so why doesn’t Earth move an equal amount as the diver?

In these examples, you can think of each force in these pairs as a reaction to the other force, which is the origin of the phrase “action and reaction.”

ASTRONOMY DOWN TO EARTH

Newton’s Laws in Everyday Life

In our study of astronomy, we use Newton’s three laws of motion to help us understand the motions of objects in the heavens. But you can see applications of Newton’s laws every day in the world around you. By considering these everyday applications, we can gain insight into how Newton’s laws apply to celestial events that are far removed from ordinary human experience.

Newton’s first law, or principle of inertia, says that an object at rest naturally tends to remain at rest and that an object in motion naturally tends to remain in motion. This law explains the sensations that you feel when riding in an automobile. When you are waiting at a red light, your car and your body are both at rest. When the light turns green and you press on the gas pedal, the car accelerates forward but your body attempts to stay where it was. Hence, the seat of the accelerating car pushes forward into your body, and it feels as though you are being pushed back in your seat.

Once the car is up to cruising speed, your body wants to keep moving in a straight line at this cruising speed. If the car makes a sharp turn to the left, the right side of the car will move toward you. Thus, you will feel as though you are being thrown to the car’s right side (the side on the outside of the turn). If you bring the car to a sudden stop by pressing on the brakes, your body will continue moving forward until the seat belt stops you. In this case, it feels as though you are being thrown toward the front of the car.

Newton’s second law states that the net outside force on an object equals the product of the object’s mass and its acceleration. You can accelerate a crumpled-up piece of paper to a pretty good speed by throwing it with a moderate force. But if you try to throw a heavy rock by using the same force, the acceleration will be much less because the rock has much more mass than the crumpled paper. Because of the smaller acceleration, the rock will leave your hand moving at only a slow speed.

Automobile airbags are based on the relationship between force and acceleration. It takes a large force to bring a fast-moving object suddenly to rest because this requires a large acceleration. In a collision, the driver of a car not equipped with airbags is jerked to a sudden stop and the large forces that act can cause major injuries. But if the car has airbags that deploy in an accident, the driver’s body will slow down more gradually as it contacts the airbag, and the driver’s acceleration will be less. (Remember that acceleration can refer to slowing down as well as to speeding up.) Hence, the force on the driver and the chance of injury will both be greatly reduced.

Newton’s third law, the principle of action and reaction, explains how a car can accelerate at all. It is not correct to say that the engine pushes the car forward, because Newton’s second law tells us that it takes a force acting from outside the car to make the car accelerate. Rather, the engine makes the wheels and tires turn, and the tires push backward on the ground. (You can see this backward force in action when a car drives through wet ground and sprays mud backward from the tires.) From Newton’s third law, the ground must exert an equally large forward force on the car, and this is the force that pushes the car forward.

You use the same principles when you walk: You push backward on the ground with your foot, and the ground pushes forward on you. Icy pavement or a freshly waxed floor have greatly reduced friction. In these situations, your feet and the surface under you can exert only weak forces on each other, and it is much harder to walk.

Newton realized that because the Sun is exerting a force on each planet to keep it in orbit, each planet must also be exerting an equal and opposite force on the Sun. However, the planets are much less massive than the Sun (for example, Earth has only 1/300,000 of the Sun’s mass). Therefore, although the Sun’s force on a planet is the same as the planet’s force on the Sun, the planet’s much smaller mass gives it a much larger acceleration, according to Newton’s second law. This is why the planets circle the Sun instead of vice versa. Thus, Newton’s laws reveal the reason for our heliocentric solar system.

The Law of Universal Gravitation

Newton’s law of gravitation is truly universal: It applies to falling apples as well as to planets and galaxies

Tie a ball to one end of a piece of string, hold the other end of the string in your hand, and whirl the ball around in a circle. As the ball “orbits” your hand, it is continuously accelerating because its velocity is changing. (Even if its speed is constant, its direction of motion is changing.) In accordance with Newton’s second law, this can happen only if the ball is continuously acted on by an outside force—the pull of the string. The pull is directed along the string toward your hand. In the same way, Newton saw, the force that keeps a planet in orbit around the Sun is a pull that always acts toward the Sun. That pull is gravity, or gravitational force.

Newton’s discovery about the forces that act on planets led him to suspect that the force of gravity pulling a falling apple straight down to the ground is fundamentally the same as the force on a planet that is always directed straight at the Sun. In other words, gravity is the force that shapes the orbits of the planets. What is more, he was able to determine how the force of gravity depends on the distance between the Sun and the planet. His result was a law of gravitation that could apply to the motion of distant planets as well as to the flight of a football on Earth. Using this law, Newton achieved the remarkable goal of deducing Kepler’s laws from fundamental principles of nature.

To see how Newton reasoned, think again about a ball attached to a string. If you use a short string, so that the ball orbits in a small circle, and whirl the ball around your hand at a high speed, you will find that you have to pull fairly hard on the string (Figure 4-19a). But if you use a longer string, so that the ball moves in a larger orbit, and if you make the ball orbit your hand at a slow speed, you only have to exert a light tug on the string (Figure 4-19b). The orbits of the planets behave in the same way: The larger the size of the orbit, the slower the planet’s speed (Figure 4-19c, d). By analogy to the force of the string on the orbiting ball, Newton concluded that the force that attracts a planet toward the Sun must decrease with increasing distance between the Sun and the planet.

Figure 4-19: An Orbit Analogy (a) To make a ball on a string move at high speed around a small circle, you have to exert a substantial pull on the string. (b) If you lengthen the string and make the same ball move at low speed around a large circle, much less pull is required. (c) Similarly, a planet that orbits close to the Sun moves at high speed and requires a substantial gravitational force from the Sun, while (d) a planet in a large orbit moves at low speed and requires less gravitational force to stay in orbit.

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Using his own three laws and Kepler’s three laws, Newton succeeded in formulating a general statement that describes the nature of the gravitational force. Newton’s law of universal gravitation is as follows:

Two objects attract each other with a force that is directly proportional to the mass of each object and inversely proportional to the square of the distance between them.

This law states that any two objects exert gravitational pulls on each other. Normally, you notice only the gravitational force that Earth exerts on you, otherwise known as your weight. In fact, you are gravitationally attracted to all the objects around you. For example, a book exerts a gravitational force on you as you read it. But because the force exerted on you by this book is proportional to the book’s mass, which is very small compared to Earth’s mass, the force is too small to notice. (It can actually be measured with sensitive equipment.)

The farther apart two objects are, the weaker the gravitational force between them. Since the gravitational force weakens by the square of the distance between two objects, doubling their distance reduces their attraction by

and tripling their distance reduces their attraction by

Newton’s law of universal gravitation can be stated as an equation. If two objects have masses m1 and m2 and are separated by a distance r, then the gravitational force F between these two objects is given by the following equation:

Newton’s law of universal gravitation

If the masses are measured in kilograms and the distance between them in meters, then the force is measured in newtons. In this formula, G is a number called the universal constant of gravitation. Laboratory experiments have yielded a value for G of

G = 6.67 × 10−11 newton · m2/kg2

We can use Newton’s law of universal gravitation to calculate the force with which any two objects attract each other. For example, to compute the gravitational force that the Sun exerts on Earth, we substitute values for Earth’s mass (m1 = 5.98 × 1024 kg), the Sun’s mass (m2 = 1.99 × 1030 kg), the distance between them (r = 1 AU = 1.5 × 1011 m), and the value of G into Newton’s equation. We get

If we calculate the force that Earth exerts on the Sun, we get exactly the same result. (Mathematically, we just let m1 be the Sun’s mass and m2 be Earth’s mass instead of the other way around. The product of the two numbers is the same, so the force is the same.) This is in accordance with Newton’s third law: Any two objects exert equal gravitational forces on each other.

Your weight is just the gravitational force that Earth exerts on you, so we can calculate it using Newton’s law of universal gravitation. Earth’s mass is m1 = 5.98 × 1024 kg, and the distance r to use is the distance between the centers of Earth and you. This distance is just the radius of Earth, which is r = 6378 km = 6.378 × 106 m. If your mass is m2 = 50 kg, your weight is

This value is the same as the weight of a 50-kg person that we calculated in Section 4-6. This example shows that your weight would have a different value on a planet with a different mass m1 and a different radius r.

CONCEPT CHECK 4-18

How much does the gravitational force of attraction change between two asteroids if the two asteroids drift 3 times closer together?

CALCULATION CHECK 4-2

How much would a 75-kg astronaut, weighing about 165 pounds on Earth, weigh in newtons and in pounds if he were standing on Mars, which has a mass of 6.4 × 1023 kg and a radius of 3.4 × 106 m?