3.4 Newton’s Three Laws Help Explain How Celestial Bodies Move
Empirical laws, such as Kepler’s laws, describe what happens, but they do not explain why. Kepler described the orbits of planets as ellipses, but he did not explain why they should be so. To take that next step in the scientific process, scientists use basic physical principles and the tools of mathematics to derive the empirically determined laws. Alternatively, a scientist might start with physical laws and predict relationships, which are then verified or falsified through experiment and observation. If those predictions are verified, the scientist may have determined something fundamental about how the universe works.
Sir Isaac Newton (1642–1727—Figure 3.20) took that next step in explaining the nature of motion. Newton was a student of mathematics at Cambridge University when it closed because of the Great Plague and students were sent home to the safer countryside. Over the next 2 years, he studied on his own, and at the age of 23 he invented calculus, which would become crucial to his development of the physics of motion. (The German mathematician Gottfried Leibniz independently developed calculus around the same time.)
Figure 3.20 Sir Isaac Newton formulated three laws of motion.
unanswered questions
Would the history of scientific discoveries in physics and astronomy have been different if the Inquisition had not prosecuted Galileo? Galileo wrote the Dialogo after the Catholic Church in 1616 ordered him not to “hold or defend” the idea that Earth moves and the Sun is still. And he wrote his equally famous Discorsi e Dimostrazioni Matematiche (often shortened in English to “Two New Sciences”) while under house arrest after his trial. However undeterred Galileo appeared to be, the effects of the decrees, prohibitions, and prosecutions might have dissuaded other scientists in Catholic countries from pursuing that type of work. Indeed, after Galileo’s experiences, the center of the scientific revolution moved north to Protestant Europe.
Building on the work of Kepler, Galileo, and others, Newton proposed three physical laws that govern the motions of all objects in the sky and on Earth. To understand how the planets and all other celestial bodies move, you must understand these three laws.
Newton’s First Law: Objects at Rest Stay at Rest; Objects in Motion Stay in Motion
A force (F) is a push or a pull on an object. Two or more forces can oppose one another such that they are perfectly balanced and cancel out. For example, gravity pulls down on you as you sit in your chair, but the chair pushes up on you with an exactly equal and opposite force. As a result, you stay motionless. Forces that cancel out do not affect an object’s motion. When forces combine to produce an effect, we often use the term net force, or sometimes just force.
Imagine that you are driving a car, and your book is on the seat next to you. A rabbit runs across the road in front of you, so you hit the brakes hard. You feel the seat belt tighten to restrain you. At the same time, your book flies off the seat and hits the dashboard. You have just experienced what Newton describes in his first law of motion. Inertia is an object’s tendency to maintain its state—either of uniform motion or of rest—until a net force pushes or pulls it. In the stopping car, you did not hit the dashboard because the force of the seat belt slowed you down. The book, however, hit the dashboard because no such force acted upon it.
Newton’s first law of motion describes inertia and states that an object in motion tends to stay in motion, in the same direction, until a net force acts upon it; and an object at rest tends to stay at rest until a net force acts upon it. Galileo’s law of inertia became the cornerstone of physics as Newton’s first law.
Recall from Section 2.1 the concept of a frame of reference. Within a frame of reference, only the relative motions between objects have any meaning. Without external clues, you cannot tell the difference between sitting still and traveling at constant speed in a straight line. For example, if you close your eyes while riding in the passenger seat of a quiet car on a smooth road, you feel as though you are sitting still. For you, the book on the seat beside you was “at rest,” whereas a person standing by the side of the road would see the book moving past at the same speed as you and the car. People in a car approaching you would see the book moving even faster than the person by the side of the road—namely, at the speed they are traveling plus the speed you are traveling! All those perspectives are equally valid, and all those speeds of the book are correct when measured in the appropriate reference frame.
A reference frame moving in a straight line at a constant speed is called an inertial frame of reference. Any inertial frame of reference is as good as another. In the inertial frame of reference of a cup of coffee, for example, Figure 3.21a shows the cup is at rest in its own frame even if the car is moving quickly down the road.
Newton’s Second Law: Motion Is Changed by Forces
What happens if a net force does act? In the earlier example, you were traveling in the car, and your motion slowed when the force of the seat belt acted upon you. Forces change an object’s motion—by changing either the speed or the direction. That effect reflects Newton’s second law of motion: if a net force acts on an object, the object’s motion changes.
In the driver’s seat of a car, you have several controls, including an accelerator and a brake pedal, which you use to make the car speed up or slow down. A change in speed is one way the car’s motion can change. But you also have the steering wheel in your hands. When you are moving down the road and you turn the wheel, your speed does not necessarily change, but the direction of your motion does. A change in direction also is a kind of change in motion.
Together, an object’s speed and direction are called velocity (v). “Traveling at 50 kilometers per hour (km/h)” indicates speed, whereas “traveling north at 50 km/h” indicates velocity. Acceleration (a) describes the changes in an object’s velocity. For example, if you go from 0 to 100 km/h in 4 seconds, you feel a strong push from the seat back as it shoves your body forward, causing you to accelerate along with the car. However, if you take 2 minutes to go from 0 to 100 km/h, the acceleration is so slight that you hardly notice it.
Partly because a car’s gas pedal is often called the accelerator, some people think acceleration always means that an object is speeding up. In physics, however, any change in speed or direction is an acceleration. Figure 3.21b illustrates that point by showing what happens to the coffee in a cup as the car speeds up, slows down, or turns. Slamming on your brakes and going from 100 to 0 km/h in 4 seconds is just as much an acceleration as going from 0 to 100 km/h in 4 seconds. Similarly, the acceleration you experience as you go through a tight turn at a constant speed is every bit as real as the acceleration you feel when you slam your foot on the accelerator or the brake. Whether speeding up, slowing down, turning left, or turning right—if you are not moving in a straight line at a constant speed, you are accelerating.
Figure 3.21a. An object moving in a straight line at a constant speed is at rest in its own inertial frame of reference. b. Any change in an object’s velocity is an acceleration. When you are driving, for example, any time your speed changes or you follow a curve in the road, you are accelerating. (Throughout the text, velocity arrows are red and acceleration arrows are green.)
Newton’s second law of motion says that a net force causes acceleration. An object’s acceleration depends on two things. First, as shown in Figure 3.22, the acceleration depends on the strength of the net force acting on the object to change its motion. If the forces acting on the object do not add up to zero, a net force is present and the object accelerates (Figure 3.22a). The stronger the net force, the greater the acceleration (Figure 3.22b). If you push on something twice as hard, it experiences twice as much acceleration. Push on something three times as hard and its acceleration will be three times as great. The acceleration occurs in the direction the net force points. Push an object away from you, and it will accelerate away from you.
Figure 3.22 According to Newton’s second law of motion, an object’s acceleration is the force acting on the object divided by the object’s mass. (Throughout the text, force arrows are blue.)
An object’s acceleration also depends on its inertia. You can push some objects easily, such as an empty box from a new refrigerator. But you can’t easily shove an actual refrigerator around, even though it is about the same size as the box. Figure 3.22c shows that the greater the mass, the greater the inertia, and the less acceleration that will occur in response to the same net force. That relationship among acceleration (a), force (F), and mass (m) is expressed mathematically in Working It Out 3.3.
working it out 3.3
Using Newton’s Laws
Your acceleration is calculated by dividing how much your velocity changes by how long that change takes to happen:
For example, if an object’s speed goes from 5 to 15 meters per second (m/s), the change in velocity is 10 m/s. If that change happens in 2 seconds, the acceleration is given by
To determine how an object’s motion is changing, we need to know two things: what net force is acting on the object and the object’s resistance to that force. We can put that idea into equation form as follows:
Newton’s second law above is often written as Force = mass × acceleration, or F=ma. The units of force are called newtons (N), so 1 N = 1 kg m/s2.
Suppose you are holding two blocks of the same size, but the block in your right hand has twice the mass of the block in your left hand. When you drop the blocks, they both fall with the same acceleration, as Galileo showed, and they hit your two feet at the same time. Which will hit with more force: the block falling onto your right foot or the one falling onto your left foot? The block in your right hand, with twice the mass, will hit your right foot with twice the force that the other block hits your left foot.
To see how Newton’s three laws of motion work together, study Figure 3.24. An astronaut is adrift in space, motionless with respect to the nearby space station. With no tether to pull on, how can the astronaut get back to the station? Suppose the 100-kg astronaut throws a 1-kg wrench directly away from the station at a speed of 10 m/s. Newton’s second law says that to change the wrench’s motion, the astronaut must apply a force to it in the direction away from the station. Newton’s third law says that the wrench must therefore push back on the astronaut with as much force but in the opposite direction. The force of the wrench on the astronaut causes the astronaut to begin drifting toward the station. How fast will the astronaut move? Turn to Newton’s second law again. Because the astronaut has more mass, he or she will accelerate less than the wrench will. A force that causes the 1-kg wrench to accelerate to 10 m/s will have much less effect on the 100-kg astronaut. Because acceleration equals force divided by mass, the 100-kg astronaut will experience only 1/100 as much acceleration as the 1-kg wrench. The astronaut will drift toward the station, but only at the leisurely rate of 1/100 × 10 m/s, or 0.1 m/s.
Figure 3.24 According to Newton’s laws, if an astronaut adrift in space throws a wrench, the two will move in opposite directions. Their speeds will depend on their masses: the same force will produce a smaller acceleration of a more massive object than of a less massive object. (Acceleration and velocity arrows not drawn to scale.)
Newton’s Third Law: Whatever Gets Pushed Pushes Back
Imagine that you are standing on a skateboard and pushing yourself along with your foot. Each shove of your foot against the ground sends you faster along your way. But why? You accelerate because as you push on the ground, the ground pushes back on you.
Part of Newton’s genius was his ability to see patterns in such everyday events. Newton realized that every time one object exerts a force on another, the second object exerts a matching force on the first. That second force is as strong as the first force but is in the opposite direction. When you accelerate yourself on the skateboard, you push backward on Earth, and Earth pushes you forward. As shown in Figure 3.23, when a woman moves a load on a cart by pulling a rope, the rope pulls back, and when a car tire pushes back on the road, the road pushes forward on the tire. Similarly, when Earth pulls on the Moon, the Moon pulls on Earth, and when a rocket engine pushes hot gases out of its nozzle, those hot gases push back on the rocket, propelling it into space.
Figure 3.23 Newton’s third law states that for every force, an equal and opposite force is always present. Those opposing forces always act on the two objects in the same pair.
what if . . .
What if you are designing a rocket ship intending to reach Mars? For a given mass of fuel to be burned and ejected from the tail of the rocket, should you look for fuel with higher or lower ejection velocity? Are you better off burning the fuel early in the journey, late in the journey, or does it matter?
All those force pairs exemplify Newton’s third law of motion, which says forces always come in pairs, with those two forces always equal in strength but opposite in direction. The forces in those pairs always act on two objects. Your weight pushes down on the floor, and the floor pushes back up on you with the same amount of force. For every force, an equal force in the opposite direction is always there.
CHECK YOUR UNDERSTANDING 3.4
If a planet moves in a perfectly circular orbit around the Sun, is that planet accelerating? (a) Yes, because it is constantly changing its speed. (b) Yes, because it is constantly changing its direction. (c) No, because its speed is not constantly changing. (d) No, because planets do not accelerate.
The law, formulated by Isaac Newton, stating that an object will remain at rest or will continue moving along a straight line at a constant speed until an unbalanced force acts on it.
1. A frame of reference moving in a straight line at constant speed, that is, not accelerating. 2. In general relativity, a frame of reference falling freely in a gravitational field.
The law that Isaac Newton formulated, stating that if an unbalanced force acts on a body, the body will accelerate in proportion to the unbalanced force and in inverse proportion to the object’s mass: a5F/m. The acceleration will be in the direction of the unbalanced force.
The rate and direction of change of an object’s position with time. Possible units include meters per second (m/s) and kilometers per hour (km/h). Compare speed.
Answer