It's a Great Day for Physics: Newton's Law

Next to E = mc², F = ma is the most famous equation in physics. Yet many people remain perplexed by this relatively simple algebraic expression. It's actually a mathematical representation of Sir Isaac Newton's second law of motion, one of the great scientist's most important contributions. The "second" implies that other laws exist, and, luckily for students and trivia hounds everywhere, there are only two additional laws of motion. All three are presented here:

1. Every object persists in its state of rest or uniform motion­ in a straight line unless it is compelled to change that state by forces impressed on it.
2. Force is equal to the change in momentum per change in time. For a constant mass, force equals mass times acceleration.
3. For every action, there is an equal and opposite reaction.

A.    The First Law

Let's restate Newton's first law in everyday terms:

An object at rest will stay at rest, forever, as long as nothing pushes or pulls on it. An object in motion will stay in motion, traveling in a straight line, forever, until something pushes or pulls on it.

The "forever" part is difficult to swallow sometimes. But imagine that you have three ramps. Also imagine that the ramps are never-endingly long and smooth. You let a marble roll down the first slope, which is set at a slight incline. The marble speeds up on its way down the ramp. Now, you give a gentle push to the marble going uphill on the second ramp. It slows down as it goes up. Finally, you push a marble on a ramp that represents the middle state ­between the first two -- in other words, a ramp that is perfectly horizontal. In this case, the marble will neither slow down nor speed up. In fact, it should keep rolling. Forever.

According to Newton's first law, the marble on that bottom ramp should just keep going. And going.

Physicists use the term inertia to describe this tendency of an object to resist a change in its motion. The Latin root for inertia is the same root for "inert," which means lacking the ability to move. So you can see how scientists came up with the word. What's more amazing is that they came up with the concept. Inertia isn't an immediately apparent physical property, such as length or volume. It is, however, related to an object's mass. To understand how, consider the sumo wrestler and the boy shown below.

Which person in this ring will be harder to move? The sumo wrestler or the little boy?

Let's say the wrestler on the left has a mass of 200 kilograms, and the boy on the right has a mass of 30 kilograms. Remember the object of sumo wrestling is to move your opponent from his position. Which person in our example would be easier to move? Common sense tells you that the boy would be easier to move.

You experience inertia in a moving car all the time. In fact, seatbelts exist in cars specifically to counteract the effects of inertia. Imagine for a moment that a car at a test track is traveling at a speed of 40 km/hour. Now imagine that a crash test dummy is inside that car, riding in the front seat. If the car slams into a wall, the dummy flies forward into the dashboard. Why? Because, according to Newton's first law, an object in motion will remain in motion until an outside force acts on it. When the car hits the wall, the dummy keeps moving in a straight line and at a constant speed until the dashboard applies a force. Seatbelts hold dummies (and passengers) down, protecting them from their own inertia.

B.     The Second Law

You may be surprised to learn that Newton wasn't the genius behind the law of inertia. But Newton himself wrote that he was able to see so far only because he stood on "the shoulders of Giants." And see far he did. Although the law of inertia identified forces as the actions required to stop or start motion, it didn't quantify those forces. Newton's second law supplied the missing link by relating force to acceleration.

When a force acts on an object, the object accelerates in the direction of the force. If the mass of an object is held constant, increasing force will increase acceleration. If the force on an object remains constant, increasing mass will decrease acceleration. In other words, force and acceleration are directly proportional, while mass and acceleration are inversely proportional.

Technically, Newton equated force to the differential change in momentum per unit time. Momentum is a characteristic of a moving body determined by the product of the body's mass and velocity. To determine the differential change in momentum per unit time, Newton developed a new type of math -- differential calculus. His original equation looked something like this:

F = (m)(Δv/Δt)

where the delta symbols signify change. Because acceleration is defined as the instantaneous change in velocity in an instant of time (Δv/Δt), the equation is often rewritten as:

F = ma

The equation form of Newton's second law allows us to specify a unit of measurement for force. Because the standard unit of mass is the kilogram (kg) and the standard unit of acceleration is meters per second squared (m/s²), the unit for force must be a product of the two -- (kg)(m/s²). This is a little awkward, so scientists decided to use a Newton as the official unit of force. One Newton, or N, is equivalent to 1 kilogram-meter per second squared.

So what can you do with Newton's second law? As it turns out, F = ma lets you quantify motion of every variety. Let's say, for example, you want to calculate the acceleration of the dog sled shown below.

If you want to calculate the acceleration, first you need to modify the force equation to get a = F/m. When you plug in the numbers for force (100 N) and mass (50 kg), you find that the acceleration is 2 m/s².

Now let's say that the mass of the sled stays at 50 kg and that another dog is added to the team. If we assume the second dog pulls with the same force as the first (100 N), the total force would be 200 N and the acceleration would be 4 m/s².

           

Finally, let's imagine that a second dog team is attached to the sled so that it can pull in the opposite direction.

If two dogs are on each side, then the total force pulling to the left (200 N) balances the total force pulling to the right (200 N). That means the net force on the sled is zero, so the sled doesn’t move.

This is important because Newton's second law is concerned with net forces. We could rewrite the law to say: When a net force acts on an object, the object accelerates in the direction of the net force. Now imagine that one of the dogs on the left breaks free and runs away. Suddenly, the force pulling to the right is larger than the f­orce pulling to the left, so the sled accelerates to the right.

What's not so obvious in our examples is that the sled is also applying a force on the dogs. In other words, all forces act in pairs. This is Newton's third law.

C.     The Third Law

Newton's third law is probably the most familiar. Everyone knows that every action has an equal and opposite reaction, right? Unfortunately, this statement lacks some necessary detail. This is a better way to say it:

A force is exerted by one object on another object. In other words, every force involves the interaction of two objects. When one object exerts a force on a second object, the second object also exerts a force on the first object. The two forces are equal in strength and oriented in opposite directions.

Many people have trouble visualizing this law because it's not as intuitive. In fact, the best way to discuss the law of force pairs is by presenting examples. Let's start by considering a swimmer facing the wall of a pool. If she places her feet on the wall and pushes hard, what happens? She shoots backward, away from the wall.

Clearly, the swimmer is applying a force to the wall, but her motion indicates that a force is being applied to her, too. This force comes from the wall, and it's equal in magnitude and opposite in direction.

Next, think about a book lying on a table. What forces are acting on it? One big force is Earth's gravity. In fact, the book's weight is a measurement of Earth's gravitational attraction. So, if we say the book weighs 10 N, what we're really saying is that Earth is applying a force of 10 N on the book. The force is directed straight down, toward the center of the planet. Despite this force, the book remains motionless, which can only mean one thing: There must be another force, equal to 10 N, pushing upward. That force is coming from the table.

If you're catching on to Newton's third law, you should have noticed another force pair described in the paragraph above. Earth is applying a force on the book, so the book must be applying a force on Earth. Is that possible? Yes, it is, but the book is so small that it cannot appreciably accelerate something as large as a planet.

You see something similar, although on a much smaller scale, when a baseball bat strikes a ball. There's no doubt the bat applies a force to the ball: It accelerates rapidly after being struck. But the ball must also be applying a force to the bat. The mass of the ball, however, is small compared to the mass of the bat, which includes the batter attached to the end of it. Still, if you've ever seen a wooden baseball bat break into pieces as it strikes a ball, then you've seen firsthand evidence of the ball's force.

These examples don't show a practical application of Newton's third law. Is there a way to put force pairs to good use? Jet propulsion is one application. Used by animals such as squid and octopi, as well as by certain airplanes and rockets, jet propulsion involves forcing a substance through an opening at high speed. In squid and octopi, the substance is seawater, which is sucked in through the mantle and ejected through a siphon. Because the animal exerts a force on the water jet, the water jet exerts a force on the animal, causing it to move. A similar principle is at work in turbine-equipped jet planes and rockets in space.