Understanding how Newton's Laws of Motion operate

Along with E = mc, F = ma stands as one of the most iconic equations in physics. However, its meaning often confuses many. This simple algebraic equation is actually a reflection of Isaac Newton's second law of motion, which remains one of his most significant discoveries. The word 'second' suggests that there are additional laws, and fortunately for students and trivia enthusiasts, there are just two more. Here they are:

These three laws lay the groundwork for classical mechanics, the branch of science that deals with the motion of objects and the forces that influence them. These objects can range from massive celestial bodies like moons and planets to everyday items on Earth's surface, such as moving cars or speeding projectiles. Even stationary objects are included in the scope of these laws.

The limitations of classical mechanics become apparent when it attempts to describe the motion of incredibly small particles, like electrons. To address this, physicists developed a new framework called quantum mechanics, which is used to explain the behavior of objects at the atomic and subatomic scale.

Quantum mechanics, however, is not the focus of this article. Here, we will concentrate on classical mechanics and Newton's three laws. We will delve into each law in detail from both a theoretical and practical perspective. Additionally, we'll explore the history behind Newton's laws, since understanding how Newton arrived at his conclusions is just as vital as the laws themselves. Naturally, we'll begin with Newton's first law.

Newton's First Law (Law of Inertia)

Let's rephrase Newton's first law using simpler language:

The idea of 'forever' can be hard to grasp. But let's picture three ramps set up like the one below. Imagine these ramps are infinitely long and perfectly smooth. You release a marble down the first ramp, which is slightly inclined. As the marble rolls down, it gains speed.

Next, you give a gentle push to a marble rolling uphill on the second ramp. As expected, it slows down as it climbs. Finally, you push a marble on a ramp that's perfectly horizontal, representing a middle ground between the first two ramps. In this case, the marble will neither accelerate nor decelerate. In fact, it should keep rolling. Forever.

Physicists refer to the property of an object resisting a change in its motion as inertia. The word 'inertia' comes from the same Latin root as 'inert,' meaning unable to move. This origin helps explain how the term came about, but what's truly fascinating is the concept itself. Inertia isn't an obvious property like size or shape. Instead, it's connected to an object's mass. To illustrate this, let's consider the sumo wrestler and the young boy shown below.

Suppose the wrestler on the left weighs 136 kilograms, while the boy on the right weighs 30 kilograms (mass is measured in kilograms). The goal in sumo wrestling is to move your opponent. So, which person in this scenario would be easier to move? Common sense tells us that the boy, with his smaller mass, would be less resistant to inertia and thus easier to move.

You feel inertia every time you're in a moving car. In fact, seat belts are specifically designed to combat inertia's effects. Imagine a test track where a car is speeding along at 55 mph (80 kph). Picture a crash test dummy sitting in the front seat. If the car crashes into a wall, the dummy will continue moving forward, slamming into the dashboard, thanks to inertia.

Why? Because, as stated in Newton's first law, an object in motion will continue moving until an external force intervenes. When the car crashes into the wall, the dummy keeps traveling in a straight line at the same speed until the dashboard applies a force. Seatbelts are there to restrain dummies (and passengers), protecting them from their own inertia.

Interestingly, Newton wasn't the first to formulate the law of inertia. That credit goes to Galileo and René Descartes. In fact, Galileo is credited with the marble-and-ramp thought experiment we discussed earlier. Newton built upon the work of those who came before him. Before delving into his other two laws, let's take a look at some of the key historical developments that influenced them.

A Brief History of Newton's Laws

For centuries, the Greek philosopher Aristotle's views dominated scientific thinking. His theories on motion were widely accepted because they appeared to align with what people observed in nature. For example, Aristotle believed that the weight of an object influenced how fast it fell, arguing that a heavier object would hit the ground more quickly than a lighter one dropped simultaneously from the same height. He also denied the idea of inertia, claiming instead that a force had to be applied continuously to keep an object in motion. While both of these ideas were wrong, it took many years and bold thinkers to prove them incorrect.

The first major challenge to Aristotle's theories came in the 16th century, when Nicolaus Copernicus introduced his heliocentric model of the universe. Aristotle had theorized that the sun, the moon, and the planets all revolved around Earth on fixed celestial spheres. In contrast, Copernicus proposed that the planets orbited the sun, not the Earth. Although his heliocentric theory wasn't directly related to mechanics, it exposed the flaws in Aristotle's scientific ideas.

Galileo Galilei was the next to challenge Aristotle's views. Galileo conducted two famous experiments that became foundational to modern science. In his first experiment, he dropped a cannonball and a musket ball from the Leaning Tower of Pisa. According to Aristotelian theory, the heavier cannonball should fall faster and land first. However, Galileo discovered that both objects hit the ground at the same time, proving Aristotle's theory wrong.

Some historians question whether Galileo actually performed the Pisa experiment, but his subsequent work is well-documented. In these later experiments, he used bronze balls of different sizes rolling down a slanted wooden plane. Galileo measured the distance each ball traveled in one-second intervals. He found that the size of the ball didn't affect its rate of descent — the rate of acceleration was constant. From this, he concluded that all freely falling objects experience uniform acceleration, regardless of mass, as long as external forces like air resistance and friction are minimized.

However, it was René Descartes, the renowned French philosopher, who deepened our understanding of inertial motion. In his "Principles of Philosophy," Descartes laid out three laws of nature. The first law asserts that everything, as long as it can, remains in the same state; and once it is in motion, it continues to move. The second law states that all motion occurs along straight lines. This idea was essentially Newton's first law, articulated in a 1644 publication — when Newton was still an infant!

It's clear that Isaac Newton studied the work of Descartes carefully. He used this knowledge to single-handedly usher in the modern era of scientific thought. Newton's contributions to mathematics led to the development of integral and differential calculus. His work in optics resulted in the invention of the first reflecting telescope. Yet, his most renowned legacy comes in the form of three relatively simple laws that, with remarkable predictive power, can describe the motion of objects both on Earth and throughout the cosmos. The first law was inspired by Descartes, but the other two were uniquely Newton's.

Newton outlined all three laws in his groundbreaking work, "The Mathematical Principles of Natural Philosophy," also known as the Principia, which was published in 1687. To this day, the Principia stands as one of the most influential books in human history. A large part of its significance lies in the elegantly simple second law, F = ma, which we will explore in the next section.

Newton's Second Law (Law of Motion)

You might be surprised to discover that Newton wasn't the original mind behind the law of inertia. However, Newton himself wrote that his ability to see further was due to standing on the "shoulders of Giants." And indeed, he saw far. While the law of inertia identified forces as what was needed to initiate or halt motion, it didn't specify the magnitude of those forces. Newton's second law provided the missing link by connecting force to acceleration. Here is what it stated:

Technically, Newton described force as the rate of change of momentum over time. Momentum is a property of a moving object, determined by multiplying its mass by its velocity. To calculate the rate of change of momentum per unit time, Newton introduced a new form of mathematics — differential calculus. His original equation looked something like this:

F = (m)(Δv/Δt)

In this equation, the delta symbols represent change. Since acceleration is defined as the instantaneous change in velocity over a brief period (Δv/Δt), the equation is often rewritten as:

F = ma

In Newton's equation, the F stands for force, which is a push or pull acting on an object. The m represents mass, indicating the amount of matter within an object. The a symbolizes acceleration, showing how an object's velocity changes over time. Velocity, similar to speed, is defined as the distance an object covers within a given time frame.

The formula form of Newton's second law helps define the unit of force. Since mass is measured in kilograms (kg) and acceleration in meters per second squared (m/s²), the unit of force must be the product of these — (kg)(m/s). This unit, though, can be awkward, so scientists chose to name it a Newton (N). One Newton equals 1 kilogram meter per second squared. To give you perspective, 1 pound is roughly 4.448 N.

What can you achieve with Newton's second law? As it turns out, F = ma enables you to measure the motion of anything. For instance, let's say you want to calculate the acceleration of the dog sled shown on the left.

Now, if we assume the sled's mass remains at 50 kilograms and add another dog to the team, with each dog exerting a force of 100 N, the total force would be 200 N. This would result in an acceleration of 4 m/s. However, if the sled's mass were doubled to 100 kilograms, the acceleration would be reduced to 2 m/s.

Finally, imagine attaching another team of dogs to the sled, pulling in the opposite direction.

This is crucial because Newton's second law is focused on net forces. We can restate the law as follows: When a net force acts on an object, the object accelerates in the direction of that net force.

Now, imagine one of the dogs on the left breaks free and runs off. Suddenly, the force pulling to the right becomes stronger than the force pulling to the left, and the sled begins to accelerate to the right.

What isn't immediately clear in our examples is that the sled also exerts a force on the dogs. In other words, all forces occur in pairs. This concept is described by Newton's third law, which we will explore in the next section.

Newton's Third Law (Law of Force Pairs)

Newton's third law is probably the most well-known. We all know the saying, 'Every action has an equal and opposite reaction.' However, this simple version misses some important details. Here’s a more accurate way to express it:

A lot of people find this law challenging to visualize, as it's not always intuitive. In fact, the best way to illustrate the law of force pairs is with examples. Let’s begin with a swimmer facing the wall of a pool. If she presses her feet against the wall and pushes hard, what happens? She moves backward, away from the wall.

It’s clear that the swimmer is applying a force to the wall, but her movement shows that a force is also being applied to her. This force comes from the wall, and it's equal in magnitude and opposite in direction.

Now, consider a book resting on a table. What forces are acting on it? One significant force is Earth's gravity. The weight of the book is essentially a measure of Earth's gravitational pull. So, if the book weighs 10 N, we are essentially saying that Earth is exerting a force of 10 N on the book. This force points straight down toward the Earth's center. Despite this downward force, the book remains still. This can only mean that another force, equal in magnitude (10 N), must be pushing upwards. That force is coming from the table.

If you're following along with Newton's third law, you may have noticed another force pair in the previous example. Since Earth exerts a force on the book, the book must also be exerting a force on Earth. Can this really happen? Yes, but since the book is so small, it can't significantly accelerate something as massive as a planet.

A similar situation occurs, though on a much smaller scale, when a baseball bat hits a ball. It's clear that the bat exerts a force on the ball, causing it to accelerate quickly. However, the ball is also exerting a force back on the bat. The ball's mass is relatively small compared to that of the bat, which also includes the batter's mass. Still, if you've ever seen a wooden bat break upon impact with the ball, you've witnessed the ball's force in action.

While these examples are insightful, they don’t really highlight practical uses of Newton's third law. So, is there a real-world application for force pairs? One example is jet propulsion. Seen in animals like squid and octopuses, as well as in airplanes and rockets, jet propulsion works by forcing a substance out of an opening at high speed. In squid and octopuses, this substance is seawater, drawn into the mantle and expelled through a siphon. The force the animal applies to the water results in the water exerting a force back, propelling the animal forward. This same principle powers turbine-driven jet planes and rockets in space.

Speaking of space, Newton's laws apply there as well. By using these laws to study planetary motion in space, Newton was able to develop the universal law of gravitation.

Applications and Limitations of Newton's Laws

While the three laws of motion alone are a monumental achievement, Newton didn’t stop there. He applied these principles to solve a long-standing mystery: the motion of the planets. Copernicus had placed the sun at the center of a solar system with planets and moons revolving around it, and the German astronomer Johannes Kepler had proven that planetary orbits were elliptical, not circular. However, no one could explain the mechanics behind this motion. Legend has it that Newton was inspired by seeing an apple fall to the ground, which led him to wonder if the falling apple and the orbiting planets might be related. Here’s how he thought about it to prove his theory:

Newton's revelation was groundbreaking, ultimately leading to the universal law of gravitation. This law states that any two objects in the universe exert a gravitational pull on each other, with the strength of this force depending on two factors: the masses of the objects involved and the distance between them. Larger masses generate stronger gravitational attractions, while increased distance weakens this attraction. Newton formalized this relationship mathematically with the following equation:

F = G(m1m2/r)

In this equation, F represents the gravitational force between two masses, m1 and m2, G is a universal constant, and r is the distance between the centers of both masses.

For centuries, scientists across various fields have tested Newton’s laws of motion, finding them to be remarkably accurate and reliable. However, there are two scenarios where Newtonian physics no longer applies. The first occurs when objects approach the speed of light. The second arises when applying Newton’s laws to incredibly small objects, such as atoms or subatomic particles, which belong to the realm of quantum mechanics.