Introduction to Newton's Laws of Motion

Sir Isaac Newton portrait.
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Each law of motion Newton developed has significant mathematical and physical interpretations that are needed to understand motion in our universe. The applications of these laws of motion are truly limitless.

Essentially, Newton's laws define the means by which motion changes, specifically the way in which those changes in motion are related to force and mass.

Origins and Purpose of Newton's Laws of Motion

Sir Isaac Newton (1642-1727) was a British physicist who, in many respects, can be viewed as the greatest physicist of all time.

Though there were some predecessors of note, such as Archimedes, Copernicus, and Galileo, it was Newton who truly exemplified the method of scientific inquiry that would be adopted throughout the ages.

For nearly a century, Aristotle's description of the physical universe had proven to be inadequate to describe the nature of movement (or the movement of nature, if you will). Newton tackled the problem and came up with three general rules about the movement of objects which have been dubbed as "Newton's three laws of motion."

In 1687, Newton introduced the three laws in his book "Philosophiae Naturalis Principia Mathematica" (Mathematical Principles of Natural Philosophy), which is generally referred to as the "Principia." This is where he also introduced his theory of universal gravitation, thus laying the entire foundation of classical mechanics in one volume.

Newton's Three Laws of Motion

  • Newton's First Law of Motion states that in order for the motion of an object to change, a force must act upon it. This is a concept generally called inertia.
  • Newton's Second Law of Motion defines the relationship between acceleration, force, and ​mass.
  • Newton's Third Law of Motion states that any time a force acts from one object to another, there is an equal force acting back on the original object. If you pull on a rope, therefore, the rope is pulling back on you as well.

Working With Newton's Laws of Motion

  • Free body diagrams are the means by which you can track the different forces acting on an object and, therefore, determine the final acceleration.
  • Vector mathematics is used to keep track of the directions and magnitudes of the forces and accelerations involved.
  • Variable equations are used in complex physics problems.

Newton's First Law of Motion

Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it.
- Newton's First Law of Motion, translated from the "Principia"

This is sometimes called the Law of Inertia, or just inertia. Essentially, it makes the following two points:

  • An object that is not moving will not move until a force acts upon it.
  • An object that is in motion will not change velocity (or stop) until a force acts upon it.

The first point seems relatively obvious to most people, but the second may take some thinking through. Everyone knows that things don't keep moving forever. If I slide a hockey puck along a table, it slows and eventually comes to a stop. But according to Newton's laws, this is because a force is acting on the hockey puck and, sure enough, there is a frictional force between the table and the puck. That frictional force is in the direction that is opposite the movement of the puck. It's this force which causes the object to slow to a stop. In the absence (or virtual absence) of such a force, as on an air hockey table or ice rink, the puck's motion isn't as hindered.

Here is another way of stating Newton's First Law:

A body that is acted on by no net force moves at a constant velocity (which may be zero) and zero ​acceleration.

So with no net force, the object just keeps doing what it is doing. It is important to note the words net force. This means the total forces upon the object must add up to zero. An object sitting on my floor has a gravitational force pulling it downward, but there is also a normal force pushing upward from the floor, so the net force is zero. Therefore, it doesn’t move.

To return to the hockey puck example, consider two people hitting the hockey puck on exactly opposite sides at exactly the same time and with exactly identical force. In this rare case, the puck would not move.

Since both velocity and force are vector quantities, the directions are important to this process. If a force (such as gravity) acts downward on an object and there's no upward force, the object will gain a vertical acceleration downward.

The horizontal velocity will not change, however.

If I throw a ball off my balcony at a horizontal speed of 3 meters per second, it will hit the ground with a horizontal speed of 3 m/s (ignoring the force of air resistance), even though gravity exerted a force (and therefore acceleration) in the vertical direction. If it weren't for gravity, the ball would have kept going in a straight line...at least, until it hit my neighbor's house.

Newton's Second Law of Motion

The acceleration produced by a particular force acting on a body is directly proportional to the magnitude of the force and inversely proportional to the mass of the body.
(Translated from the "Princip​ia")

The mathematical formulation of the second law is shown below, with F representing the force, m representing the object's mass and a representing the object's acceleration.

∑​ F = ma

This formula is extremely useful in classical mechanics, as it provides a means of translating directly between the acceleration and force acting upon a given mass. A large portion of classical mechanics ultimately breaks down to applying this formula in different contexts.

The sigma symbol to the left of the force indicates that it is the net force, or the sum of all the forces. As vector quantities, the direction of the net force will also be in the same direction as the acceleration. You can also break the equation down into x and y (and even z) coordinates, which can make many elaborate problems more manageable, especially if you orient your coordinate system properly.

You'll note that when the net forces on an object sum up to zero, we achieve the state defined in Newton's First Law: the net acceleration must be zero. We know this because all objects have mass (in classical mechanics, at least). If the object is already moving, it will continue to move at a constant velocity, but that velocity will not change until a net force is introduced. Obviously, an object at rest will not move at all without a net force.

The Second Law in Action

A box with a mass of 40 kg sits at rest on a frictionless tile floor. With your foot, you apply a 20 N force in a horizontal direction. What is the acceleration of the box?

The object is at rest, so there is no net force except for the force your foot is applying. Friction is eliminated. Also, there's only one direction of force to worry about. So this problem is very straightforward.

You begin the problem by defining your coordinate system. The mathematics is similarly straightforward:

F =  m *  a

F / m = ​a

20 N / 40 kg = a = 0.5 m / s2

The problems based on this law are literally endless, using the formula to determine any of the three values when you are given the other two. As systems become more complex, you will learn to apply frictional forces, gravity, electromagnetic forces, and other applicable forces to the same basic formulas.

Newton's Third Law of Motion

To every action there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

(Translated from the ​​"Principia")

We represent the Third Law by looking at two bodies, A and B, that are interacting. We define FA as the force applied to body A by body B, and FA as the force applied to body B by body A. These forces will be equal in magnitude and opposite in direction. In mathematical terms, it is expressed as:

FB = - FA

or

FA + FB = 0

This is not the same thing as having a net force of zero, however. If you apply a force to an empty shoebox sitting on a table, the shoebox applies an equal force back on you. This doesn't sound right at first — you're obviously pushing on the box, and it is obviously not pushing on you. Remember that according to the Second Law, force and acceleration are related but they aren't identical!

Because your mass is much larger than the mass of the shoebox, the force you exert causes it to accelerate away from you. The force it exerts on you wouldn't cause much acceleration at all.

Not only that, but while it's pushing on the tip of your finger, your finger, in turn, pushes back into your body, and the rest of your body pushes back against the finger, and your body pushes on the chair or floor (or both), all of which keeps your body from moving and allows you to keep your finger moving to continue the force. There's nothing pushing back on the shoebox to stop it from moving.

If, however, the shoebox is sitting next to a wall and you push it toward the wall, the shoebox will push on the wall and the wall will push back. The shoebox will, at this point, stop moving. You can try to push it harder, but the box will break before it goes through the wall because it isn't strong enough to handle that much force.

Newton's Laws in Action

Most people have played tug of war at some point. A person or group of people grab the ends of a rope and try to pull against the person or group at the other end, usually past some marker (sometimes into a mud pit in really fun versions), thus proving that one of the groups is stronger than the other. All three of Newton's Laws can be seen in a tug of war.

There frequently comes a point in a tug of war when neither side is moving. Both sides are pulling with the same force. Therefore, the rope does not accelerate in either direction. This is a classic example of Newton's First Law.

Once a net force is applied, such as when one group begins pulling a bit harder than the other, an acceleration begins. This follows the Second Law. The group losing ground must then try to exert more force. When the net force begins going in their direction, the acceleration is in their direction. The movement of the rope slows down until it stops and, if they maintain a higher net force, it begins moving back in their direction.

The Third Law is less visible, but it's still present. When you pull on the rope, you can feel that the rope is also pulling on you, trying to move you toward the other end. You plant your feet firmly in the ground, and the ground actually pushes back on you, helping you to resist the pull of the rope.

Next time you play or watch a game of tug of war — or any sport, for that matter — think about all the forces and accelerations at work. It's truly impressive to realize that you can understand the physical laws that are in action during your favorite sport.