The Law of Inertia

In this topic we will take a closer look at the concept of force. We begin with Newton's first law of motion, sometimes called the Law of Inertia:

A body continues in a state of rest or uniform motion unless acted upon by a net force (or a combination of forces that do not balance to zero).

Newton didn't claim to have come up with this law on his own — Galileo and Descartes had had the same insights — and some readers of the Principia would have been familiar with the idea already. Newton stated his first law not as something new and original but, as physicist Arnold Arons put it in A Guide to Introductory Physics Teaching, "as a declaration of independence from Aristotelian schools of thought." Recall that Aristotle said that a force must continually be applied to keep an object in uniform motion.

Rene Descartes (1596-1650) was a contemporary of Galileo. His first two laws of motion, taken together, are like Newton’s first. Descartes I: Each thing, as far as is in its power, always remains in the same state, and consequently, when it is once moved, it always continues to move. Descartes II: All movement is, of itself, along straight lines. Learn more about Descartes. Source: Wikipedia.

Zero force and net force

Newton’s first law is actually a definition rather than a physical law. It provides a qualitative idea of what a force is. It tells us how to recognize the case of zero force: a body with no net forces on it remains at rest or in uniform motion in a straight line. We need to know what a force is to completely understand the motion of a body, but by itself the first law does not give us complete information on what a force is.

The condition for a body to change from its state of rest or uniform motion is that there is no net force on it. As we will see in more detail later, two or more forces may act simultaneously on a body, and the result is the cumulative effect of all the forces. It is possible for two forces to balance each other in terms of their effect (one force acting to push an object to the right, for example, and another, equal, force acting to push the object to the left). Newton’s first law says that a body may continue at rest or in uniform motion if there is no force, or if the forces balance to zero.

What does it mean to change an object’s state of rest or uniform motion?

Before we define “force” operationally, we will give some examples of forces at work on objects that are initially at rest or in uniform motion. The examples apply in an idealized case with no friction.

Example 1:

At an ice rink, a hockey puck glides over ice in a frictionless way. It is gliding at a constant speed and direction. Its velocity is v (say 30 km/h, north) and its acceleration is 0. It has uniform motion.

Now a hockey player, moving in the same direction as the puck, comes along and places his stick in front of the puck and holds it there to slow the puck down. The puck has not changed direction, but its velocity is changing because its speed is being reduced. During the time the hockey player holds the stick against the puck, the puck has a negative acceleration. The puck no longer has uniform motion.

The steady holding of the stick against the puck is the application of a force on the puck. The application of the force changed the puck’s acceleration from 0 to a negative value. Its velocity is decreasing from (30 km/h, north), to (lesser speeds, north).

Example 2:

At an ice skating competition, a fan in the audience tosses a rose. It lands on the ice. The rose has 0 velocity and 0 acceleration; it is at rest.

An ice resurfacing truck (Zamboni), cleaning the ice between acts pushes the rose from left to right so that it has a constant velocity of 5 miles per hour to the right. (For a while, the rose was in contact with the Zamboni; then it and the Zamboni proceeded to the right at the same speed, with the rose in front.)

The rose now has uniform motion and a constant velocity of 5 miles per hour to the right, but it started at 0 miles per hour. For some short interval of time, during the contact with the Zamboni, the rose was accelerating from 0 to 5 miles per hour.

The steady pushing of the ice resurfacing vehicle applied a force to the rose during the time of contact.

Example 3:

A paperclip is at rest on a table. A magnet is brought close and the paperclip rises through the air and attaches itself to the magnet.

The magnet exerted a force on the paperclip, which caused it to accelerate from 0 to some velocity, during the short time that it took the paper clip to move from the table to the magnet.

Example 4:

A sledder travels in a straight line at constant speed over level (frictionless) ground. She then encounters a smooth curved wall that forms a quarter-circle section. At the end of the section, she slides in a straight line at constant speed.

The curved wall making contact with the sled imparted a force that changed the direction of the sled's velocity. When the wall ended, the force was no longer imparted, so the sled continued its journey in a straight line at constant speed.

What can we learn from these examples?

You may have noticed in these examples that when an object experienced a change from a state of rest or uniform motion, the change was caused by contact or interaction with another object. In the first example, the hockey player, through his stick, exerted a steady force to decelerate the puck. In the second example, the Zamboni pushed the rose and accelerated it until it reached the same speed as the Zamboni. In the third example, the magnet attracted the paper clip in a form of “action at a distance.” (The magnet did not need to touch the paper clip in order to change its velocity.) In the fourth example, the smooth curved wall exerted a force on the sled that changed its direction. In each case, the force was something external to the object which caused the object that was initially at rest or in uniform motion to change its velocity or, to put it another way, to undergo acceleration.

At the elementary level, students learn that a force is a push or a pull. For example, one uses force to pull open a drawer. At a more advanced level, we see that a force is something that can change an object’s speed or direction. In other words, we need to define force using the concept of acceleration.

Definition of force

An operational definition gives instructions or operations to carry out to measure the thing being defined quantitatively (just as you would do in a recipe). Our operational definition of force is as follows:

Take a frictionless puck that has been defined to have a mass of 1 kg, and place it on a level frictionless tabletop. We do not need to know yet what "mass" is; suffice it to say the puck is a standard one.

Attach this standard puck to a spring which has a display needle to indicate the degree to which the spring is stretched. Pull on the spring (by hand or by some apparatus) in one direction (say, to the right) and measure the acceleration of the puck (in m/s²). (One can verify at this point that any constant stretching of the spring leads to some constant value of acceleration.)

Note the position of the needle on the display for various values of acceleration. Adjust the degree of pulling until the puck has a uniform acceleration of 1 m/s².

Now define a force of 1 Newton (symbol N) as the force required to get the standard 1 kg puck to accelerate at 1 m/s².

1 Newton = force required to accelerate a 1 kg standard object to 1 m/s².

If we pull on the standard puck until it has an acceleration of 2 m/s², that is a force of 2 N. Similarly, an acceleration of the standard puck to 3 m/s² is a force of 3 N, etc.

Remember the story of the apple that fell on Newton's head? Coincidentally, a small apple, falling under the influence of gravity, would exert a force on your head of about 1 N! This should give you a "feeling" for the magnitude of 1 N.

Force is a vector quantity

Vector quantities have a magnitude and direction, and also add together according to the law of vector addition.

Addition of vectors. The vector shown in black is the sum of the vector shown in purple and the one shown in green.

In the previous section we saw how the magnitude of a force is defined. We can similarly define the direction of the force to be the direction of the acceleration:

The direction of F is the direction of a.

We can check experimentally to see if force, as we have defined it, behaves like a vector. We can attach two springs on different sides of the puck and pull on those springs simultaneously. Say we pull with a force of 1 N to the east and 1 N to the north. We find experimentally that the puck accelerates at 1.4 m/s² to the northeast, just as we expect from vector addition.

With this experiment, we can confirm that force, as we have defined it, is a vector quantity. And we can also demonstrate an important phenomenon: each force acting on a body can produce acceleration independently, and the resulting acceleration of the body is the vector sum of each independent acceleration. To put it another way, when an object is subject to two or more forces at once, the result is the cumulative effect of all the forces.

If we attach springs on opposite sides of the puck and pull with a force of 1 N to the left and 1 N to the right, the puck will not move. The vector sum of the forces is zero and the accelerations also "cancel out."

Newton's first law states that an object will move only if acted upon by a force or, as it is sometimes stated, an unbalanced or net force. When the vector sum of all the forces acting on an object is not zero, we are left with an unbalanced force or net force.

A statement that follows from Newton's first law is:
Unbalanced forces cause changes in velocity.

If the forces represented by the two arrows are of the same magnitude, they will balance and the object will be at rest. If the force represented by the pink arrow is larger, there will be a net or unbalanced force to the right.