Newton’s Laws of Motion

Before listing Newton’s laws, it will be worthwhile to clarify what is meant by the word “law.” In science today, the words law, theory, and hypothesis are roughly synonymous. A theory is a descriptive or explanatory account of something. The word hypothesis usually refers to a theory that has not yet been extensively tested and confirmed by observations or experiments. The word law typically refers to a well-confirmed theory that describes some regularity in nature. However, these terms are not always used in a precise or consistent way (for example, “string theory” is an entirely untested hypothesis), so don’t rely on these distinctions too much. For now, just bear in mind that the words theory, hypothesis, and law are essentially synonyms: they all refer to descriptive or explanatory accounts (though they connote different degrees of observational support).

On the other hand, when Newton used the word “law” to describe the mathematical regularities he had discovered in nature, he wasn’t using the word merely as a synonym for “theory.” Like his predecessors Kepler and Galileo, Newton was a devout Christian, and regarded natural regularities as the result of divinely-instituted principles that govern creation. At the end of his magnum opus, The Mathematical Principles of Natural Philosophy (1846), Newton explicitly credits God for the order that is found in nature:

This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being. And if the fixed stars are the centres of other like systems, these, being formed by the like wise counsel, must be all subject to the dominion of One… This Being governs all things, not as the soul of the world, but as Lord over all… And from his true dominion it follows that the true God is a living, intelligent, and powerful Being; and, from his other perfections, that he is supreme, or most perfect. He is eternal and infinite, omnipotent and omniscient; that is, his duration reaches from eternity to eternity; his presence from infinity to infinity; he governs all things, and knows all things that are or can be done.The entire text is available here.

Newton’s laws of motion concern the effects of forces on physical objects. A force is simply a push or a pull. Forces are vector quantities: they have both magnitude and direction. The net force (total force) exerted on an object is the sum of all the forces acting on it. (See the previous page for an explanation of how vectors can be added together.)

Newton’s first law of motion says that an object’s velocity (speed and direction of motion) does not change so long as the net force on the object is zero. Newton’s first law is also known as the law of inertia.

Newton’s first law is logically entailed by the second law, since the latter implies that acceleration is zero when the net force is zero. So the first law can be regarded as merely a special case of the second law: the case where F = 0.

Newton’s second law of motion says that the net force on an object is equal to the object’s mass times its acceleration: F = ma In the above equation, F is the magnitude (strength) of the net force, m is the mass of the object, and a is the magnitude (rate) of its acceleration.

The standard unit of force, called the newton (symbolized with an uppercase “N”), is the amount of force needed to accelerate one kilogram of mass at a rate of one meter per second per second (i.e., its velocity increases by 1 m/s every second): 1 N = 1 kg m/s2

Newton’s third law of motion says that whenever one object exerts a force on another, the second object simultaneously exerts a force on the first object. These two forces are equal in magnitude but opposite in direction.

For example, the earth pulls on a falling apple with a downward gravitational force; the apple simultaneously pulls the earth upward with a force of equal magnitude. (Of course, this upward pull doesn’t budge the earth noticeably, because the earth has so much mass.)

Although the forces are equal and opposite, they do not cancel each other out, since they act on different objects.

Newton’s three laws together imply the law of conservation of momentum, which says that the total momentum of a physical system (any collection of physical objects) is conserved (doesn’t change), so long as no external forces act upon the system. In other words, the total momentum of the system (the sum of the momentum vectors for each object in the system) always stays the same unless an object outside the system exerts a force on one or more objects within the system. This is true for the total linear momentum and for the total angular momentum of the system.

Here’s a simple example to illustrate how the conservation of momentum follows from Newton’s laws. Suppose a 2 kg rock is moving to the right with a velocity of 5 m/s, and a 3 kg rock is moving to the left with a velocity of 4 m/s. The momentum of the smaller rock (its mass × velocity) is 2 kg × 5 m/s = 10 kg m/s to the right. The momentum of the larger rock is 3 kg × 4 m/s = 12 kg m/s to the left. Since the two momentum vectors are pointing in opposite directions, adding those vectors together yields a vector of length 2, pointing to the left. (See the previous page for an explanation of vector addition.) So, the total momentum of the two-rock system, initially, is 2 kg m/s to the left.

Now suppose the rocks exert forces on each other. By Newton’s third law, those forces must be equal in magnitude but opposite in direction. For example, if the large rock exerts 30 newtons of force pushing the small rock to the left, the small rock must exert 30 newtons pushing the large rock to the right. Let’s see what happens if the rocks exert that much force on each other for a duration of 1 second. First, consider what happens to the small rock, which has a mass of 2 kg. According to Newton’s second law:

force = mass × acceleration
30 N = 2 kg × acceleration
15 m/s2 = acceleration

The small rock will accelerate at a rate of 15 m/s2 to the left—the direction in which it is being pushed. In other words, its leftward velocity will increase by 15 m/s every second. Since it was initially traveling to the right at 5 m/s, after 1 second it will be going 10 m/s to the left. Therefore, its final momentum will be 2 kg × 10 m/s = 20 kg m/s to the left.

Next, let’s see what happens to the big rock, which has a mass of 3 kg:

force = mass × acceleration
30 N = 3 kg × acceleration
10 m/s2 = acceleration

The big rock will accelerate at a rate of 10 m/s2 to the right. Since it was initially moving to the left at 4 m/s, after 1 second it will be going 6 m/s to the right. Therefore, its final momentum will be 3 kg × 6 m/s = 18 kg m/s to the right.

So, the final momentum of the little rock is 20 kg m/s to the left, and the final momentum of the big rock is 18 kg m/s to the right. What is the total momentum of the two-rock system now, after the rocks have exerted forces on each other? Adding these final momentum vectors together yields a vector of length 2, pointing to the left. In other words, the final momentum of the two-rock system is 2 kg m/s to the left—the same as it was at the beginning! Although both rocks are moving in different directions and at different speeds than they were initially, and their individual momenta have changed, the total momentum of the two-rock system hasn’t changed at all.