Newton's Laws

By: Ashwin Vadivel

Isaac Newton is one of the fathers of physics. His laws of motion and universal gravitation are what has allowed humanity to technologically progress so much in the past few centuries. It started when Sir Edmund Halley(well known for his discovery of Halley's comet) approached Newton regarding a puzzling question that he had. He wanted to find out the mathematical connection that would describe why the planets orbited the sun in well defined elliptical orbits. Thomas Hooke(well known for Hooke's Law, describing spring force) had promised to solve this problem for Halley but had taken too much time and could not arrive at a reasonable answer. When Halley approached Newton, he was surprised to find out that Newton had already solved this problem and defined the law of gravitation. Halley was fascinated and took Newton's work to the Royal Society of London to be published. Unfortunately, the Royal Society was bankrupt at the time and could not publish it. In desperation, Halley published Newtons's Mathematical Principles of Natural Philosophy with his own money and distributed it for the world to admire. It was the work of Newton, and the persistence of Halley that allowed for the great body of work to be shared and which have allowed for the great progress that we have made (Eggen).

Portrait of Sir Isaac Newton

For this wiki, I have read the first chapter of the first three sections of Newton's Principia, available at the GT Archives Dept. All of the quotations that I have used come from there.

Preface

Newton describes the scientific process well in his book The Principia with this quote

"The constitution of particular things is known by observations and experiments; and when that is done, it is by this rule that we judge universally of the nature of such things in general."

Definitions

Density

"Quantity of Matter is the measure of the same, arising from its density and bulk conjunctly" (Newton). Newton proved that mass of an object is proportional to its weight through experiments with pendulums. With this in mind, Newton defined mass to be the density of matter in a given space. Later it has been shown to be the number of concentric lines of force that are present in a unit of volume. The density of an object can be shown with the equation: Density=Mass/Volume. The mass of an object refers to the product of the density of an object in a given volume. Thus, porous bodies such as sponges are lighter in mass when the occupy the same volume as more rigid bodies such as wood.

Motion

"The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjunctly" (Newton). Newton defines motion to be the absolute translation of all the matter in an object. The motion of the whole body is the sum of the motion of all of its parts. Thus, a body with mass M and velocity V the motion is equivalent to MV. A body double in mass but with equal velocity has absolute motion of 2MV. This is the principle of momentum. The motion of the whole.

Inertia

Drawing used by Newton

"Vis insita, or innate force of matter, is a power of resisting by which everybody, as much as it lies, endeavors to preserve its present stat, whether it be of rest or of moving uniformly forward in a straight line" (Newton). This is the basis of the first law of motion that we will be exploring soon. Inertia is the amount of resistance that a body has to a change in velocity and this is quantified by its mass. An object in motion will maintain its rectilinear (straight) motion unless a force changes its motion in a different direction. When this is done, each component of the motion of the object will be added/subtracted separately, which is why it is useful to understand vector notations. The diagonal to the right is directly from Newton's Principia. If an object starts at location C. And if a force M is applied to the object in direction AB and another force N is applied in AC, then the object will end up at point D at the same time as it would take if each force was applied individually. This is because each component is added separately to the motion of the object.

Force

A force is an action that is exerted on a body that makes it change its state, either of rest of of uniform rectilinear motion. The force only consists in its action. Once the action is completed, it no longer remains in the body that is was impressed upon. This is because a body maintains every new state that it acquires. This can be easily seen by observing the momentum principle: P(f)=P(i)+FnetΔt. The momentum of an object, which we earlier described to be the sum of the motion of its parts and thus its overall motion, is affected by an added force. Some forces work over a greater length of time, such as the continuously changing gravitational force that our sun exerts on the earth and the same that the earth exerts on its moon. The motion of an object is arithmetically affected by added forces. In cases where non-constant forces act over a long time interval, it is useful to use programs such as VPython that can model this quickly.

Centripetal Force

"A centripetal force is that by which bodies are drawn or impelled, or any way tend towards a point as to a center" (Newton). A centripetal force changes the straight motion of an object and makes it move in curvilinear orbits. For example, when throwing a stone in the air, if one wishes to maximize the distance it travels then they can do two things. One, throw it on a planet with less gravitational force for its quantity of matter. Or, throw it with a greater velocity, thus giving it more time for the constant centripetal force of gravity to chip away at its motion and bring it to the ground. Both of these methods will cause the stone to deviate less from its rectilinear course and thus the body will travel farther.

Absolute Quantity

Newton described the absolute quantity of centripetal force as the "center of force " itself. Such as the earth is the center of the gravitational force. The more massive that the earth is, the greater the centripetal force(gravity) it will exert on matter.

Accelerative Quantity

This quantity is what Newton described to be "the place of body". Such as the fact that the gravitational force is much stronger in valleys than on the tops of high mountains. The more distant from the center of force that an object is, the less centripetal force towards the center that it will experience.

Motive Quantity

"The body itself". Newton described the motive quantity of a force to be measured by a change in the quantity of motion, thus the change in momentum of an object. The force that causes this change in motion can be split into a component that is parallel to the motion and one that is perpendicular. These affect the motion of the body individually.

Relative vs. Absolute Motion

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The origin of space-time since the big bang

Newton's descriptions of relative and absolute motion provided the basis for Einstein to further develop the principles of relativity.

Absolute translation: Absolute translation is absolute motion in absolute space and time. Space-time is a continually expanding entity since the birth of the universe with the big bang. Below is an image of the space time continuum, since its inception. Movement along the space time continuum is absolute, at least so far in our understanding. For example, according to the multiverse theory, our universe is just one of many coming in and out of existence. However, as far as we are concerned, movement along this absolute continuum is not relative to other movements that we know about.

Relative translation: Relative translation is movement within a medium that is relative to the movement of another. For example, if you throw a baseball in a moving train at 50 mph and the train is moving at 100 mph, to an observer outside, the baseball will be moving at 50+100=150 mph. However that is relative to the movement of the earth, which is spinning at 1000 mph and travelling around the sun at 67,000 mph! It all depends on the reference frame form which you analyze motion, a concept that Newton and later physicists understood.

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Motion is relative to the reference frame from which you are analyzing it

Laws of Motion

Law I

"Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon" (Newton). This is the first law of motion. An object will continue in its straight(rectilinear) path or in its state of rest unless it is forced to change that path by a force that is exerted upon it.

Law II

"The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed" (Newton). Earlier we described the motive quantity to be a change in momentum of an object. The change in momentum of an object can be arithmetically calculated by the force that is exerted on an object, thus altering its motion in the direction of the force for a specified time interval. This is the basis for the derivation of the glorious Momentum Principle.

Law III

"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" (Newton). This is huge. This is the basis for the conservation of momentum, the conservation of energy and the conservation of angular momentum. This is why Wolfgang Pauli hypothesized that a particle(now known as the neutrino) must exist around the nucleus. Energy of an atom during certain nuclear decays were not conserved. Later the neutrino was discovered. For every action that occurs within a system, there must exist an equal reaction. The entire system must conserve its energy and momentum. This is the fundamental basis of physics.

References

Eggen, Olin Jeuck. "Edmond Halley." Encyclopedia Britannica Online. Encyclopedia Britannica, n.d. Web. 05 Dec. 2015.

Newton, Isaac, and John Carr. The First Three Sections of Newton's Principia: With Copious Notes and Illustrations, and a Great Variety of Deductions and Problems. Designed for the Use of Students. 2d ed., improved and enlarged. London: Printed for Baldwin, Cradock and Joy, 1825. Print.