# Newton's Second Law

## Information about Newton's Second Law

Newton's First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica.
Newton's laws of motion are three physical laws which provide relationships between the forces acting on a body and the motion of the body, first compiled by Sir Isaac Newton. Newton's laws were first published together in his work Philosophiae Naturalis Principia Mathematica (1687). The laws form the basis for classical mechanics. Newton used them to explain many results concerning the motion of physical objects. In the third volume of the text, he showed that the laws of motion, combined with his law of universal gravitation, explained Kepler's laws of planetary motion.

Briefly stated, the three laws are:
1. An object in motion will remain in motion unless acted upon by a net force.
2. Force equals mass multiplied by acceleration.
3. To every action there is an equal and opposite reaction.

## The statements of the laws

Newton's laws of motion describe the acceleration of massive objects. The modern understanding of Newton's three laws of motion is:
First Law
If no net force acts on a particle, then it is possible to select a set of reference frames, called inertial reference frames, observed from which the particle moves without any change in velocity.
Second Law
Observed from an inertial reference frame, the net force on a particle is proportional to the time rate of change of its linear momentum: . Momentum is the product of mass and velocity. This law is often stated as (the force on an object is equal to its mass multiplied by its acceleration).
Third Law
Whenever a particle A exerts a force on another particle B, B simultaneously exerts a force on A with the same magnitude in the opposite direction. The strong form of the law further postulates that these two forces act along the same line.

In the given interpretation mass, acceleration and most importantly force are assumed to be externally defined quantities. This is the most common, but not the only interpretation: one can consider the laws to be a definition of these quantities. Notice that the second law only holds when the observation is made from an inertial reference frame, and since an inertial reference frame is defined by the first law, asking a proof of the first law from the second law is a logical fallacy.

### Newton's first law: law of inertia

Lex I: Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare.

Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed.[1]

This law is also called the law of inertia.

This is often paraphrased as "zero net force implies zero acceleration", but this is an over-simplification. As formulated by Newton, the first law is more than a special case of the second law. Newton arranged his laws in hierarchical order for good reason (e.g. see Gailili & Tseitlin 2003).[2] Essentially, the first law establishes frames of reference for which the other laws are applicable, such frames being called inertial frames. To understand why this is required, consider a ball at rest within an accelerating body: an aeroplane on a runway will suffice for this example. From the perspective of anyone within the aeroplane (that is, from the aeroplane's frame of reference when put in technical terms) the ball will appear to move backwards as the plane accelerates forwards (the same feeling of being pushed back into your seat as the plane accelerates). This appears to contradict Newton's second law as, from the point of view of the passengers, there appears to be no force acting on the ball which would cause it to move. The reason why there is in fact no contradiction is because Newton's second law (without modification) is not applicable in this situation because Newton's first law was never applicable in this situation (i.e. the stationary ball does not remain stationary). Thus, it is important to establish when the various laws are applicable or not since they are not applicable in all situations. On a more technical note, although Newton's laws are not applicable on non-inertial frames of reference, such as the accelerating aeroplane, they can be made to do so with the introduction of a "fictitious force" acting on the entire system: basically, by introducing a force that quantifies the anomalous motion of objects within that system (such as the ball moving without an apparent influence in the example above).

The net force on an object is the vector sum of all the forces acting on the object. Newton's first law says that if this sum is zero, the state of motion of the object does not change. Essentially, it makes the following two points:
• An object that is not moving will not move until a net force acts upon it.
• An object that is in motion will not change its velocity (accelerate) until a net force acts upon it.
The first point seems relatively obvious to most people, but the second may take some thinking through, because we have no experience in every-day life of things that keep moving forever (except celestial bodies). If one slides a hockey puck along a table, it doesn't move forever, 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 frictional force between the table and the puck, and that frictional force is in the direction opposite the movement. It is this force which causes the object to slow to a stop. In the absence of such a force, as approximated by an air hockey table or ice rink, the puck's motion would not slow. Newton's first law is just a restatement of what Galileo had already described and Newton gave credit to Galileo. It differs from Aristotle's view that all objects have a natural place in the universe. Aristotle believed that heavy objects like rocks wanted to be at rest on the Earth and that light objects like smoke wanted to be at rest in the sky and the stars wanted to remain in the heavens.

However, a key difference between Galileo's idea and Aristotle's is that Galileo realized that force acting on a body determines acceleration, not velocity. This insight leads to Newton's First Law—no force means no acceleration, and hence the body will maintain its velocity.

The Law of Inertia apparently occurred to many different natural philosophers independently. Inertia of motion was described in the third century BC in the Mo Tzu, a collection of Chinese philosophical texts, and the 17th century philosopher René Descartes also formulated the law, although he did not perform any experiments to confirm it.

There are no perfect demonstrations of the law, as friction usually causes a force to act on a moving body, and even in outer space gravitational forces act and cannot be shielded against, but the law serves to emphasize the elementary causes of changes in an object's state of motion: forces.

### Newton's second law: law of acceleration

Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.

The rate of change of momentum of a body is proportional to the resultant force acting on the body and is in the same direction.

In Motte's 1729 translation (from Newton's Latin), the Second Law of Motion reads:

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. — If a force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.

Using modern symbolic notation, Newton's second law can be written as a vector differential equation:

where:
is the net force vector
is mass
is the velocity vector
is time.

The product of the mass and velocity is the momentum of the object (which Newton himself called "quantity of motion"). The use of algebraic expressions became popular during the 18th Century, after Newton's death, while vector notation dates to the late 19th Century. The Principia expresses mathematical theorems in words and consistently uses geometrical rather than algebraic proofs.

If the mass of the object in question is constant, this differential equation can be rewritten as:

where:
is the acceleration.

A verbal equivalent of this is "the acceleration of an object is proportional to the force applied, and inversely proportional to the mass of the object". If momentum varies nonlinearly with velocity (as it does for high velocities—see special relativity), then this last version is not accurate.

Taking Special Relativity into consideration, the equation becomes

where:
is the rest mass or invariant mass.
is the speed of light.

Note that force depends on speed of the moving body, acceleration, and its rest mass. However, when the speed of the moving body is much lower than the speed of light, the equation above reduces to the familiar .

Mass must always be taken as constant in classical mechanics. So-called variable mass systems like a rocket can not be directly treated by making mass a function of time in the second law. The reasoning, given in An Introduction to Mechanics by Kleppner and Kolenkow and other modern texts, is excerpted here:

Newton's second law applies fundamentally to particles. In classical mechanics, particles by definition have constant mass. In case of well-defined systems of particles, Newton's law can be extended by integrating over all the particles in the system. In this case, we have to refer all vectors to the center of mass. Applying the second law to extended objects implicitly assumes the object to be a well-defined collection of particles. However, 'variable mass' systems like a rocket or a leaking bucket do not consist of a set number of particles. They are not well-defined systems. Therefore Newton's second law can not be applied to them directly. The naïve application of F = dp/dt will usually result in wrong answers in such cases. However, applying the conservation of momentum to a complete system (such as a rocket and fuel, or a bucket and leaked water) will give unambiguously correct answers.

### Newton's third law: law of reciprocal actions

Newton's third law. The skaters' forces on each other are equal in magnitude, and in opposite directions
Lex III: Actioni contrariam semper et æqualem esse reactionem: sive corporum duorum actiones in se mutuo semper esse æquales et in partes contrarias dirigi.

All forces occur in pairs, and these two forces are equal in magnitude and opposite in direction.

This law of motion is commonly paraphrased as: "To every action force there is an equal, but opposite, reaction force".

A more direct translation is:
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. — Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. If a body impinge upon another, and by its force change the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, toward the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of the bodies; that is to say, if the bodies are not hindered by any other impediments. For, because the motions are equally changed, the changes of the velocities made toward contrary parts are reciprocally proportional to the bodies. This law takes place also in attractions, as will be proved in the next scholium.

In the above, as usual, motion is Newton's name for momentum, hence his careful distinction between motion and velocity.

As shown in the diagram opposite, the skaters' forces on each other are equal in magnitude, and opposite in direction. Although the forces are equal, the accelerations are not: the less massive skater will have a greater acceleration due to Newton's second law. It is important to note that the action/reaction pair act on different objects and do not cancel each other out. The two forces in Newton's third law are of the same type, e.g., if the road exerts a forward frictional force on an accelerating car's tires, then it is also a frictional force that Newton's third law predicts for the tires pushing backward on the road.

Newton used the third law to derive the law of conservation of momentum;[3] however from a deeper perspective, conservation of momentum is the more fundamental idea (derived via Noether's theorem from Galilean invariance), and holds in cases where Newton's third law appears to fail, for instance when force fields as well as particles carry momentum, and in quantum mechanics.

## Importance and range of validity

Newton's laws were verified by experiment and observation for over 200 years, and they are excellent approximations at the scales and speeds of everyday life. Newton's laws of motion, together with his law of universal gravitation and the mathematical techniques of calculus, provided for the first time a unified quantitative explanation for a wide range of physical phenomena.

These three laws hold to a good approximation for macroscopic objects under everyday conditions. However, Newton's laws (combined with Universal Gravitation and Classical Electrodynamics) are inappropriate for use in certain circumstances, most notably at very small scales, very high speeds (in special relativity, the Lorentz factor must be included in the expression for momentum along with rest mass and velocity) or very strong gravitational fields. Therefore, the laws cannot be used to explain phenomena such as conduction of electricity in a semiconductor, optical properties of substances, errors in non-relativistically corrected GPS systems and superconductivity. Explanation of these phenomena requires more sophisticated physical theory, including General Relativity and Relativistic Quantum Mechanics.

According to the principle of relativity, there is no preferred frame of reference. The laws of physics are equally valid in all frames of reference. Motion can only be measured relative to a frame of reference. According to the equivalence principle, an observer on the surface of the Earth could not find any difference between the gravitational attraction of earth and the inertial force that he feels when he is in a rocket in outer space that accelerates upwards (from the standpoint of the observer) at g. In other words, he may regard any inertial force as a gravitational force. Consequently, Newton's laws of motion are only valid in an inertial frame of reference. Notice that the surface of the Earth does not define an inertial frame of reference because it is rotating and orbiting and because of Earth's gravity. However, since the rotation and revolution are relatively slow, the inertial force is tiny. Therefore, Newton's laws of motion remain a good approximation on earth. In a non-inertial frame of reference, inertial forces must be considered for Newton's laws to remain valid.

In quantum mechanics concepts such as force, momentum, and position are defined by linear operators that operate on the quantum state; at speeds that are much lower than the speed of light, Newton's laws are just as exact for these operators as they are for classical objects. At speeds comparable to the speed of light, the second law holds in the original form , which says that the force is the derivative of the momentum of the object with respect to time, but some of the newer versions of the second law (such as the constant mass approximation above) do not hold at relativistic velocities.

## Relationship to the conservation laws

The laws of conservation of momentum, energy, and angular momentum are of more general validity than Newton's laws, since they apply to both light and matter, and to both classical and non-classical physics.

Because force is the time derivative of momentum, the concept of force is redundant and subordinate to the conservation of momentum, and is not used in fundamental theories (e.g. quantum mechanics, quantum electrodynamics, general relativity, etc.). The standard model explains in detail how the three fundamental forces known as gauge forces originate out of exchange by virtual particles. Other forces such as gravity and fermionic degeneracy pressure arise from conditions in the equations of motion in the underlying theories.

Newton stated the third law within a world-view that assumed instantaneous action at a distance between material particles. However, he was prepared for philosophical criticism of this action at a distance, and it was in this context that he stated the famous phrase "I feign no hypotheses". In modern physics, action at a distance has been completely eliminated, except for subtle effects involving quantum entanglement.

Conservation of energy was discovered nearly two centuries after Newton's lifetime, the long delay occurring because of the difficulty in understanding the role of microscopic and invisible forms of energy such as heat and infra-red light.

## References

1. ^ Isaac Newton, The Principia, A new translation by I.B. Cohen and A. Whitman, University of California press, Berkeley 1999.
2. ^ Galili, I. & Tseitlin, M. (2003), "Newton's first law: text, translations, interpretations, and physics education.", Science and Education 12 (1): 45-73
3. ^ Newton, Principia, Corollary III to the laws of motion
• Marion, Jerry and Thornton, Stephen. Classical Dynamics of Particles and Systems. Harcourt College Publishers, 1995. ISBN 0-03-097302-3
• Fowles, G. R. and Cassiday, G. L. Analytical Mechanics (6ed). Saunders College Publishing, 1999. ISBN 0-03-022317-2

A physical law, scientific law, or a law of nature is a scientific generalization based on empirical observations of physical behavior.
In mathematics and statistics, a positive or direct relationship is a relationship between two variables in which they both increase or decrease in conjunction.

## Characteristics of a direct (or positive) relationship

In physics, force is an action or agency that causes a body of mass m to accelerate. It may be experienced as a lift, a push, or a pull. The acceleration of the body is proportional to the vector sum of all forces acting on it (known as net force or resultant force).
In physics, a physical body (sometimes called simply a body or even an object) is a collection of masses, taken to be one. For example, a cricket ball can be considered an object but the ball also consists of many particles (pieces of matter).
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Sir Isaac Newton

Isaac Newton at 46 in
Godfrey Kneller's 1689 portrait
Born 4 January 1643 [OS: 25 December 1642]
Philosophiæ Naturalis Principia Mathematica (Latin: "mathematical principles of natural philosophy", often Principia or Principia Mathematica for short) is a three-volume work by Isaac Newton published on July 5, 1687.
8th century - 9th century - 10th century
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Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies.
Isaac Newton's theory of universal gravitation is a physical law describing the gravitational attraction between massive bodies. It is a part of classical mechanics and was first formulated in Newton's work Philosophiae Naturalis Principia Mathematica, published in 1687.
Kepler's laws of planetary motion are three mathematical laws that describe the motion of planets in the Solar System. German mathematician and astronomer Johannes Kepler (1571–1630) discovered them.
acceleration is defined as the rate of change of velocity, or, equivalently, as the second derivative of position. It is thus a vector quantity with dimension length/time². In SI units, acceleration is measured in metres/second² (m·s-²).
Mass is a fundamental concept in physics, roughly corresponding to the intuitive idea of "how much matter there is in an object". Mass is a central concept of classical mechanics and related subjects, and there are several definitions of mass within the framework of relativistic
In physics, force is an action or agency that causes a body of mass m to accelerate. It may be experienced as a lift, a push, or a pull. The acceleration of the body is proportional to the vector sum of all forces acting on it (known as net force or resultant force).
A frame of reference is a particular perspective from which the universe is observed. Specifically, in physics, it refers to a provided set of axes from which an observer can measure the position and motion of all points in a system, as well as the orientation of objects in it.
An inertial frame of reference, or inertial reference frame, is one in which Newton's first and second laws of motion are valid. Newton's laws are valid in any reference frame that is neither rotating nor accelerating relative to the sun and other stars.
velocity is defined as the rate of change of position. It is a vector physical quantity, both speed and direction are required to define it. In the SI (metric) system, it is measured in meters per second (m/s). The scalar absolute value (magnitude) of velocity is speed.
momentum (pl. momenta; SI unit kg m/s, or, equivalently, N•s) is the product of the mass and velocity of an object. For more accurate measures of momentum, see the section "modern definitions of momentum" on this page.
A fallacy is a component of an argument that is demonstrably flawed in its logic or form, thus rendering the argument invalid in whole. In logical arguments, fallacies are either formal or informal.
Inertia is a property of matter by which it remains at rest or in uniform motion in the same straight line unless acted upon by some external force The principle of inertia is one of the fundamental principles of classical physics which are used to describe the motion of matter and
A frame of reference is a particular perspective from which the universe is observed. Specifically, in physics, it refers to a provided set of axes from which an observer can measure the position and motion of all points in a system, as well as the orientation of objects in it.
An inertial frame of reference, or inertial reference frame, is one in which Newton's first and second laws of motion are valid. Newton's laws are valid in any reference frame that is neither rotating nor accelerating relative to the sun and other stars.
A fictitious force, also called a pseudo force[1] or d'Alembert force[2], is an apparent force that acts on all masses in a non-inertial frame of reference such as a rotating reference frame.
velocity is defined as the rate of change of position. It is a vector physical quantity, both speed and direction are required to define it. In the SI (metric) system, it is measured in meters per second (m/s). The scalar absolute value (magnitude) of velocity is speed.
Galileo Galilei

Portrait of Galileo Galilei by Giusto Sustermans
Born January 15 1564[1]
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