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Newton's laws of motion

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Newton's laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. These laws are fundamental to classical mechanics.

Newton first published these laws in Philosophiae Naturalis Principia Mathematica (1687) and used them to prove many results concerning the motion of physical objects. In the third volume (of the text), he showed how, combined with his law of universal gravitation, the laws of motion would explain Kepler's laws of planetary motion.

Note:In this article, vector quantities are written in bold whereas scalar ones are in italics.
Contents

Importance of Newton's laws of motion

Nature and Nature's laws lay hid in night;
God said, Let Newton be! And all was light.Alexander Pope

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 such as: the motion of spinning bodies, motion of bodies in fluids; projectiles; motion on an inclined plane; motion of a pendulum; the tides; the orbits of the Moon and the planets. The law of conservation of momentum, which Newton derived as a corollary of his second and third laws, was the first conservation law to be discovered.

Newton's laws were verified by experiment and observation for over 200 years. They describe the kinematics of the world on our scale (from 10e-6 m to 10e4, at speeds ranging from 0 to 100 000 000 m/s) beyond what can be accurately measured.

As a rule of thumb, Newton's Laws apply for any speed up to a third of the speed of light, after which point the error becomes too big to be ignored (see Einstein's correction factor).

Newton's First Law: Law of Inertia

This law is also called the Law of Inertia or Galileo's Principle.

Alternative formulations:

  • Every body's center of mass continues in its state of rest, or of uniform motion in a right [straight] line, unless it is compelled to change that state by forces impressed upon it.
  • A body's center of mass remains at rest, or moves in a straight line (at a constant velocity, v), unless acted upon by a net outside force.

In calculus notation, this may be expressed as: <math>\frac{d}{dt}\mathbf{v} = \mathbf{0}<math>

Despite the fact that Newton's First Law appears to be a special case of Newton's Second Law, the First Law defines the reference frames in which the other two laws are valid. These reference frames are called inertial reference frames or Galilean reference frames, and are moving at constant velocity, that is to say, without acceleration. (Note that an object may have a constant speed and yet have a non-zero acceleration, as in the case of uniform circular motion. This means that the surface of the Earth is not an inertial reference frame, since the Earth is rotating on its axis and orbits around the Sun. However, for many experiments, the Earth's surface can safely be assumed to be inertial. The error introduced by the acceleration of the Earth's surface is minute.)

In less formal terms, Aristotle thought that things stood still if you left them alone, that to be at rest was natural, and that movement needed a cause. It would be natural to think thus, as any movement (except for that of celestial objects, which were deemed perfect) that one observes eventually stops because of friction. But Galileo's experiments, with a ball rolling down an inclined plane, found that "Things travel naturally at a steady speed (which may or may not be zero), if left alone".

Moving from Aristotle's "A body's natural state is at rest" to Galileo's discovery (Newton's First Law) was one of the most profound and important discoveries in physics. In everyday life, the force of friction usually acts upon moving objects, slowing them down and eventually bringing them to rest. Newton described a mathematical model from which one could derive the motions of bodies from elementary causes: forces.

Newton's Second Law: Fundamental law of dynamics

Alternative formulations:

  • The rate of change in momentum is proportional to the net force acting on the object and takes place in the direction of the force.
  • The acceleration of an object of constant mass is proportional to the resultant force acting upon it.

These formulations may be expressed mathematically in the following ways:

<math> \mathbf{F} \propto \frac{d}{dt}\mathbf{p} = \frac{d}{dt}m\mathbf{v}<math> 
or 
<math>\mathbf{F} = m\frac{d\mathbf{v}}{dt} = m\mathbf{a}<math> if m is constant.

where

  • <math> \mathbf{F} <math> is the force acting,
  • <math> m <math> is the mass of the object in question,
  • <math> \mathbf{a} <math> is the object's acceleration,
  • <math> \mathbf{v} <math> is the object's velocity, and
  • <math> \mathbf{p} = m\mathbf{v} <math> collectively is called the object's momentum.

This equation expresses that

  • the more net force acts on an object, the greater the change in its momentum will be.

The quantity m, or mass, in the above equation is is a characteristic of the object. For an object of constant mass m (a constant of proportionality) the more net force acts on an object, the greater the change in its acceleration will be. This equation, therefore, indirectly defines the concept of mass.

In the equation, F = ma, a is directly measurable but F is not. The second law only has meaning if we are able to assert, in advance, the value of F. Rules for calculating force include Newton's law of universal gravitation.

But <math> \mathbf{F} = m\mathbf{a}<math> is not always valid. In general both the mass of the object and its velocity can be variable. For this case:

<math>\mathbf{F} = \frac{d}{dt}(m\mathbf{v}) 
= m\frac{d\mathbf{v}}{dt} + \mathbf{v}\frac{dm}{dt} 
= m\mathbf{a} + \mathbf{v}\frac{dm}{dt} <math>

This equation works in cases when the mass is variable. This equation is also valid in special relativity if we express the momentum as <math>\mathbf{p}=\gamma m\mathbf{v}<math>, where <math>\gamma<math> is the well known <math>\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}<math>.

The physical meaning behind this equation is important as it implies that objects interact by exchanging momentum, and they do this via a force.

Taken together with Newton's Third Law of Motion, Newton's Second Law implies the Law of Conservation of Momentum.

Newton's Third Law: Law of reciprocal actions

Alternative formulations:

  • Whenever one body exerts force upon a second body, the second body exerts an equal and opposite force upon the first body.
  • Momentum is conserved.

The very common formulation "for every action there is an equal and opposite reaction" should be avoided, as it is, at best, ambiguous and confusing. A better formulation would be that when there exists a force acting on a body A, due to another body B, there exists also a reciprocal force, acting on body B, due to the existence of body A.

These formulations imply that if you strike an object with a force of 200 N, then the object also strikes you (with a force of 200 N). Not only do planets accelerate toward stars; but, stars accelerate toward planets. The reaction force has the opposite direction of action, and is of the same type and magnitude as the original force. However, it doesn't necessarily "line up" in space with the action. One example of this is a force on an electric dipole due to a point charge, when the dipole points in a direction perpendicular to the line connecting the point charge and the dipole. The force on the dipole due to the point charge is perpendicular to the line connecting them, so there is a reaction force on the point charge in the opposite direction, but these two force vectors are parallel and, even when extended to a line, they never cross each other in space.

It is often contended that Newton's third law is incorrect when electromagnetic forces are included: if a body A exerts a force on body B, then body B will in general exert a different force on body A (the force considered is the Lorentz force, generated by electric and magnetic fields). Modern theory predicts that the electromagnetic field generated by such interactions itself transports momentum via electromagnetic radiation. Newton's third law is valid if the momentum of the field is included in the calculations.

Also see: Physics Study Guide (http://wikibooks.org/wiki/Force_(Physics_Study_Guide))

Weak and strong forms of Newton's third law

The so-called "weak form" of Newton's Third Law applies for classical physical forces (Marion and Thorton, 1995, pp. 333-337). In a system of particles, let <math>\mathbf{F}_{ab}<math> represent the force exerted on particle <math>a<math> due to particle <math>b<math>. The weak form requires that:

<math>\mathbf{F}_{ab} = -\mathbf{F}_{ba}<math>

All classical physical forces satisfy this condition.

The "strong form" of Newton's Third Law requires that, in addition to being equal and opposite, the forces must be directed along the line connecting the two particles. Gravitational force satisfies the strong form, while electromagnetic forces satisfy the weak form. For an example in electrostatics where the strong form is not obeyed, consider the interaction between a point charge and a perfect dipole aligned in a direction perpendicular to the line connecting the charge and the dipole.

The weak form is a valuable mathematical abstraction, because it allows one to study concepts such as the center of mass in the presence of arbitrary forces.

Range of validity

In 1916, Einstein's theory of relativity extended the scale to which we can make predictions. But at non-relativistic (low-enough) speeds, his relativistic model reduces to the classical one presented in this article.

<math>\lim_{v\rightarrow 0} \sqrt{1-\frac{v^2}{c^2}} = 1<math>

Or, put more simply, the multiplying correction factor (called <math>\gamma<math>) approaches one, for speeds less than a third of the speed of light.

See also

References

  • Marion, Jerry and Thornton, Stephen. Classical Dynamics of Particles and Systems. Harcourt College Publishers, 1995.ca:Lleis_de_Newton

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