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Physics VI: Newton's Laws
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In the given interpretation mass, acceleration, momentum, 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.
Some authors interpret the first law as defining what an inertial reference frame is; from this point of view, the second law only holds when the observation is made from an inertial reference frame, and therefore the first law cannot be proved as a special case of the second. Other authors do treat the first law as a corollary of the second.^{[3]} The explicit concept of an inertial frame of reference was not developed until long after Newton's death.
At speeds approaching the speed of light the effects of special relativity must be taken into account.^{[note 2]}
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 persists 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.^{[4]}
Newton's first law is also called the law of inertia. It states that if the vector sum of all forces (that is, the net force) acting on an object is zero, then the acceleration of the object is zero and its velocity is constant. Consequently:
The first point needs no comment, but the second seems to violate everyday experience. For example, a hockey puck sliding along ice does not move forever; rather, it slows and eventually comes to a stop. According to Newton's first law, the puck comes to a stop because of a net external force applied in the direction opposite to its motion. This net external force is due to a frictional force between the puck and the ice, as well as a frictional force between the puck and the air. If the ice were frictionless and the puck were traveling in a vacuum, the net external force on the puck would be zero and it would travel with constant velocity so long as its path were unobstructed.
Implicit in the discussion of Newton's first law is the concept of an inertial reference frame, which for the purposes of Newtonian mechanics is defined to be a reference frame in which Newton's first law holds true.
There is a class of frames of reference (called inertial frames) relative to which the motion of a particle not subject to forces is a straight line.^{[5]}
Newton placed the law of inertia first to establish frames of reference for which the other laws are applicable.^{[5]}^{[6]} To understand why the laws are restricted to inertial frames, consider a ball at rest inside an airplane on a runway. From the perspective of an observer within the airplane (that is, from the airplane's frame of reference) the ball will appear to move backward as the plane accelerates forward. This motion appears to contradict Newton's second law (F = ma), since, from the point of view of the passengers, there appears to be no force acting on the ball that would cause it to move. However, Newton's first law does not apply: the stationary ball does not remain stationary in the absence of external force. Thus the reference frame of the airplane is not inertial, and Newton's second law does not hold in the form F = ma.^{[note 3]}
Newton's first law is 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 Lawâ€"no force means no acceleration, and hence the body will maintain its velocity.
The law of inertia apparently occurred to several different natural philosophers and scientists independently. The inertia of motion was described in the 3rd century BC by the Chinese philosopher Mo Tzu, and in the 11th century by the Muslim scientists Alhazen^{[7]} and Avicenna.^{[8]} The 17th century philosopher RenÃ© Descartes also formulated the law, although he did not perform any experiments to confirm it.
The first law was understood philosophically well before Newton's publication of the law.^{[note 4]}
Newton's second law states that the force applied to a body produces a proportional acceleration; the relationship between the two is
where F is the force applied, m is the mass of the body, and a is the body's acceleration. If the body is subject to multiple forces at the same time, then the acceleration is proportional to the vector sum (that is, the net force):
The second law can also be shown to relate the net force and the momentum p of the body:
Therefore, Newton's second law also states that the net force is equal to the time derivative of the body's momentum:
Consistent with the first law, the time derivative of the momentum is non-zero when the momentum changes direction, even if there is no change in its magnitude (see time derivative). The relationship also implies the conservation of momentum: when the net force on the body is zero, the momentum of the body is constant.
Both statements of the second law are valid only for constant-mass systems,^{[9]}^{[10]}^{[11]} since any mass that is gained or lost by the system will cause a change in momentum that is not the result of an external force. A different equation is necessary for variable-mass systems.
Newton's second law requires modification if the effects of special relativity are to be taken into account, since it is no longer true that momentum is the product of inertial mass and velocity.
An impulse I occurs when a force F acts over an interval of time Î"t, and it is given by^{[12]}^{[13]}
Since force is the time derivative of momentum, it follows that
This relation between impulse and momentum is closer to Newton's wording of the second law.^{[14]}
Impulse is a concept frequently used in the analysis of collisions and impacts.^{[15]}
Variable-mass systems, like a rocket burning fuel and ejecting spent gases, are not closed and cannot be directly treated by making mass a function of time in the second law.^{[10]} The reasoning, given in An Introduction to Mechanics by Kleppner and Kolenkow and other modern texts, is that Newton's second law applies fundamentally to particles.^{[11]} In classical mechanics, particles by definition have constant mass. In case of a well-defined system of particles, Newton's law can be extended by summing over all the particles in the system:
where F_{net} is the total external force on the system, M is the total mass of the system, and a_{cm} is the acceleration of the center of mass of the system.
Variable-mass systems like a rocket or a leaking bucket cannot usually be treated as a system of particles, and thus Newton's second law cannot be applied directly. Instead, the general equation of motion for a body whose mass m varies with time by either ejecting or accreting mass is obtained by rearranging the second law and adding a term to account for the momentum carried by mass entering or leaving the system:^{[9]}
where u is the relative velocity of the escaping or incoming mass with respect to the center of mass of the body. Under some conventions, the quantity uâ€‰dm/dt on the left-hand side is defined as a force (the force exerted on the body by the changing mass, such as rocket exhaust) and is included in the quantity F. Then, by substituting the definition of acceleration, the equation becomes
Newton's Latin wording for the second law is:
Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.
This was translated quite closely in Motte's 1729 translation as:
LAW II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.
According to modern ideas of how Newton was using his terminology,^{[note 5]} this is understood, in modern terms, as an equivalent of:
Motte's 1729 translation of Newton's Latin continued with Newton's commentary on the second law of motion, reading:
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.
The sense or senses in which Newton used his terminology, and how he understood the second law and intended it to be understood, have been extensively discussed by historians of science, along with the relations between Newton's formulation and modern formulations.^{[note 6]}
A more direct translation than the one just given above 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. â€" 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 impinges upon another, and by its force changes 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, as 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.^{[note 7]}
In the above, as usual, motion is Newton's name for momentum, hence his careful distinction between motion and velocity.
The Third Law means that all forces are interactions, and thus that there is no such thing as a unidirectional force. If body A exerts a force on body B, simultaneously, body B exerts a force of the same magnitude body A, both forces acting along the same line. As shown in the diagram opposite, the skaters' forces on each other are equal in magnitude, but act in opposite directions. 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 and reaction 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;^{[16]} 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.
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.
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 F = dp/dt, 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.
In modern physics, 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.
This can be stated simply, "Momentum, energy and angular momentum cannot be created or destroyed."
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 also arise from the momentum conservation. Indeed, the conservation of 4-momentum in inertial motion via curved space-time results in what we call gravitational force in general relativity theory. Application of space derivative (which is a momentum operator in quantum mechanics) to overlapping wave functions of pair of fermions (particles with semi-integer spin) results in shifts of maxima of compound wavefunction away from each other, which is observable as "repulsion" of fermions.
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. However in modern engineering in all practical applications involving the motion of vehicles and satellites, the concept of action at a distance is used extensively.
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.
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