Wikis
| Classical mechanics | ||||||||
History of ...
|
||||||||
Newton's laws of motion are three physical laws that form the basis for classical mechanics. They have been expressed in several different ways over nearly three centuries,[1] and can be summarised as follows:
These laws describe the relationship between the forces acting on a body and the motion of that body. They were first compiled by Sir Isaac Newton in his work Philosophiæ Naturalis Principia Mathematica, first published on July 5, 1687.[2] Newton used them to explain and investigate the motion of many physical objects and systems.[3] For example, in the third volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation, explained Kepler's laws of planetary motion.
Contents |
Newton's laws of motion are often defined as:
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.[4][5] 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.[6]
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.[7]
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:
In the first point, the phrase unbalanced force refers to a set of forces which do not have a zero sum (net force zero) or whose torques about the center of mass of the object do not have a zero sum. Indeed, without the torque requirement, a net force of zero will not accelerate the center of mass of an extended object, but may cause the object to rotate.
The second point 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.[8]
Newton placed the law of inertia first to establish frames of reference for which the other laws are applicable.[8][9] 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.[10]
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 Law—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[11] and Avicenna.[12] The 17th century philosopher René 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.[13]
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. This can be said easily. Net force is equal to rate of change of momentum for those who are unfamiliar with calculus. This definition holds even when the speed of the object approaches the speed of light.
Both statements of the second law are valid only for constant-mass systems,[14][15][16] 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 Δt, and it is given by[17][18]
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.[19]
Impulse is a concept frequently used in the analysis of collisions and impacts.[20]
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.[15] 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.[16] 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 Fnet is the total external force on the system, M is the total mass of the system, and acm 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:[14]
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 u 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,[21] 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.[22]
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. — 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.[23]
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, body B simultaneously exerts a force of the same magnitude on 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;[24] 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.
That when a thing lies still, unless somewhat else stir it, it will lie still forever, is a truth that no man doubts. But [the proposition] that when a thing is in motion it will eternally be in motion unless somewhat else stay it, though the reason be the same (namely that nothing can change itself), is not so easily assented to. For men measure not only other men but all other things by themselves. And because they find themselves subject after motion to pain and lassitude, [they] think every thing else grows weary of motion and seeks repose of its own accord, little considering whether it be not some other motion wherein that desire of rest they find in themselves, consists.
| Navigation | |
|---|---|
| Next: | Work, Power, and Energy |
| Previous: | Motion in a straight line |
| Up: | Topic: High School Physics |
Kinematics is the study of the motion of objects. Basic kinematic problems are approached using Isaac Newton's laws of motion.
Contents |
Newton attempted to explain how objects move using as few assumptions as possible. These assumptions are what we call Newton's Laws today. They are remarkable in that they have stood the test of time for almost all motion except those at the smallest scales (quantum mechanics) and the largest scales (general relativity). Even then, we have mostly just added or modified some assumptions that we had thought reasonable to assume about the nature of the universe, but the three laws have mostly remained. The laws are used to deduce the fundamental equation we will be using to study almost all of classical physics below. The wording of the laws has been altered slightly in order to be less jarring to a modern student. Another translation of the original laws can be found here.
Every body remains in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed thereon.
At this point, Newton has not told us what a force is, so this law isn't a description of how bodies move. You may think "Hey, I know what a force is, I force something to move when I push on it." However, that "contact force" is not the only type of force Newton means to deal with. He has basically used this statement to define what a force is: anything that changes the linear motion (including zero motion) of a body. This allows us to include gravity as a force. As a matter of fact, we will see that classical physics successfully describes all forces in terms of fields (similar to the way gravity works). Your pushing on an object is actually a result of electromagnetic field interactions between the particles in your hand and the particles in whatever you're pushing. Your hand never actually comes into direct contact with anything, in the sense of there being no space between your hand and another object.
The change in quantity of motion is proportional to the magnitude of the force; and is made in the direction of the straight line in which that force is directed.
This law is extremely rich in content, as Newton used devices similar to what we refer to today as Newton's cradle to note that the "quantity of motion" given to the end ball (assuming perfect collisions where all the "motive force" of the ball you lift on one end is transferred to the ball at the other end) is proportional to the final velocity of the ball you initiate. He also noted that different types of balls imparted larger arcs to other balls and thus he came up with the product of mass (a quantity proportional to the object's weight) and velocity as being the "quantity of motion" of an object that should be preserved in perfect collisions. Today, we call this product the momentum of an object and usually denote it as
where p and v have both a
magnitude and direction, as described by Newton's laws.
Manipulating objects that have both magnitude and direction (called
vectors) is covered by vector algebra, which will be talked about
in a supplementary review guide. Now, if we denote the force vector
by 
,
then Newton's second law tells us that
The 
is just the ordinary time derivative from calculus. If you have not
yet studied calculus, the above can also be written as
where a is the change in velocity with respect to time, also known as acceleration.
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.
This law is primarily for idealized objects such as spheres and
points that act on each other's centers of motion. However, it is
the impetus that allowed us to separate mass from weight. In
symbolic terms, it states that if one body exerts a force 
on another body, then the second body exerts a force 
on the first body, and in vector terms 
.
Using the equation we derived from the second law, it is not hard
to see that the mass can then be measured by defining one mass to
be a standard, and using the ratio of accelerations as the mass of
the second object.
Vectors are Physical Quantities that have both magnitude and
direction.
Some example of vectors are 'force', 'velocity', 'acceleration',
etc.
Free body diagrams are extremely useful tools when analyzing a physics problem. When approaching a kinematics problem, the first thing you should do is draw a free body diagram. What is a free body diagram? It is a simple sketch that shows all of the forces acting on an object ( a "free body"). When drawing a free body diagram, first you must isolate the object in question. Then, you must identify all of the forces acting on it. Remember, if an object is at rest, the vector sum of all forces must be zero. Newton's laws also tell us that every action has an equal and opposite reaction. If an object is sitting on a table, its weight is pushing down on the table, but the table is also pushing up on it (this force against the weight of an object is called the normal force. It is a very important concept in kinematics). Since both forces are equal and opposite, the object neither falls through the table nor flies up into the air.
We are now going to apply Newton's laws to an idealized object, a single particle. A particle in classical physics has a very well-defined meaning: it is an object without any spatial dimensions; mathematically it is a single point. We study these simple objects first, and later we consider larger objects as being made up of a lot of particles. We will see that spheres (objects like pool balls) behave a lot like particles (classically) and so do objects that interact along their centers of mass, and thus we will talk about sphere-like objects, simple center-of-mass interactions, and particles freely. We will show that the sentence above is accurate later when we have developed the tools and familiarity with physics to do so.
Now, take a ball and hold it in your hand some distance above the floor. If you stop holding the ball, the ball will fall to the floor. According to Newton's law of inertia, the ball could not have done so unless it was acted upon by a force (inertially, it should have remained where you were holding it). Since you did not force the ball downwards, some other force besides yourself must have acted on the ball. Due to our experience with other objects falling to the ground, we hypothesize that the Earth is exerting a force on your ball. According to Newton's third law, the ball must also exert a force on the Earth of equal magnitude and opposite direction.
Derivation of basic kinematics being written...
Isaac Newton (1642-1727), the father of the dynamics, – the study of motion – developed three sets of laws that are believed to be true because the results agree with the laws he produced.
Contents |
If a body is at rest it remains at rest or if it is in motion it moves with uniform velocity until it is acted on by a resultant force. (Duncan, 1995)
In other words, the first law says that an object that is not moving or moving in a constant speed in a straight line, will stay like that until something pushes it or blocks its path. As we all know, nothing in the visual world ever stays in constant speed, but the object itself is moving at constant speed, while a force is stopping it from moving at constant speed, friction. That does not change the law. However, in space, an object can move in a constant speed as long as it does not get close to any other objects, and stays in open space. This is why rockets use less fuel in space than they do getting to it.
Force is equal to mass times acceleration.
This law provides the definition and calculation of force through mass and acceleration.
To help the understanding of this concept, replace force with weight. Weight is a force that we feel on Earth, caused by gravity and our mass. Since gravity is calculated using the units of , therefore it is an acceleration constant. We could come to the conclusion that:
Newton's third law is:
For every action, there is an equal and opposite reaction.
The statement means that in every interaction, there is a pair of forces acting on the two interacting objects. The size of the forces on the first object equals the size of the force on the second object. The direction of the force on the first object is opposite to the direction of the force on the second object. Forces always come in pairs - equal and opposite action-reaction force pairs.
A variety of action-reaction force pairs are evident in nature. Consider the propulsion of a fish through the water. A fish uses its fins to push water backwards. But a push on the water will only serve to accelerate the water. Since forces result from mutual interactions, the water must also be pushing the fish forwards, propelling the fish through the water. The size of the force on the water equals the size of the force on the fish; the direction of the force on the water (backwards) is opposite the direction of the force on the fish (forwards). For every action, there is an equal (in size) and opposite (in direction) reaction force. Action-reaction force pairs make it possible for fish to swim.
Consider the flying motion of birds. A bird flies by use of its wings. The wings of a bird push air downwards. Since forces result from mutual interactions, the air must also be pushing the bird upwards. The size of the force on the air equals the size of the force on the bird; the direction of the force on the air (downwards) is opposite the direction of the force on the bird (upwards). For every action, there is an equal (in size) and opposite (in direction) reaction. Action-reaction force pairs make it possible for birds to fly.
Consider the motion of a car on the way to school. A car is equipped with wheels which spin backwards. As the wheels spin backwards, they grip the road and push the road backwards. Since forces result from mutual interactions, the road must also be pushing the wheels forward. The size of the force on the road equals the size of the force on the wheels (or car); the direction of the force on the road (backwards) is opposite the direction of the force on the wheels (forwards). For every action, there is an equal (in size) and opposite (in direction) reaction. Action-reaction force pairs make it possible for cars to move along a roadway surface.
Duncan, Tom. Advanced Physics for Hong Kong: Volume 1 Mechanics & Electricity. John Murray Ltd, 1995.
|
|
Search
