Newton's Laws

The motion of any mechanical system (for example, a single particle of mass, a complex flexible structure, or the solar system) is described by Newton's laws of motion.

Type of physical science: Classical physics

Field of study: Mechanics

Overview

Sir Isaac Newton's laws of motion describe how an object moves under the influence of one or more forces. In their exact form, these laws apply to particles of mass; that is, objects whose mass is concentrated at single points in space. It is relatively straightforward, however, to extend them to real objects with measurable dimensions. Although referred to as laws, they are really observations about the mechanical relations of objects within the realm of everyday experience. First published by Newton in Philosophae Naturalis Principia Mathematica (1687; Mathematical Principles of Natural Philosophy, 1729), they formed the basis of all systematic study of motion that followed. Only when one considers extremely small or large masses, or extremely high speeds, do Newton's laws require modification (using the theories of quantum mechanics and relativity).

Newton's first law of motion states that an object's velocity (speed and direction) will be constant unless a force external to the object acts on the mass. (An example of an internal force would be the atomic forces holding the object together.) Therefore, an object will move along a straight path with constant speed unless an external force acts upon it. Indeed, the fundamental definition of force is derived from the first law: Force is a quantity that causes objects to change velocity. An object that is stationary will remain so in the absence of an external force; this is known as the principle of inertia. When more than one external force is present, it is the total (or net) force that acts on the object.

Applying an external force to a mass causes it to accelerate--its velocity will change either in speed and/or direction. Newton's second law states that the relation between the force and the resulting acceleration is described by the equation F = ma, where ma is the object's mass times its acceleration. The direction of the acceleration will be the same as that of the force. A comparison of the first two laws reveals that the second law is a more exact statement of the first: If there is no force, then the acceleration is zero, and the object maintains its velocity. If that initial velocity is zero, then it will remain zero.

Newton's third law of motion explains that mechanical forces always occur in pairs.

When one object exerts a force on a second, the second object exerts a force of equal magnitude on the first; this is sometimes called the reaction force. The reaction force will be in exactly the opposite direction of the force that acts on the second object. It is common to refer to this law as the "action-reaction principle," and to state it in the following terms: For every action, there is an opposite and equal reaction. An example of this phenomenon is observed by pushing on a massive object. The reaction force of the object is clearly felt by the pusher. If this experiment is conducted on a smooth surface, such as ice, both the object and the pusher will be accelerated from a condition of rest to some speed, each by the force exerted by the other. The forces and, hence, the accelerations will be in opposite directions.

Two useful quantities that help to describe motion are energy and momentum. In applying a force to a stationary object, causing it to begin moving, the force is doing work on the object. This work is manifested in the continued motion of the object, even after the force is removed (according to Newton's first law, the object's speed and direction will now remain constant unless another force acts); the object is said to possess kinetic energy (energy of motion). Alternatively, the work done by the force may be stored in a nonmoving form. For example, using a force to compress a spring results in energy being stored in the spring. This is called potential energy (energy of position). If one end of the compressed spring is kept stationary and the other end is allowed to push an object, then releasing the spring will result in a force acting on the mass, which then accelerates to some speed. The potential energy in the spring is thus converted to kinetic energy in the object. If the spring is attached to the object, then the spring will continue to extend, eventually starting to pull back on the object. At some point, the object will come to rest, as the spring reaches the limit of its extension. The kinetic energy of motion has now been converted back into potential energy and stored in the extended spring. In the absence of other forces, this alternating exchange of kinetic and potential energies will persist indefinitely as the mass oscillates back and forth.

A moving object is said to have momentum, a quantity whose magnitude is given by its speed times its mass and whose direction is the same as that of the object's motion. In terms of Newton's first law of motion, the momentum is constant unless an external force acts on the object. Newton's second law may be restated in terms of momentum: The rate at which an object's momentum changes is equal to the external force acting on the object. The greater the momentum, the stronger the force required to change it; for example, stopping a large truck moving at high speed requires a much greater force than stopping a small car moving at the same speed.

A real object—whose mass is distributed over measurable dimensions—is a collection of particles. The forces holding the collection together (atomic and molecular binding forces) are internal and do not affect the overall motion of the object. Newton's third law of motion indicates that, since every conceivable pair of particles within the object generates pairs of forces having equal magnitude and opposite direction, adding together all of these forces thus gives zero net force on the object as a whole.

In order to apply the second law of motion to such an object, it is necessary to use the concept of center of mass, which is the average location of all the mass in the object. For example, a uniform sphere has its center of mass located at the sphere's geometric center; a flat triangular plate has its center of mass at a point two-thirds the distance from each corner to the center of the opposite edge and halfway between the two flat sides. The net effect of all forces acting on the real object may be studied by adding all the forces in such a way as to consider that they are being exerted against the center of mass.

Applications

Newton's laws of motion apply both to static (stationary) and dynamic (moving) systems. A system may consist of a single object or a complex arrangement of objects. The application of Newton's third law of motion to stationary systems constitutes the subject of statics, in which the set of forces acting among several objects, or among portions of a large object, is determined. The design of buildings and bridges, for example, depends upon the ability to predict the distribution of forces throughout the structure, and the concept of action and reaction forces plays a critical role.

A classic example of a dynamic system that illustrates all three laws of motion is that of a rocket being launched from Earth's surface. Prior to the launch, the rocket experiences only the force of gravity, pulling it downward onto the launch platform and the reaction force of the platform (the force of support), acting upward against the rocket. By the third law of motion, these forces have equal magnitude and opposite direction, giving a net force of zero on the rocket. Both the first and second laws then indicate that the rocket is not accelerating, since no net force acts on it. When the rocket engine ignites, though, a new force is introduced. The chemical reaction (burning fuel) in the engine's combustion chamber produces hot gases composed of highly energetic particles that collide with the chamber's walls; each collision exerts a small force against that particular side of the chamber. One end of the chamber is open to an exhaust nozzle, which allows the gases to escape to the outside. Inside the chamber, the large number of collisions per unit time is distributed randomly over the interior surface. On average, there are as many collisions on one side of the chamber as on the opposite side, resulting in no net force being exerted on the chamber as a whole, except on the chamber wall opposite the exhaust nozzle. Particles that collide here exert forces in the forward direction (away from the exhaust nozzle), but particles leaving through the nozzle do not exert any forces on the chamber. The result is a net force, called thrust, in the forward direction; the combustion chamber transmits this force to the structure of the rocket.

With the thrust acting upward and the gravitational force (weight) downward, the net force on the rocket is now the difference between the thrust and weight. According to Newton's second law, the rocket will accelerate upward at a value given by this net force divided by the rocket's mass. (Once the rocket begins to move, the platform no longer exerts a support force against it.) In order to continue producing sufficient amounts of thrust, the rocket must expel large quantities of gases, whose mass is simply the converted form of the original propellants. Therefore, the rocket is continually losing mass in the form of the exhaust gases. Most rocket engines are designed to produce a relatively constant level of thrust, and so, again by the second law, the rocket's acceleration will be increasing. Alternatively, one may view this entire process in terms of an exchange of energy. The combustion process releases energy, a considerable amount of which is lost as heat, but some of which is transformed into the kinetic energy of the rocket's motion and the potential energy of its position above Earth (produced by the thrust doing work on the rocket to move it against the pull of gravity).

A second example is that of a satellite in orbit around Earth. With its engines off, the satellite experiences only the force of gravity from Earth. From the first law of motion, it is necessary, then, that the satellite's velocity change in either speed, direction, or both quantities.

Giving a satellite exactly the right speed at a given distance from Earth will result in a circular orbit; under these conditions, gravity provides just the right acceleration to change only the satellite's direction, but not its speed. It thus maintains a constant speed at a fixed distance from Earth, with its direction of motion constantly changing to produce a circular path. For a somewhat different initial speed, the satellite describes an elliptical path, alternately closer, then farther from Earth. On a circular path, both the kinetic and potential energies are constant, since the satellite's speed and distance from Earth are unchanging. Yet, the elliptical path demonstrates an exchange of the two forms, with the potential energy increasing as the satellite moves farther from Earth (and the kinetic energy correspondingly decreasing), and the reverse occurs as the satellite moves closer to Earth.

A third example that illustrates the laws of motion is that of a wooden block that has been set sliding horizontally on a flat surface of ice. Here, the forces in the vertical direction are the block's weight (a downward force) and the reaction force of support from the ice, acting upward on the block. In the horizontal direction, the block experiences two forces, both directed opposite to the block's motion: air resistance, and friction between the block and the ice. Since the net force in the vertical direction is zero, the block will always have the same speed in the vertical direction, namely, zero. If it were not for the air resistance and friction with the ice, the block would (by the first law) continue with its original horizontal speed and direction indefinitely (or until it moved off the ice onto a rougher surface). Indeed, if the surface of the ice extended forever (and there were no horizontal forces), the block would never slow down. In reality, the air resistance and friction, which depend on the block's speed, decelerate it at a value given by the second law, namely, the total force of air resistance plus friction at any given moment, divided by the block's mass. As the speed decreases, the forces of air resistance and friction also decrease, until finally the block comes to rest with no horizontal forces acting upon it at all. In this example, the potential energy is constant (no work is done against gravity by lifting the block up off the ice), but the kinetic energy diminishes as the air resistance and friction do work against the block. This loss of energy takes the form of heat, which is dissipated into the surrounding air and the ice.

Context

Following their publication in 1687, Newton's laws of motion were to have a profound effect on how scientists viewed the mechanical world. Indeed, they represent the first case of a capability to predict the future behavior of a mechanical system, based only upon knowledge of its initial behavior (positions and velocities of its constituent masses) and application of the laws of motion. Two other accomplishments—his invention of the calculus and formulation of the law of universal gravitation—enhanced this predictive capability immeasurably. Despite their simple form, the laws of motion constitute the basis for the entire field of classical mechanics. Writing of Newton's great achievements, Edmond Halley, who encouraged him to publish his ideas in the Principia, penned "Nearer the gods no mortal may approach."

It was not until the end of the nineteenth century that physicists encountered phenomena that could not be predicted accurately by classical mechanics. Understanding the interaction of radiant energy with matter requires quantum mechanics, which recognizes the wavelike nature of matter on the atomic scale. In addition, Albert Einstein's theory of special relativity further modifies the Newtonian view by predicting changes in the length and mass of an object (properties that are absolute in classical mechanics) when it moves at some significant fraction of the speed of light. Yet another revision is required, by Einstein's theory of general relativity, if the motion occurs in close proximity to such massive objects as stars. Nevertheless, Newton's laws still maintain their accuracy and importance in the realm of human scales of mass, length, and speed; the new theories predict the same motions when applied to that domain.

The classical laws of motion continue to provide insights to mechanical and dynamical phenomena. Their application to some nonlinear systems has spawned the study of chaotic behavior, which deals with the sensitivity of a dynamical system to slight changes in its initial conditions. This has applications in such diverse systems as a simple pendulum with a nonlinear spring (that is, a spring whose force is not directly proportional to its compression or extension) connected at the hinge, and meteorological systems. In either example, minuscule changes in the initial conditions of motion can cause enormous changes in the system's behavior at a later time.

In many situations, the complexity of motion requires computer modeling to predict future behavior. Such matters as the long-term prediction of the solar system's motion or the analysis of systems with many interacting flexible components (for example, a robotic arm with several flexible structural members connected by hinged joints), are examples of problems for which the solutions are only approximate. Increasingly faster and higher-capacity computers will permit such systems to be studied more accurately and thoroughly, but here, as with much simpler cases, Newton's laws of motion will continue to hold the same importance as in the previous three centuries.

Principal terms

ACCELERATION: the rate at which an object's velocity changes (in speed and/or direction)

ENERGY: the amount of work done in moving an object to its current position (potential energy) or in getting an object to move with a particular speed (kinetic energy)

FORCE: the amount of push or pull exerted against an object

MASS: the amount of matter in an object

MOMENTUM: the mass of an object multiplied by its speed; momentum also has a direction, given by the direction of the velocity

VELOCITY: the speed and direction of a moving object

Bibliography

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