Newton’s laws
Newton’s laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
- A body remains at rest, or in motion at a constant speed in a straight line, unless acted upon by a force.
- When a body is acted upon by a net force, the body’s acceleration multiplied by its mass is equal to the net force.
- If two bodies exert forces on each other, these forces have the same magnitude but opposite directions.
The three laws of motion were first stated by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), originally published in 1687. Newton used them to investigate and explain the motion of many physical objects and systems, which laid the foundation for classical mechanics. In the time since Newton, the conceptual content of classical physics has been reformulated in alternative ways, involving different mathematical approaches that have yielded insights which were obscured in the original, Newtonian formulation. Limitations to Newton’s laws have also been discovered; new theories are necessary when objects move at very high speeds (special relativity), are very massive (general relativity), or are very small (quantum mechanics).
Prerequisites
Newton’s laws are often stated in terms of point or particle masses, that is, bodies whose volume is negligible. This is a reasonable approximation for real bodies when the motion of internal parts can be neglected, and when the separation between bodies is much larger than the size of each. For instance, the Earth and the Sun can both be approximated as pointlike when considering the orbit of the former around the latter, but the Earth is not pointlike when considering activities on its surface.
The mathematical description of motion, or kinematics, is based on the idea of specifying positions using numerical coordinates. Movement is represented by these numbers changing over time: a body’s trajectory is represented by a function that assigns to each value of a time variable the values of all the position coordinates. The simplest case is one-dimensional, that is, when a body is constrained to move only along a straight line. Its position can then be given by a single number, indicating where it is relative to some chosen reference point. For example, a body might be free to slide along a track that runs left to right, and so its location can be specified by its distance from a convenient zero point, or origin, with negative numbers indicating positions to the left and positive numbers indicating positions to the right. If the body’s location as a function of time is s ( t ), then its average velocity over the time interval from t 0to t 1 is
Here, the Greek letter Δ (delta) is used, per tradition, to mean “change in”. A positive average velocity means that the position coordinate s increases over the interval in question, a negative average velocity indicates a net decrease over that interval, and an average velocity of zero means that the body ends the time interval in the same place as it began. Calculus gives the means to define an instantaneous velocity, a measure of a body’s speed and direction of movement at a single moment of time, rather than over an interval. One notation for the instantaneous velocity is to replace Δ with the symbol d, for example,
This denotes that the instantaneous velocity is the derivative of the position with respect to time. It can roughly be thought of as the ratio between an infinitesimally small change in position d s to the infinitesimally small time interval d t over which it occurs. More carefully, the velocity and all other derivatives can be defined using the concept of a limit. A function f ( t ) has a limit of L at a given input value t 0 if the difference between f and L can be made arbitrarily small by choosing an input sufficiently close to t 0. One writes,
Instantaneous velocity can be defined as the limit of the average velocity as the time interval shrinks to zero:
Acceleration is to velocity as velocity is to position: it is the derivative of the velocity with respect to time. Acceleration can likewise be defined as a limit:
Consequently, the acceleration is the second derivative of position, often written
Position, when thought of as a displacement from an origin point, is a vector: a quantity with both magnitude and direction. Velocity and acceleration are vector quantities as well. The mathematical tools of vector algebra provide the means to describe motion in two, three or more dimensions. Vectors are often denoted with an arrow, as in
, or in bold typeface, such as s. Often, vectors are represented visually as arrows, with the direction of the vector being the direction of the arrow, and the magnitude of the vector indicated by the length of the arrow. Numerically, a vector can be represented as a list; for example, a body’s velocity vector might be
, indicating that it is moving at 3 metres per second along a horizontal axis and 4 metres per second along the vertical axis. The same motion described in a different coordinate system will be represented by different numbers, and vector algebra can be used to translate between these alternatives.
The physics concept of force makes quantitative the everyday idea of a push or a pull. Forces in Newtonian mechanics are often due to strings and ropes, friction, muscle effort, gravity, and so forth. Like displacement, velocity, and acceleration, force is a vector quantity.
Laws
First law
Newton’s first law reads,
Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it.
Newton’s first law expresses the principle of inertia: the natural behavior of a body is to move in a straight line at constant speed. In the absence of outside influences, a body’s motion preserves the status quo.
The modern understanding of Newton’s first law is that no inertial observer is privileged over any other. The concept of an inertial observer makes quantitative the everyday idea of feeling no effects of motion. For example, a person standing on the ground watching a train go past is an inertial observer. If the observer on the ground sees the train moving smoothly in a straight line at a constant speed, then a passenger sitting on the train will also be an inertial observer: the train passenger feels no motion. The principle expressed by Newton’s first law is that there is no way to say which inertial observer is “really” moving and which is “really” standing still. One observer’s state of rest is another observer’s state of uniform motion in a straight line, and no experiment can deem either point of view to be correct or incorrect. There is no absolute standard of rest.
Second law
The change of motion of an object is proportional to the force impressed; and is made in the direction of the straight line in which the force is impressed.
By “motion”, Newton meant the quantity now called momentum, which depends upon the amount of matter contained in a body, the speed at which that body is moving, and the direction in which it is moving. In modern notation, the momentum of a body is the product of its mass and its velocity:
Newton’s second law, in modern form, states that the time derivative of the momentum is the force:
If the mass m does not change with time, then the derivative acts only upon the velocity, and so the force equals the product of the mass and the time derivative of the velocity, which is the acceleration:
As the acceleration is the second derivative of position with respect to time, this can also be written
The forces acting on a body add as vectors, and so the total force on a body depends upon both the magnitudes and the directions of the individual forces. When the net force on a body is equal to zero, then by Newton’s second law, the body does not accelerate, and it is said to be in mechanical equilibrium. A state of mechanical equilibrium is stable if, when the position of the body is changed slightly, the body remains near that equilibrium. Otherwise, the equilibrium is unstable.
A common visual representation of forces acting in concert is the free body diagram, which schematically portrays a body of interest and the forces applied to it by outside influences. For example, a free body diagram of a block sitting upon an inclined plane can illustrate the combination of gravitational force, “normal” force, friction, and string tension
Newton’s second law is sometimes presented as a definition of force, i.e., a force is that which exists when an inertial observer sees a body accelerating. In order for this to be more than a tautology — acceleration implies force, force implies acceleration — some other statement about force must also be made. For example, an equation detailing the force might be specified, like Newton’s law of universal gravitation. By inserting such an expression for
into Newton’s second law, an equation with predictive power can be written. Newton’s second law has also been regarded as setting out a research program for physics, establishing that important goals of the subject are to identify the forces present in nature and to catalogue the constituents of matter.
Third law
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.
Overly brief paraphrases of the third law, like “action equals reaction” might have caused confusion among generations of students: the “action” and “reaction” apply to different bodies. For example, consider a book at rest on a table. The Earth’s gravity pulls down upon the book. The “reaction” to that “action” is not the support force from the table holding up the book, but the gravitational pull of the book acting on the Earth.[note 8]
Newton’s third law relates to a more fundamental principle, the conservation of momentum. The latter remains true even in cases where Newton’s statement does not, for instance when force fields as well as material bodies carry momentum, and when momentum is defined properly, in quantum mechanics as well. In Newtonian mechanics, if two bodies have momenta
and
respectively, then the total momentum of the pair is
, and the rate of change of
is
By Newton’s second law, the first term is the total force upon the first body, and the second term is the total force upon the second body. If the two bodies are isolated from outside influences, the only force upon the first body can be that from the second, and vice versa. By Newton’s third law, these forces have equal magnitude but opposite direction, so they cancel when added, and p → is constant. Alternatively, if p is known to be constant, it follows that the forces have equal magnitude and opposite direction.
