Newton’s Laws

Newton’s Laws , also known as Newton’s Laws of Motion or Laws of Dynamics , are three principles from which most of the problems posed by dynamics are explained , in particular that related to the movement of bodies, that is The movement of the bodies was explained as well as its effects and causes. Newton’s Laws allow us to explain both the movement of the stars and the movements of artificial projectiles created by humans, as well as all the mechanics of machine operation.


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  • 1 History
  • 2 Theoretical foundations of the laws
  • 3 The laws
    • 1 Newton’s first law or law of inertia
      • 1.1 Example
    • 2 Newton’s second law or law of force
    • 3 Newton’s third law or law of action and reaction
      • 3.1 Example
    • 4 Limits of validity of Newton’s laws
    • 5 See also
    • 6 References
    • 7 Sources
    • 8 Bibliography


The mathematical formulation was published by Isaac Newton in 1687 , in his Philosophiae Naturalis Principia Mathematica . Newton’s laws form, together with Galileo’s transformation , the basis of classical mechanics. In the third volume of the Principia Newton showed that, combining these laws with his Law of universal gravitation , Kepler’s Laws on planetary motion can be deduced and explained .

Theoretical foundations of the laws

Newton stated that all movements abide by three main laws formulated in mathematical terms and involving concepts that must first be rigorously defined. Newton’s first concept is that of mass, which identifies with “quantity of matter” another concept is force, cause of motion; both are usually named by the letters F and m .


  • Cause of movement ( F).


  • Measurement of the amount of matter put into motion ( m).

Newton then assumes that the amount of motion is the result of the product of mass times velocity. Third, it points out the importance of distinguishing between the absolute and the relative whenever we speak of time, space, place or movement.

In this sense, Newton, who understands movement as a translation of a body from one place to another, to arrive at the absolute and true movement of a body composes the (relative) movement of that body in the (relative) place in which it is he considers it, with the (relative) movement of the place itself in another place in which it is located, and so on, step by step, until reaching an immobile place, that is, the reference system of absolute movements.

Accordingly, Newton states that apparent movements are the differences from true movements and that forces are causes and effects of these. Consequently, force in Newton is absolute, not relative.


Newton’s first law or law of inertia

Every body continues in its state of rest, or of uniform movement in a straight line, unless it is forced to change that state by forces applied to it. [one]

Movement is relative, that is, it depends on which observer describes the movement. Thus, for a passenger A sitting a train, another passenger B also sitting is at rest, while for someone who sees the train pass from the platform of a station, both passenger A and B are moving at high speed. Therefore, a reference system is needed to refer the movement to.

Newton’s first law serves to define a special type of reference systems known as inertial reference systems, which are those reference systems from which it is observed that a body on which no net force acts moves with constant speed.

In reality, it is impossible to find an inertial reference system, since there are always some kind of forces acting on the bodies, but it is always possible to find a reference system in which the problem being studied can be treated as if it were in an inertial system. In many cases, assuming a fixed observer on Earth is a good approximation of the inertial system.


A good example in which this force is appreciated, is when a bus stops abruptly, if people on the bus are not restrained they will continue their rectilinear movement (they will move forward), if on the contrary the bus is stopped and begins to move abruptly the tendency will be to maintain the state of rest (they will move backwards)

Newton’s second law or law of force

Force defines the direction in which the body moves or changes that movement. Both force and mass determine the speed with which the body changes its rest or movement: the greater the force applied and the less the body mass, the greater the speed. [one]

Second Law Newton.jpg

This law is responsible for quantifying the concept of force. In mathematical terms it is expressed through the relationship:

F = m  a

Both force and acceleration are vector quantities, that is, they have, in addition to a value, a direction and a sense. Thus, Newton’s Second Law must be expressed as:
→ →
F = m  a

The unit of force in the International System is the Newton and is represented by N . A Newton is the force that must be exerted on a body of one kilogram of mass so that it acquires an acceleration of 1 m / s 2 , that is,

1 N = 1 Kg  1 m / s 2

The expression of Newton’s Second Law that we have given is valid for bodies whose mass is constant. If the mass varies, such as a rocket burning fuel, the relation F = m • a is not valid .

We are going to generalize Newton’s Second Law to include the case of systems in which the mass can vary. For this we will first define a new physical quantity. This physical quantity is the amount of movement that is represented by the letter p and is defined as the product of the mass of a body by its speed, that is:

p = m  v

The momentum is also known as linear momentum. It is a vector magnitude and, in the International System, it is measured in Kg  m / s. In terms of this new physical magnitude, Newton’s Second Law is expressed as follows:

The Force acting on a body is equal to the temporal variation of the amount of movement of that body, that is,

F = d p / dt

In this way we also include the case of bodies whose mass is not constant

Newton’s third law or law of action and reaction

If a body A exerts a force on another B, then, body B will exert a force on A, of equal value; but in the opposite direction. [one]

Third Law Newton.gif

Mathematically Newton’s third law of motion is usually expressed as follows:

F1 = F2 ‘

where F1 is the force acting on the body 1 and F2 ‘is the reactive force acting on the body 2


When a swimmer A pushes another swimmer B in a swimming pool , both move in the opposite direction, even if the latter does not attempt to push the first swimmer . This is because the reaction the bather B makes the bather on A .

It is important to note that this principle of action and reaction relates two forces that are not applied to the same body, producing different accelerations, depending on their masses. For the rest, each of these forces obeys the second law separately, that is, although the pairs of action and reaction have the same value and opposite directions, they do not cancel each other, since they act on different bodies.

Limits of validity of Newton’s laws

The laws that constitute the bases of dynamics, are known under the name of laws of mechanical motion and were formulated in 1687 by Isaac Newton, these laws allowed to understand the behavior of mechanical phenomena and explain others such as the motion of the Earth around the Sun, the movement of pendulums, of bodies suspended by docks, such as accurately determining the movement of space vehicles and even predicting their behavior, but these laws are limited in their validity.

Newton’s first law or law of inertia

This first law is only fulfilled for an Inertial System and a material particle-point, to understand this approach it must be known that to study the movement of a body, a reference system is first analyzed. The same movement seems different if viewed from different reference systems. A system is defined as inertial if it is at rest or in Uniform Rectilinear Motion . Material point, it is the idealization of a body that we assume with mass but without occupying volume, which means assigning it an infinite density (d = m / v).

Newton’s second law or law of force

This law is only fulfilled in Inertial Systems. (In non-inertial systems the valid formula is: F + Fi = m • a), for not very small masses (that do not have quantum implications) and for small speeds v <<< c (speed of light). According to classical dynamics, a force acting on a body communicates an acceleration a = cte, but the speed increases indefinitely v = a • t. If this were so in an infinite time, the speed would be infinite, which is in disagreement with experience and is explained in relativistic mechanics that puts a limit to V = 3 • 10 8 m / s.

Newton’s third law or law of action and reaction

The forces come from an interaction and always appear two by two. Each one is applied to one of the interacting bodies (if the two were applied to the same body they would produce rest). To obtain balance, two or more interactions on a body are required for the originated forces to cancel.

The third Law is only fulfilled if the interaction time is long enough for the response to the action to be established.

When solving the dynamics problems, Newton’s laws are applied without thinking if they are valid in all cases, nor is it taken into account if the reference system in which the movement of bodies is analyzed can influence when operating with these laws. , or if the values ​​of the speeds at which the bodies move, can limit the application of them, there is even talk of bodies that are considered as a material point, which as a result of the interactions only experience variations in their translational movement


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