Second law of Newton

Newton’s Second Law, also known as the Fundamental Law of Dynamics , is the one that determines a proportional relationship between force and variation of the amount of movement or linear momentum of a body. In other words, force is directly proportional to the mass and acceleration of a body.


The Newton ‘s First Law tells us that for a body to alter its movement there must be something that causes the change. That something is what we know as forces. These are the result of the action of some bodies on others.

Newton’s Second Law is responsible for quantifying the concept of force. It tells us that the net force applied to a body is proportional to the acceleration that the body acquires. The constant of proportionality is the mass of the body, so that we can express the relationship as follows:

F = ma

This law explains what happens if a net force acts on a body in motion (whose mass does not have to be constant): the force will modify the state of motion, changing the speed in module or direction. Specifically, the changes experienced in the amount of movement of a body are proportional to the driving force and develop in its direction; that is, forces are causes that cause accelerations in the bodies.

Example: If a moving train carriage with a load suddenly stops on its tracks, because it ran into an obstacle, its load tends to keep moving with the same speed and direction that it had at the time of the crash.

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 said body, that is,

F = dp / dt

In this way we also include the case of bodies whose mass is not constant. In the case that the mass is constant, remembering the definition of momentum and how a product is derived we have: F = d (m • v) / dt = m • dv / dt + dm / dt • v As the mass is constant

dm / dt = 0

and remembering the definition of acceleration, we have

F = ma

as previously seen.

Another consequence of expressing Newton’s Second Law using the quantity of movement is what is known as Principle of conservation of the quantity of movement. If the total force acting on a body is zero, Newton’s Second Law tells us that:

0 = dp / dt

That is, the derivative of momentum with respect to time is zero. This means that the momentum must be constant over time (the derivative of a constant is zero). This is the Principle of conservation of the amount of movement: if the total force acting on a body is zero, the amount of movement of the body remains constant over time.


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