# Newton's Second Law of Motion

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A Newton's Second Law of Motion is a physical law that is a second-order differential equation which establishes the relationship between an object's mass its acceleration and the applied force.

**AKA:**F=ma.**Context:**- It is usually stated as follows:

*"The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object."*

- It can be expressed as

- [math]\displaystyle{ m\;\frac{d^2y}{dt^2}=F(y,y, \frac{dy}{dt}) }[/math]
- where [math]\displaystyle{ m }[/math], [math]\displaystyle{ \frac{dy}{dt}=v(t) }[/math] and [math]\displaystyle{ \frac{d^2y}{dt^2}=a(t) }[/math] are the object's mass, velocity and acceleration, repectively, and [math]\displaystyle{ F }[/math] is the applied Force

**Example(s):****Counter-Example(s):****See:**Philosophiæ Naturalis Principia Mathematica, Kepler's Laws of Planetary Motion, Classical Mechanics, Force, Motion (Physics), Elegant Model, Inertial Reference Frame, Navier-Stokes Equation.

## References

### 2015

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Newton's_laws_of_motion Retrieved:2015-8-23.
**Second law**: The vector sum of the forces**F**on an object is equal to the mass*m*of that object multiplied by the acceleration vector**a**of the object:**F**=*m***a**.

- (NASA Website, 2016) ⇒ http://www.grc.nasa.gov/www/k-12/airplane/newton.html
- The
**second law**explains how the velocity of an object changes when it is subjected to an external force. The law defines a force to be equal to change in momentum (mass times velocity) per change in time. Newton also developed the calculus of mathematics, and the "changes" expressed in the second law are most accurately defined in differential forms. (Calculus can also be used to determine the velocity and location variations experienced by an object subjected to an external force.) For an object with a constant mass m, the second law states that the force F is the product of an object's mass and its acceleration a:

- The

[math]\displaystyle{ F = m \times a }[/math]

- For an external applied force, the change in velocity depends on the mass of the object. A force will cause a change in velocity; and likewise, a change in velocity will generate a force. The equation works both ways.

### 2005

**Newton's Second Law**as stated below applies to a wide range of physical phenomena, but it is not a fundamental principle like the Conservation Laws. It is applicable only if the force is the net external force. It does not apply directly to situations where the mass is changing, either from loss or gain of material, or because the object is traveling close to the speed of light where relativistic effects must be included. It does not apply directly on the very small scale of the atom where quantum mechanics must be used.

### 1996

- (The Physics Classroom,1996) ⇒ http://www.physicsclassroom.com/class/newtlaws/Lesson-3/Newton-s-Second-Law
**Newton's second law of motion**pertains to the behavior of objects for which all existing forces are not balanced. The second law states that the acceleration of an object is dependent upon two variables - the net force acting upon the object and the mass of the object. The acceleration of an object depends directly upon the net force acting upon the object, and inversely upon the mass of the object. As the force acting upon an object is increased, the acceleration of the object is increased. As the mass of an object is increased, the acceleration of the object is decreased.

- Newton's second law of motion can be formally stated as follows:
- The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.

- This verbal statement can be expressed in equation form as follows:

- Newton's second law of motion can be formally stated as follows:

[math]\displaystyle{ a = \frac{F_{net}}{m} }[/math]

- The above equation is often rearranged to a more familiar form as shown below. The net force is equated to the product of the mass times the acceleration.

- [math]\displaystyle{ F_{net} = m \times a }[/math]
- In this entire discussion, the emphasis has been on the net force. The acceleration is directly proportional to the net force; the net force equals mass times acceleration; the acceleration in the same direction as the net force; an acceleration is produced by a net force. The NET FORCE. It is important to remember this distinction. Do not use the value of merely "any 'ole force" in the above equation. It is the net force that is related to acceleration. As discussed in an earlier lesson, the net force is the vector sum of all the forces. If all the individual forces acting upon an object are known, then the net force can be determined. If necessary, review this principle by returning to the practice questions in Lesson 2.
- Consistent with the above equation, a unit of force is equal to a unit of mass times a unit of acceleration. By substituting standard metric units for force, mass, and acceleration into the above equation, the following unit equivalency can be written.
`1 Newton = 1 kg • m/s^2`

- The definition of the standard metric unit of force is stated by the above equation. One Newton is defined as the amount of force required to give a 1-kg mass an acceleration of 1 m/s/s.

### 1963

- (Feynman et al., 1963) ⇒ Richard P. Feynman, Robert B. Leighton and Matthew Sands (1963, 1977, 2006, 2010, 2013) "The Feynman Lectures on Physics": New Millennium Edition is now available online by the California Institute of Technology, Michael A. Gottlieb, and Rudolf Pfeiffer ⇒ http://www.feynmanlectures.caltech.edu/
**Chapter 9:**(...) we shall discuss only the Second Law, which asserts that the motion of an object is changed by forces in this way: the time-rate-of-change of a quantity called momentum is proportional to the force. We shall state this mathematically shortly, but let us first explain the idea.

- Momentum is not the same as velocity. A lot of words are used in physics, and they all have precise meanings in physics, although they may not have such precise meanings in everyday language. Momentum is an example, and we must define it precisely. If we exert a certain push with our arms on an object that is light, it moves easily; if we push just as hard on another object that is much heavier in the usual sense, then it moves much less rapidly. Actually, we must change the words from “light” and “heavy” to less massive and more massive, because there is a difference to be understood between the weight of an object and its inertia (...)
- We use the term mass as a quantitative measure of inertia, and we may measure mass, for example, by swinging an object in a circle at a certain speed and measuring how much force we need to keep it in the circle. In this way we find a certain quantity of mass for every object. Now the momentum of an object is a product of two parts: its mass and its velocity. Thus Newton’s Second Law may be written mathematically this way:

[math]\displaystyle{ F=\frac{d}{dt}(mv) \quad\quad (9.1) }[/math]

- Thus at the beginning we take several things for granted. First, that the mass of an object is constant; it isn’t really, but we shall start out with the Newtonian approximation that mass is constant, the same all the time, and that, further, when we put two objects together, their masses add. (...) The most important thing to realize is that this relationship involves not only changes in the magnitude of the momentum or of the velocity but also in their direction. If the mass is constant, then Eq. (9.1) can also be written as:

[math]\displaystyle{ F=m\frac{dv}{dt}=ma.\quad\quad (9.2) }[/math]

- The acceleration
*a*is the rate of change of the velocity, and Newton’s Second Law says more than that the effect of a given force varies inversely as the mass; it says also that the direction of the change in the velocity and the direction of the force are the same.

- The acceleration