Newton’s Second Law of Motion
When a force acts upon a body, it imparts an acceleration proportional to the force and inversely proportional to the mass of the body and in the direction of the force.
This law can be formulated in many different ways, we’ll cite here a few other formulations to help in understanding it.
The acceleration of an object of constant mass is proportional to the net force acting upon it.
If an unbalanced force is acting upon an object this object accelerates. The acceleration is proportional to the net force causing it and has the direction of that force.
From the last formulation we deduce that unbalanced force means the same thing as net force. Acceleration caused by the acting force always has the direction of that force and this is obvious from the formula defining Newton’s Second Law of Motion.
If on both sides of equal sign there are vector quantities, as in equation (1), these two vectors must have the same direction.
In paragraph Force in Physics we defined force qualitatively as
“something” which can change the state of motion.
Here we will define force quantitatively. To do this we must first explain the term “mass”. In a dictionary we will find, among other definitions, “the amount of matter in an object”. Such definition is not precise enough for a physicist. But, happily enough, Newton’s Second Law of Motion will define mass precisely.
In this law we deal with force 
, mass
m, and acceleration 
, which are related by
equation
(1)
We wrote quantities in this equation in such order on purpose to exactly follow the text of Newton’s Second Law given at the beginning of this paragraph:
ACCELERATION is proportional to FORCE and inversely proportional to MASS.
Any algebraic rearrangement of equation (1) is allowed, and very often it is written in the form
(2)
which consequently should be read:
The force acting on a mass m is proportional to the acceleration a caused by this force. The coefficient of proportionality is a property of the object, called mass. In other words, the larger the acceleration of a given object with constant mass, the larger is the force acting on this object. The order in which acceleration, force, and mass appear in equation (1) is more convincing. It is a force that causes acceleration not vice versa.
The Newton’s Second Law of Motion can also be considered as a definition of mass.
In most textbook on physics vector quantities are written without arrows above the symbols, but with bold face letters. The equation defining Newton’s Second Law of Motion will look like
a = F/m (1a)
or
F = ma (2a)
From time to time we will also use this notation of vectors to remind you that they represent exactly the same vector quantities.
Units of Force
From formula (2) we can find
[F] = [m] [a]
[F] = kg m/s2 (3)
The unit force in the SI system is force needed to accelerate one kilogram of mass by one meter per square second. It is called a
newton and the symbol used for it is N
The Newton as a unit of force is still not very common in everyday life. More popular are:
1kgf – one kilogram of force,
unit of force equal to the gravitational force on a mass of one kilogram.
1 lbf – one pound of force,
unit of force equal to the gravitational force on a mass of one pound.
There are many other units of force, all derived from the formula (2) describing Newton’s Second Law of Motion, but these three listed above are the mostly used in science and in everyday life. In this Tutorial we will mainly use newton as a unit of force as it is recommended officially for any scientific application. It also slowly coming into use in everyday life.
Comprehensive understanding of Newton’s Second Law of Motion will come after solving numerous problems located at the end of this Chapter. As mentioned in the Introduction to this tutorial we add-on to it on an every day basis, so nearly each time you visit it you will find more paragraphs, and more problems which are solved and carefully explained.
The formulation of Newton’s Second law of Motion given at the beginning of this paragraph does not state that the mass of the body must be constant during the acceleration. Nevertheless we defined and explained such a case, where the mass of an object in question is constant. This is a simplest case for introducing Newton’s Second Law of Motion. There will be numerous problems given for such situation.
But…. what about a rocket launched from the surface of Earth? It accelerates due to the force produced by its engine, but at the same time its mass decreases because the fuel which constitutes a large part of its mass is burned at a very high rate.
This situation requires a more general formulation of Newton’s Second law of Motion which will be the subject of next paragraph, but most of the problems will be devoted to “constant mass conditions” because they are more common in everyday life.