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If a net external force acts on a body, the body accelerates. The direction of acceleration is the same as the direction of the net force. The net force vector is equal to the mass of the body times the acceleration of the body.——If a net external force acts on a body, the body accelerates. The direction of acceleration is the same as the direction of the net force. The net force vector is equal to the mass of the body times the acceleration of the body.——If the resultant force acting on a particle is not zero, the particle will have an acceleration proportional to the magnitude of the resultant and in the direction of this resultant force.——The force acting on a body is proportional to the product of the mass and the acceleration in the direction of the force.
Equations
Nomenclature
Details
If a particle of a particular mass is acted on by a force 
F |
1, the particle with respond with a corresponding acceleration a1. If the same particle is instead acted on by a different force
F |
2 of magnitude F2, the mass will respond again with a corresponding acceleration a2. It turns out that for any magnitude of force acting on the particle, dividing by its corresponding acceleration will equal the quotient of any other magnitude of force and its corresponding acceleration to that particle. This quotient also equals that particle's mass. Consider the following relationship:
The constant value obtained for the ratio of the magnitudes of the forces and accelerations is a characteristic of the particle under consideration, which is the mass m of the particle. Whe a particle of mass m is acted upon by a force
F |
, the force
F |
and the acceleration
a |
of the particle must therefore satisfy the relation:
| (Eq1) |
F |
= m
a |
|
This relation provides a complete formulation of Newton's second law; it expresses not only that the magnitudes of
F |
and
a |
are proportional but also (since m is a positive scalar) that the vectors
F |
and
a |
have the same direction. It should be noted that Eq1 still holds when
F |
is not constant but varies with time in magnitude or direction. The magnitudes of
F |
and
a |
remain proportional, and the two vectors have the same direction at any instant. However, they will not, in general, be tangent to the path of the particle.
When a particle is subjected simultaneously to several forces, Eq1 should be replaced by:
| (Eq2) | Σ
F |
= m
a |
|
where Σ
F |
represents the sum, or resultant, of all the forces acting on the particle.
It should be noted that the system of axes with respect to which the acceleration
a |
is determined is not arbitrary. These axes must have a constant orientation with respect to the stars, and their origin must either be attached to the sun (more accurately, to the mass center of the solar system) or move with a constant velocity with respect to the sun. Such a system of axes is called a newtonian frame of reference. Since stars are not actually fixed, a more rigorous definition of a newtonian frame of reference (also called an inertial system) is one with respect to which Eq2 holds. A system of axes attached to the earth does not constitute a newtonian frame of reference, since the earth rotates with respect to the stars and is accelerated with respect to the sun. However, in most engineering applications, the acceleration
a |
can be determined with respect to axes attached to the earth and Eq1 and Eq2 used without any appreciable error. On the other hand, these equations do not hold if
a |
represents a relative acceleration measured with respect to moving axes, such as axes attached to an accelerated car or to a rotating piece of machinery.
It is observed that if the resultant Σ
F |
of the forces acting on the particle is zero, it follows from Eq2 that the acceleration
a |
of the particle is also zero. If the particle is initially at rest (
v |
0 = 0) with respect to the newtonian frame of reference used, it will thus remain at rest (
v |
0 = 0). If originally moving with a velocity
v |
0, the particle will maintain a constant velocity
v |
=
v |
0; that is, it will move with the constant speed v0 in a straight line. This is the statement of Newton's first law. Thus, Newton's first law is a particluar case of Newton's second law and can be omitted from the fundamental principles of mechanics.