3.1    Newton's Laws

Newton's three laws of motion, and his law of gravitation, are the foundation upon which the rest of this chapter is built.  They may be stated as follows:

The First Law of Motion:  A body at rest will remain at rest, and a body in motion will continue in motion with constant velocity, unless it is compelled to change by forces acting upon it.

The Second Law of Motion:  The net force acting upon a body is equal to the product of its mass and acceleration; the direction of the force is the same as that of the acceleration.  In equation form:

[math]

(3.1)

where:

 F  is the net force acting upon the body.

 m  is the mass of the body.

 R_dot_dot  is the acceleration of the body in inertial space.

Figure 3.1

Figure 3.1:  Second Law of Motion.

The Third Law of Motion:  The mutual actions of two bodies upon each other are always equal and opposite.

The Law of Gravitation:  Every body in the universe is attracted to every other body by a force directly proportional to each of their masses and inversely proportional to the square of the distance between them.  The net gravitational force exerted on a body is the sum of the gravitational forces exerted on it by every other body.  In equation form:

[math]

(3.2)

where:

 Fk  is the net gravitational force acting upon body k.

 G  is the gravitational constant:  6.67 × 10-11 N·m2/kg2.

 mj, mk  are the masses of bodies j, k.

 Rj, Rk  are the positions of bodies j, k.

 Rjk  is the absolute distance between bodies j, k[math].

 [math]  is a unit vector directed from k toward j.

Figure 3.2

Figure 3.2:  Law of Gravitation.

The "bodies" referred to by these four laws are actually infinitesimal particles of mass.  Larger bodies must be treated as the sums of their constituent particles.

It can be shown (by integrating equation 3.2 with respect to R over the volume of a sphere) that the gravitational attraction exerted on or by a homogeneous sphere is the same as if its entire mass were concentrated at its center.  This is true for bodies that are both homogeneous and spherical, and is approximately true for the Earth.  In general, a body's center of gravity is not the same as its center of mass, because the intensity of a gravitational field varies with distance squared.  Homogeneous spheres represent the exception, not the rule.

The second law of motion and the law of gravitation show the equality of inertial and gravitational mass:  the force required to accelerate a body, and the gravitational force that compels it to accelerate, are both directly proportional to its mass.  Setting equations 3.1 and 3.2 equal to each other and dividing by mass:

[math]

(3.3)

The acceleration of a body under the influence of gravity is independent of its mass; its acceleration depends only on the masses and relative positions of other bodies.  Furthermore, in common human experience, the gravitational attraction of all bodies other than the Earth is insignificant.  Substituting the mass and radius (distance to the center) of the Earth, and ignoring all other bodies, gives the acceleration due to gravity at the Earth's surface.  Its magnitude is:

[math]

(3.4)

and its direction is downward, toward the center of the Earth.

The motion of a body can be measured only by reference to some other body.  For example: the motion of a car can be measured relative to the Earth; the motion of the Earth can be measured relative to the sun; and so on.  In applying Newton's laws, the choice of an appropriate reference is essential.  An "inertial" reference system is defined, somewhat reflexively, as one in which Newton's laws correctly describe the motion of a body.  A reference that moves with constant velocity relative to an inertial system is also inertial; a reference that accelerates relative to an inertial system is not.  The choice of reference system depends on the scale of the problem at hand.  In everyday Earthly life, the Earth itself serves as a suitable reference.  When launching an object into space, a reference system that accounts for the Earth's rotation - and perhaps even its orbital motion - must be used instead.