Mathematics of Physics and Engineering

40 Kinematics and Dynamics of a Point Mass

University.

Newton's First Law: Unless acted upon by a force, a point mass is either
not moving or moves in a straight line with constant speed. This law is also
called the Law of Inertia or Galileo's Principle.
Newton's Second Law: The acceleration of the point mass is directly
proportional to the net force exerted and inversely proportional to the mass.
Newton's Third Law: For every action, there is an equal and opposite
reaction.

Mathematically, the Second Law is

o9r(t) _ F

dt^2 m

where r = r(t) is the position of the point mass at time t.

EXERCISE 2.1.1. c Show that the Second Law implies the First Law. In
other words, show that if (2.1.1) holds, then the point mass m acted on by
zero external force will move with constant velocity v. Hint: find the general
solution of equation (2.1.1) when F = 0.

The notion of momentum provides an alternative formulation of New-
ton's Second Law. Consider a point mass m moving along the path r — r(t)
with velocity r(t) relative to a reference frame with origin at O. The (linear)
momentum p is the vector

p = mr. (2.1-2)

If the reference frame is inertial, then (2.1.1) becomes

p = F, (2.1.3)

and the force is now interpreted as the rate of change of the linear momen-
tum. Incidentally, the Latin word momentum means "motion" or "cause of
motion." One advantage of (2.1.3) over (2.1.1) is the possibility of variable
mass.
Similarly, the study of circular motion suggest the definition of the
angular momentum about the point O as the vector

Lo = mr x r. (2-1-4)

Note that both p and Lo depend on the reference point O, but do not
depend on the coordinate system.

(2.1.1)