To derive the equation of motion, we express Newton's law of motion
for each particle (string segment) solely in terms of the solution
qy(x,t) and constants specified in the problem. We therefore
consider the free-body diagram for a single individual segment, which
we draw in Figure 2 for the segment
of string between positions x and 
.
Figure 2: Free-body diagram for a segment of string
Only two forces act on a given string segment, the tensions from the
segments neighboring to the left and eight: no significant long-range
forces act
and the only other contact with the string is
with the surrounding air.
These tension forces act along the direction tangent to the string.
Anticipating coming developments, the figure brakes the tension forces
into x and y components. The tensions on either side of the
segment need not be equal, and so we further identify the components of the
tension as being measured either at point x ( 
and 
)
or at point ( 
and 
).
Newton's Law for the states
where we use the version of Newton's law which applies to finite
bodies so that we need not assume that the segment is a small point.
In this version of Newton's law, we need only consider external forces
acting on the system, namely the tensions acting on either end, 
is the acceleration of the center of mass of
the segment, and 
is the mass of the entire segment. Because
the mass per unit length of the string is 
and the length of the
segment is 
, we
have 
.
