What force is necessary to
stretch
or
compress
a spring? is that force constant? The answers to these questions can be
obtained experimentally. Robert Hooke, 1635-1703, found the answers to
the previous question when establishing what is known today as Hooke's law.
The force necessary to stretch a spring is proportional to the
elongation of the spring; at the same time, the force necessary to
compress the spring is proportional to the shortening of the spring
length. These results are valid only up to a limit, called the
elastic limit, which when
exceed, the spring extends or shortens in a non-linear form.

The graph on the left shows a straight line
representing the linear behavior of a spring together with the curve
representing the actual experimental behavior of the spring. The
experimental and the actual behaviors coincide only in the elastic
region of the graph. In the elastic region, the force necessary to
stretch the spring is proportional to the
elongation of the spring; and, it is given by
where
represent the
elongation (or compression) and
is the spring
constant. The spring constant is a proportionality constant between the
elongation and the force needed to achieve the desire elongation (or
compression). This constant depend on the manufactured characteristics
of the spring such as length, diameter, thickness, and material.

The elongation of a spring is measured from the
un-stretched natural length of the spring,
. The spring constant of an
ideal spring is studied interactively at the
Spring Constant section of these
notes.

The diagram on the left shows a spring of natural
length,
, and
spring constant
, .
In this case, the origin of coordinates is at the left most point of the
spring and the right most point of the spring is at the coordinate
. Suppose that a sufficiently strong force to stretch the spring is
applied to it, when the right most point of the spring is at the
position
, the necessary force
to maintain the spring at that position is
,
.
This just mentioned statement is an experimental result.

Thus, the spring exerts a force opposite to the external force,
,
.
In the case of elongating the spring,
and
the external force is positive (toward the right) and the spring
force is negative (toward the left). On the other side, if the spring
is compressed, the external force points toward the left,
,
and it is given again by
.
Similarly, when compressed, the direction of the spring force is toward
the right,
,
and the vectorial expression that represent the force at this point is
again
.

The previous relations relating the forces and the stretching
or compressing of the spring can be simplified when the origin of coordinates is
selected to coincide with the end of the spring where the external force is
applied. In this case,
and the
external force as well as the spring force are given by
and
respectively.

The energy stored in a spring can be defined similarly than the energy stored by
an object under the influence of gravity. In this case, the definition of the energy stored in a spring
can be equated to the work done on the
spring by an external force pulling or pushing on it to a given elongation (or
shortening). The presiding statement implicitly assume that the zero level of
elastic potential energy has been selected at the point where the spring is
completely relaxed. The work so calculated is the same as the work that eventually the spring can
deliver on other objects when dissipative forces, such as friction, are ignored.
Thus, the work done in stretching or compressing the spring can be
calculated following the technique to obtain the
work done by a force using the area under the curve of the
applied force versus the displacement.

If the analysis of the required force to elongate (or
shorten) a spring is maintained inside its elastic region
and, at the same time, the origin of coordinates is selected at
the point of the spring where the external force is applied, the value of the
external force is given by
. Then,
the stretching of the spring can be represented in a graph of the force
versus the displacement as shown in the graph on the left,
.
In this graph, when the elongation of the spring reaches the
displacement , the
necessary force to hold the spring at this point is
. The external force
have been increasing from zero, when the spring was not stretched to the
previous value accordingly with the rule governed by a straight
line with y-axis intercept at zero and slope
. The area under the
curve is represented at
.
This area corresponds to the area of a triangle with base
and height
. Calculating the
area of a triangle accordingly with the geometry formula:
,
,
the work done by the external force in elongating the spring from
to
is
.

This is also the elastic potential energy stored on
the spring,

When the origin of coordinates is shifted back to the left of the spring such
that the undistorted right most point of the spring (point where the force is
applied), the graph used for calculating the area under the curve can be seeing
in the following diagram,
.
As shown in this diagram, the necessary external force at
is
(notices that
at the force
is zero as expected) which lead to the triangle area
. Thus, in
general terms, the potential energy stored on a spring

The previous relation identified the potential energy stored on the spring with
the work done by the external force. Moreover, the work done by the spring force
is the negative of the work done by the external force,
; and,
consequently, the energy stored on the spring (elastic potential energy) can be
calculated as