Force on a Spring, Hooke's Law

 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

 1) Question a) 0.17J N b) -0.58J c) -0.1 d) e) None of the above.

 by Luis F. Sáez, Ph. D. Comments and Suggestions: LSaez@dallaswinwin.com