We will begin our discussion with an investigation of the forces exerted by a spring on a hanging mass. Consider the system shown at the right with a spring attached to a support. The spring hangs in a relaxed, unstretched position. If you were to hold the bottom of the spring and pull downward, the spring would stretch. If you were to pull with just a little force, the spring would stretch just a little bit. And if you were to pull with a much greater force, the spring would stretch a much greater extent. Exactly what is the quantitative relationship between the amount of pulling force and the amount of stretch?
To determine this quantitative relationship between the amount of force and the amount of stretch, objects of known mass could be attached to the spring. For each object which is added, the amount of stretch could be measured. The force which is applied in each instance would be the weight of the object. A regression analysis of the force-stretch data could be performed in order to determine the quantitative relationship between the force and the amount of stretch. The data table below shows some representative data for such an experiment.
By plotting the force-stretch data and performing a linear regression analysis, the quantitative relationship or equation can be determined.Â
A linear regression analysis yields the following statistics:
slope = 0.00406 m/N
y-intercept = 3.43 x10-5 (pert near close to 0.000)
regression constant = 0.999
The equation for this line is
Stretch = 0.00406•Force + 3.43x10-5
The fact that the regression constant is very close to 1.000 indicates that there is a strong fitbetween the equation and the data points. This strong fit lends credibility to the results of the experiment.
This relationship between the force applied to a spring and the amount of stretch was first discovered in 1678 by English scientist Robert Hooke. As Hooke put it:Â Ut tensio, sic vis. Translated from Latin, this means "As the extension, so the force." In other words, the amount that the spring extends is proportional to the amount of force with which it pulls. If we had completed this study about 350 years ago (and if we knew some Latin), we would be famous! Today this quantitative relationship between force and stretch is referred to as Hooke's law and is often reported in textbooks as
Fspring = -k•x
where Fspring is the force exerted upon the spring, x is the amount that the spring stretches relative to its relaxed position, and k is the proportionality constant, often referred to as the spring constant. The spring constant is a positive constant whose value is dependent upon the spring which is being studied. A stiff spring would have a high spring constant. This is to say that it would take a relatively large amount of force to cause a little displacement. The units on the spring constant are Newton/meter (N/m). The negative sign in the above equation is an indication that the direction that the spring stretches is opposite the direction of the force which the spring exerts. For instance, when the spring was stretched below its relaxed position, x is downward. The spring responds to this stretching by exerting an upward force. The x and the F are in opposite directions. A final comment regarding this equation is that it works for a spring which is stretched vertically and for a spring is stretched horizontally (such as the one to be discussed below).
Force Analysis of a Mass on a Spring
Earlier in this lesson we learned that an object that is vibrating is acted upon by a restoring force. The restoring force causes the vibrating object to slow down as it moves away from the equilibrium position and to speed up as it approaches the equilibrium position. It is this restoring force which is responsible for the vibration. So what is the restoring force for a mass on a spring?
We will begin our discussion of this question by considering the system in the diagram below.
