Hooke's Law
Introduction
The purpose of this lab is to test if springs individually and collectively obey Hooke's law. Hooke’s law
states that a stretched or compressed spring exerts a force proportional to the distance it is stretched or
compressed:
Fs = -k x
Fs = spring force
k = spring constant
x = displacement from the equilibrium position
If a mass is attached to a spring and it oscillates vertically, then Newton’s second law for the hanging
mass is the following:
ΣF = ma
-mg + -kx = ma
To predict the motion of the mass from first principles, we would need to solve this “differential
equation” which requires techniques beyond the math required for this course. We will simply use the
solution from the textbook which includes the formula for the period as follows:
T  2
m
k
[1]
Further analysis that takes into account the mass of the spring estimates the period as follows:
T  2
m
ms
3
[2]
k
m = hanging mass
ms = mass of spring
Experimental Procedures
Hooke’s law for individual springs
1) Select a spring from those provided.
2) Suspend five different masses or combinations of masses from the end of the spring and measure
the distance that it stretches. It is not important how you measure the stretch as long as you
consistently apply your method. The masses must be sufficient to stretch the spring at least a few
centimeters, but not able to damage the spring. Calculate the weights of the hanging masses (F =
mg). Be sure to use SI units (kg and newtons). Save and label this spring for later use.
3) Plot the results using Excel with the force in newtons on the vertical axis and the displacement in
meters on the horizontal axis. Plot the error bars as estimated for displacement. You may ignore
error bars for force since the error will be small. Add a trend line and display both the equation and
R2. Review and obey the general lab instructions regarding graph formats.
4) The slope of the equation given by Excel is the estimate of the spring constant, k, in N/m. You do
not need to calculate the error in the spring constant. An R2 value close to 1 indicates that Hooke’s
law should be deemed true for this spring for the range of applied forces. Show your plot to your
instructor and discuss your conclusion.
5) Select another spring and repeat steps 2 through 4.
Hooke’s law for springs in series
6) Attach the two springs used previously in series. Theory predicts that the total spring constant will
be:
ktotal = k1k2/(k1+k2)
7) Repeat steps 2 through 4 for the combined system. Be sure that each spring is stretched, but not
overstretched. How does the predicted spring constant compare to the value derived by the graph
in Excel? Calculate an encountered error = 100% * |theory – experiment|/theory. You do not
need to obtain an estimated error for any of the k values.
Period of oscillation
8) Attach a measured mass to an individual spring and displace it from equilibrium. You may need to
use tape to prevent the mass from becoming airborne. Measure the initial displacement. Measure
the time it takes to complete 10 oscillations. Calculate the experimental period (the measured time
divided by the number of oscillations) and compare it to the theoretical period (use equation [1] or
[2]). You do not need to calculate error in the theoretical period since you do not have an estimated
error in the spring constant.
9) Repeat step 8 several times. You can vary the mass, spring, or initial displacement. Which variables
affect the period?
Hooke’s law for rubber bands
10) Repeat steps 1 to 5, 8, and 9 with a rubber band.
Note: Do not use the formula k = mg/x to calculate the spring constant during any portion of this experiment.