PHYSICS EXPERIMENTS —132
2-1
Experiment 2
Simple Harmonic Motion of a Mass Suspended from a Spring
In this experiment you investigate the behavior
of a simple physical system consisting of a mass
hanging on the end of a spring.
Although
mechanically simple, this system is important
because it exhibits repetitive motion. The mass
oscillates! The frequency of oscillation depends on
physical properties of the system; the masses in
motion and the spring stiffness. You determine
whether Hooke’s Law describing an ideal spring
applies to your system and determine the spring
constant of your spring by both static and dynamic
means.
Preliminaries.
Part A. Static Determination of Spring Constant.
A spring with spring constant k hangs vertically
with a mass m attached to the lower end, as in
Figure 1.
msp
k
Figure 1. Hanging a mass on a spring
An external force F applied to a spring stretches the
spring a distance x. Hooke’s Law states that, for an
ideal spring, the applied force and stretch are
linearly related
F = kx
(eq. 1)
where k is the spring constant, a measure of spring
stiffness. The stretched spring exerts an (equal and
opposite) elastic restoring force, Fs = -kx, in the
opposite direction. If the mass is stationary, in
equilibrium, the elastic restoring force must balance
the mass’s weight,
mg = kx
(eq. 2)
where g is the acceleration due to gravity.
Part B. Dynamic Determination of Spring
Constant.
If the mass in the system of Figure 1 is pulled
away from its equilibrium position and then
released, it will oscillate up and down about its
equilibrium position. Motion in which the restoring
force is proportional to the displacement, but
oppositely directed, is called simple harmonic
motion.
In this experiment, the mass of the spring msp is
comparable to that of the hanging mass m. This
spring mass is usually ignored in model mass-spring
systems discussed in textbooks. Realistically, the
spring oscillates along with the hanging mass in a
complicated way. Only the lowest part of the spring
oscillates with the amplitude of the attached mass,
while coils near the top oscillate a very small
amplitude. For this reason, the effective mass of the
system is increased by some fraction  of the spring
mass msp.
(The fraction  is theoretically
determined to be one third ( = 1/3) for an ideal
spring.)
The period of oscillation, T, is theoretically
related to mass-spring system parameters by
T = 2p
(m + bmsp )
k
(eq. 3)
Squaring eq. 3 gives
æ 4p 2 ö
4p 2 bmsp
T =ç
÷m +
k
è k ø
2
(eq.4)
PHYSICS EXPERIMENTS — 132
Eq. 4 is the equation of a straight line in standard
slope-intercept form on a graph of "period squared"
vs. "hanging mass."
In this experiment, the period is measured for
different hanging masses. The graph of "period
squared" vs. "hanging mass" is constructed, and the
spring constant and mass fraction are determined
from the graph.
Procedure.
Part A. Static Determination of Spring Constant.
• Clamp a long vertical pole to the edge of the lab
table. Connect a horizontal bar near the top of the
pole and attach one end (the narrower one) of a
spring to the end of the horizontal bar. The spring
should hang with its narrower end on top so that, as
it stretches, the coil spacing is uniform in width.
• Hang at least five different weights on the spring
Measure the amount of spring stretch for each.
Make certain not to damage the spring by
stretching it beyond its elastic limit.
• Make a graph with the "spring stretch" on the
vertical axis and "hanging mass" on the horizontal
axis. Determine the spring constant from the graph.
Part B. Dynamic Determination of Spring
Constant.
● Measure and record the mass of the spring msp.
Attach 100 g total mass to the unattached end of
the spring, using the weights mounted on the weight
hanger. Record the total hanging mass m. Do not
forget to add the hanger mass! Pull the mass several
centimeters from equilibrium and release. Start the
timer and time five oscillations. Divide by five to
determine the period of oscillation. Repeat several
times to insure consistency and determine precision
of data. Record.
●
Add mass in 50 g increments up to about 300 g
and determine the period for each hanging mass. Be
very careful that the spring is not stretched
excessively, where it will be permanently deformed
and not able to snap back!
●
2-2
Graph the data with the "period" on the vertical
axis and "hanging mass" on the horizontal axis.
●
Make another graph with "period squared" on
the vertical axis and "hanging mass" on the
horizontal axis.
●
Calculate the spring constant and the effective
mass fraction from the graph.
●
Questions (Answer clearly and completely).
1. Does your spring obey Hooke’s Law?
2. Eq. 4 predicts a straight line if period squared
(vertical) is plotted against hanging mass
(horizontal). Derive an equation for the slope of the
line in terms of mass-spring parameters (k, msp).
Derive an equation for the vertical axis intercept in
terms of mass-spring parameters (k, msp). Derive
an equation for the horizontal axis intercept in terms
of mass-spring parameters (k, msp).
3. Is the suggested technique for determining the
oscillation period a good one? Would it be better to
time a single oscillation? Would it be better to time
a hundred oscillations and divide by a hundred?
4. What value do you determine for the spring
constant statically? What value do you determine
for the spring constant dynamically? These are
supposed to be the same. What is the percent
difference between these values?
5. What value do you determine experimentally for
the effective mass fraction of the spring, ? What is
the percent difference between this value and the
theoretical result of 1/3?
6. How would the period squared vs mass graph be
different from the one you plotted if the spring were
stiffer?
7. How would the period squared vs mass graph be
different from the one you plotted if the spring were
more massive?
rev. 8/13