EXPERIMENTS PHYSICS 132
10-1
Experiment 10
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 Hookes 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
msp
k
with a mass m attached to the lower end, as in
Figure 1.
Figure 1. Hanging a mass on a spring
An external force F applied to a spring stretches the
spring a distance x. Hookes 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 masss weight,
mg = kx
(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 through only a
very small amplitude. For this reason, the effective
mass of the system is increased by some fraction f
of the spring mass msp of the spring. (The fraction f
is theoretically determined to be one third (f=1/3)
for an ideal spring.)
The period of oscillation T is theoretically related
to mass-spring system parameters by
( m + fmsp )
T = 2
(eq. 3)
k
Squaring eq. 3 gives
(eq.4)
T 2 = (4 2 / k ) m + (4 2 / k ) fmsp
Eq. 4 is the equation of a straight line on a graph of
squared period vs. hanging mass in standard slopeintercept form.
In this experiment, the period is measured for
different hanging masses. The graph of squared
period vs. hanging mass is constructed, and the
10-2
PHYSICS EXPERIMENTS 132
spring constant and mass fraction determined from
the graph.
Procedure.
Questions (Answer clearly and completely).
Part Static A. 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, it becomes uniform in width.
Hang at least five different weights on the spring
Measure the amount of spring 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 the 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 100g 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 50g increments up to about 400g
(do not exceed 400g) 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!
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 appropriate graph.
1. Does your spring obey Hookes Law?
2. Equation 4 predicts a straight line if period
squared (on the vertical) is plotted against hanging
mass (on the horizontal). What is the equation for
the slope of the line in terms of mass-spring
parameters (k, fmsp). What is the equation for the
vertical axis intercept in terms of mass-spring
parameters (k, fmsp). Derive an equation for the
horizontal axis intercept in terms of mass-spring
parameters (k, fmsp).
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? How do you
account for any differences?
5. What value do you determine experimentally for
the effective mass fraction of the spring? What is
the percent error 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?