SHM of a Spring-Mass System
Richard Lupa
Engineering Physics II
October 10, 2007
Purpose:
To verify experimentally the relation between frequency (or period), mass, and
the effective spring constant for a simple harmonic oscillator on an air track.
Equipment:
Air track, glider, triple beam balance, masses, two springs, and a stop watch
Discussion:
Any mechanical system that obeys Hooke’s law
FR = - kx
where FR = restoring force
k = spring constant
x = displacement from equilibrium
will undergo simple harmonic motion.
Simple harmonic motion is the conceptual basis for many physical phenomena
and for many engineering applications. It can be applied to the study of mechanical
vibrations, electrical oscillations, wave phenomena (including sound and electromagnetic
waves), and the vibration of atoms in molecules and crystals. Even in cases where
Hooke’s Law is not obeyed exactly, the motion is often approximately simple harmonic.
As long as the potential energy curve can be approximated by a parabola, the motion will
be essentially simple harmonic for small oscillations about the equilibrium position.
Einstein derived an important result about the specific heat of crystals using a model in
which the atoms or ions of the crystal acted as if they were connected by springs obeying
Hooke’s Law.
For this experiment, the vibrating mass will be a glider connected to two springs
gliding over an air track.
For such a system, the restoring force is given by
FR = - k x
where k is the effective spring constant. It is easy to show that
k = k1 + k2,
the sum of the spring constants of the two individual springs.
According to the theory of simple harmonic motion, the frequency of
oscillation is
k
1
f=
2 m
and the period is the reciprocal of the frequency,
m
T = 2
.
k
Procedure:
A
Determine the spring constant of each spring by timing N oscillations
of a suitable mass attached to the spring.
k = 4π2m/T2
B
where T = tN /N
Set up the glider and the two springs on the air track as shown in the
figure. Displace the glider 10 cm from its equilibrium position and let go.
Measure the time to complete N oscillations.
Then
Texp = tN / N
Compare the experimental period Texp with the theoretical period Ttheory
Ttheory = 2 
M
k
where M = mass of glider
k = k1 + k2
Repeat the same procedure for initial displacements of 20 cm and 30 cm.
Data:
Amount of mass used in part A (mA)
Mass of glider (mB)
Equilibrium position of glider
100 g
300.4 g
800 cm
.1 kg
.3004 kg
Part A:
Data of First Spring
Oscillations (Hz) Time (s)
25
25.28
25
25.78
25
25.34
Ave.
25.47
Data of Second Spring
Oscillations (Hz) Time(s)
25
28.09
25
28.91
25
28.38
Ave.
28.46
Part B:
Glider displaced 10 cm (at 900 cm mark)
Oscillations (Hz)
Time(s)
25
31.28
25
31.25
25
30.41
Ave.
30.98
Glider displaced 20 cm (at 1000 cm mark)
Oscillations (Hz)
Time(s)
25
31.43
25
31.32
25
31.19
Ave.
31.31
Glider displaced 30 cm (at 1010 cm mark)
Oscillations (Hz)
Time(s)
25
31.25
25
31.31
25
31.31
Ave.
31.29
Calculations:
Part A:
First Spring:
Period of Oscillations: T = tn/N
T = 25.47 s /25 Hz
T = 1.019 s/Hz
Spring Constant: k1 = (4π2mA)/T2
k1 = (4π2(.1 kg))/(1.019 s/Hz)2
k1= 3.802 N/m
Second Spring:
Period of Oscillations: T = tn/N
T = 28.46 s /25 Hz
T = 1.138 s/Hz
Spring Constant: k2 = (4π2mA)/T2
k2 = (4π2(.1 kg))/(1.138 s/Hz)2
k2 = 3.048 N/m
Part B:
Effective Spring Constant: k = k1 + k2
k = 3.802 N/m + 3.048 N/m
k = 6.85 N/m
mB
Theoretical Period: Ttheory = 2π
k
Ttheory = 2π
.3004kg
6.85 N m
Ttheory = 1.316
Displacement of 10 cm:
Expected Period of Oscillations: Texp = tn/N
Texp = 30.98 s /25 Hz
Texp = 1.239 s/Hz
% difference = [(1.316-1.239)/(1.277)] × 100
= 6.03% difference
Displacement of 20 cm:
Expected Period of Oscillations: Texp = tn/N
Texp = 31.31 s /25 Hz
Texp = 1.252 s/Hz
% difference = [(1.316-1.252)/(1.284)] × 100
= 4.98% difference
Displacement of 30 cm:
Expected Period of Oscillations: Texp = tn/N
Texp = 31.29 s /25 Hz
Texp = 1.252 s/Hz
% difference = [(1.316-1.252)/(1.284)] × 100
= 4.98% difference
Error analysis:
From calculating the percent difference of each procedure, it was determined that
some error had occurred during the procedure of each lab. This error could have resulted
due to the balance beam not being correctly zeroed at the start of the experiment. Also
when performing each trial a different person would use the stop watch and each person
has a different reaction time. If the case, these errors would have caused inaccurate values
of the glider’s mass and some fluctuations in each data timed. Having these errors would
be enough to offset the correct results, to a point where the experimental data is incorrect.
Conclusions:
As stated before, the purpose of the lab was to verify a relation between the
period, mass, and the spring constant using a simple harmonic oscillator. This was made
possible by utilizing the equations above. When the calculations were determined the
difference between each test was close to each others values. This in turn proves that
there is a relation between the period, mass, and the spring constant.