Student
January 04, 2010
Formal Lab Report
Mallok P.
Physics 12 - SP4U1
Hooke’s Law and Simple Harmonic Motion
Abstract
The spring constant of a normal spring was
determined through three ways: by
measuring its elongation when subjected to
loading various masses without deformation;
by measuring through mechanical Newton’s
meter; and by measuring the period of a
mass hung from one end and set into vertical
oscillation. The resulting values of 18N/m,
13N/m, and 18N/m, respectively, show that
the normal spring functions according to
Hooke’s Law.
The above normal spring was then used to
determine the value of kinetic friction
constant of the study table and lab bench by
letting the spring’s restoring force drags a
heavy mass across their top surfaces. The
calculated value of 0.16 and 0.31,
respectively, indicate that friction is
consistent for the two surfaces.
Introduction
A periodic motion is one that repeats itself
in successive equal intervals of time, the
time required for one complete repetition of
the motion being called its period. Imagine,
for example, a particle moving back and
forth along a straight line between two fixed
points. If the particle moves in such a way
that its acceleration is always proportional to
its displacement from the midpoint of its
path, and is always directed toward that
midpoint, then the motion is said to be
simple harmonic.
If a mass undergoes simple harmonic
motion, the force acting on it must be one
which varies in just the way that the
acceleration is to vary; that is, the mass must
be acted upon by a force that is proportional
to the displacement of the mass from its
center or equilibrium position and directed
toward that position. One can, without
undue complication, apply a force that varies
in this way, and thus cause the mass to
execute a simple harmonic motion.
Suppose, for instance, that the mass is
suspended from a spring. If, when the mass
is above or below its equilibrium position,
there exists a restoring force proportional to
the displacement from equilibrium (and,
according to Hooke's law such should be the
case), then just the type of force is present
that is required to produce simple harmonic
motion.
For a Hooke's law restoring force, the
relationship between the force and the
displacement is given by F = -kx where k is
called the force (spring) constant.
Application of such a force to a mass m
yields F = -kx = ma, which is the
mathematical statement of the condition for
simple harmonic motion discussed in the
first paragraph. When x = 0 the mass is at
the center or equilibrium position.
From centripetal force, it is shown that
ac =
4𝜋 2 𝑟
𝑇2
Since centripetal force also experiences a
periodic motion over a period of T, it can be
further shown that in any simple harmonic
motion that
T = 2π
𝑚
𝑘
(1)
1
Student
January 04, 2010
Formal Lab Report
Procedure
We placed the spring hanger near the edge
of the study table to support a normal spring
of length 48cm (before being stretched or
compressed) so it can freely oscillated at its
maximum amplitude. We performed three
different methods to find the spring constant,
k.
For the first method of find k, we suspended
a load of 100g at the end of the spring (L =
48cm long before being stretched or
compressed). Because we hooked one end
of the spring directly onto the supporter, lo is
negligible. We then measure l1 after spring
being stretched by the load. We repeated
this method for 4 other loads with an
increment of 100g each. (Table 1)
In the second method we only have to place
a mechanical Newton’s meter at the end of
the spring where the load was placed. Then
we pulled the spring from its equilibrium
position to a final position. Afterward, we
recorded the force shown on the meter and
the final position l1 in Table 2.
We
performed this method for 4 other different
forces.
Force, in N, is plotted against
elongation, in m, in the graph 1.
Mallok P.
Physics 12 - SP4U1
We performed the third method of finding k
dynamically. As such we chose a single
mass of 200g and hung it from the spring
while letting it comes to rest. Afterward, we
stretched the spring by holding on to the
mass to a final position. Then we let it
oscillate 10 times in a simple harmonic
motion.
Then we record the time of
oscillation with a digital stop watch. We
repeated the oscillation for the same final
position 3 more times for accuracy. This
method was repeated with 3 other different
positions. (Table 3)
We performed the following method in
order to find the coefficient friction of the
table and the lab bench. We placed a long
ruler on the table where distance can be
easily recorded. Then we placed the spring
in parallel to the ruler with a 1kg mass
attached to its end and stretched it to a final
position.
After we recorded the final
position we let go of the mass for it to travel
20cm, while recording its time by a digital
stop watch. We repeated this method 3
more times for the same final position. Then
the time for it to travel 20cm was averaged
out. We applied the same method on the
study table to the lab bench except the
distance traveled was 35cm and the final
position that the spring stretched to is
different.
Analysis
Load (Kg)
l1 (m)
∆L (m)
0.10
0.53
0.05
0.20
0.59
0.11
0.30
0.65
0.17
0.40
0.72
0.24
0.50
0.77
0.29
Table 1. Various load freely suspended
2
Formal Lab Report
Force (N)
l1 (m)
∆L (m)
1.0
0.55
0.07
2.0
0.63
0.15
2.5
0.66
0.18
3.0
0.74
0.26
5.0
0.86
0.38
Table 2. Newton’s Meter at the end of spring
Mallok P.
Physics 12 - SP4U1
Constant k from Newton's Meter
6
5
Force (N)
Student
January 04, 2010
y = 12.473x + 0.1056
R² = 0.9808
4
3
2
1
Distance
(m)
Time1
(sec)
Time2
(sec)
Time3
(sec)
Time
Average
(sec)
0
0
0.55
7.21
6.80
6.80
6.94
0.56
6.88
7.00
6.75
6.88
0.63
6.65
6.05
7.05
6.58
0.66
7.25
7.40
7.50
7.38
Table 3. 10 Oscillations of various masses
Distance
Traveled
(m)
Time1
(sec)
Time2
(sec)
Time3
(sec)
Time
Average
(sec)
0.20
0.260 0.265 0.235
0.253
Table 4. 1kg mass on table top with final
position at 65cm.
Distance
Traveled
(m)
Time1
(sec)
Time2
(sec)
Time3
(sec)
Time
Average
(sec)
0.35
0.45
0.46
0.41
0.44
Table 5. 1kg mass on lab bench with final
position at 75cm.
From Table 1, we used the formula F = -kx
to find the spring constant k, which averaged
out to be, kaverage = 18N/m.
From Table 2, we graphed force against
elongation to produce the slope which yield
the kaverage = 13.
0.1
0.2
0.3
0.4
Distance Stretched (m)
The mass used in calculating k was not
merely the load attached to the bottom of the
spring. The reason for this is that the system
that is vibrating includes the spring itself.
However, the entire spring does not vibrate
with the same amplitude as the load and
therefore it is reasonable to assume that the
effective load is the mass hung from the end
of the spring plus some fraction of the mass
of the spring. The fraction used was 1/3
since similar experiments with other types of
springs have led to this empirical result.
[Sears, Zemansky, and Young] It was
assumed that 1/3 of the mass was a
reasonable approximation to the correct
value for this particular spring. As such
from Table 3, the spring constant k can be
calculated using formula 1as following,
Period
T
Period
T2
k
(
𝟒𝝅𝟐 𝒎
𝑻𝟐
)
0.694
0.482
17.7
0.688
0.473
18.1
0.658
0.433
19.8
0.738
0.545
15.7
Table 6. k from 200g oscillated mass
3
Student
January 04, 2010
Formal Lab Report
Mallok P.
Physics 12 - SP4U1
Thus, from Table 6, the average spring
constant is kaverage = 18.
From Table 4, the friction force for the study
table is calculated to be Ff = -1.52N and the
kinetic friction constant is µk = 0.16.
Similarly from Table 5, Ff for the lab bench
is Ff = -3.15N and the µk = 0.31.
Discussion
Incoming.
Conclusion
Incoming.
Reference
Hirsh, A. J., Martindale, D., Stewart, C.,
Barry, M., Physics 12, 1st Edition,
Nelson, Canada (2005).
Peckham, D. C. (Fall 2005). Hooke’s Law
and a Simple Spring. Retrieved
January 10, 2010, from
http://it.stlawu.edu/~koon/classes/22
1.222/221L/SampleFormalLab.pdf
Sears, F. W., Zermansky, M. and Young, H.
D., University Physics, 5th Edition,
Addison-Wesley, N. Y. (1981), as
cited in Yost, S. A., “The effect of
spring mass on the oscillation
frequency.”
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