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ERROR ANALYSIS AND SIMPLE HARMONIC MOTION
OBJECT
The object of this experiment is to study error calculation methods using a simple system, unfamiliar to the
student, so that a basic comparison between expected and measured values can be made without a deep
understanding of the theory.
The spring constant is calculated in two ways and the students verify the
theory by finding if the error ranges of the measurements overlap.
APPARATUS
Spiral spring and rigid support, weight holder and weights, platform balance, datastudio program, motion
sensor, interface box.
THEORY
Elasticity is that property of a body which causes it to return to its original shape and size after being
distorted by a force.
According to Hooke’s Law, when an elastic body, such as a spiral spring, is subjected
to a force which elongates it, the elongation, x, is proportional to the force, F, provided the elastic limit is
not exceeded.
This proportionality is expressed by the equation,
F
=

k
x
,
(
1
)
where k, the constant of proportionality, is called the force constant of the spring.
The value of k depends
upon the shape and size of the spring and upon the elastic properties of the material of the spring.
It is also true that a mass, M, suspended from the lower end of a spring (Fig. 1) – will vibrate with
harmonic motion if it is displaced a
small amount
(less than 3 cm for this lab) and then released.
If the
mass of spring is negligible in comparison with the suspended mass, the period of vibration, T, is given by
the equation
T
=
2
π
k
M
.
(
2
)
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In this experiment the mass of the spring cannot be neglected because it is comparable to the suspended
mass.
Since each part of the spring has a different displacement it is not correct to simply add the mass of
spring to the mass suspended at the end of the spring.
A complete analysis of the vibration of a uniform
spring shows that if onethird of the mass of the spring is added to the mass on the end of the spring, then
equation (2) is correct.
Therefore, we can rewrite equation (2) in the form
T
=
2
π
k
M
M
s
3
1
+
,
(3)
Where M is the mass on the end of the spring and M
s
is the mass of the spring.
(M
s
/3) is called the
equivalent mass of the spring.
You will be comparing the
k
from the displacement measurements (eq.
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This note was uploaded on 08/25/2011 for the course PHYS 164 taught by Professor Johnjames during the Fall '09 term at MO St. Louis.
 Fall '09
 JohnJames
 Simple Harmonic Motion

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