EXPERIMENT 6
THE SPRING CONSTANT DETERMINATION
Purpose: One of the goals of science is the development of physical and mathematical
models to describe physical systems by using observational and experimental data. We
then use these models to either explain previously observed data or to predict results that
have not actually been observed where the quality of the model determines its predictive
value . The objective of the experiment is to determine the spring constant of a spiral spring
by Hooke’s law and by its period of oscillatory motion in response to a weight.
Apparatus: A spiral spring, a set of weights, a weight hanger, a balance, a stop watch, and a
lab scale.
Theory: The restoring force, F, of a stretched spring is proportional to its elongation, x, if the
deformation is not too great. This relationship for elastic behavior is known as Hooke's law
and is described by
F = -kx
(eq. 1)
where k is the constant of proportionality called the spring constant. The spring’s restoring
force acts in the opposite direction to its elongation, denoted by the negative sign. For a
system such as
shown in figure 1, the spring's elongation, x – x0 , is dependent upon the
spring constant,
k, and the weight of a 0 mass, mg, that hangs on the spring. If the system
of forces is in equilibrium (i.e., it has no relative acceleration), then the sum of the forces
down (the weight) is equal and opposite to the sum of the forces acting upward (the restoring
force of the spring), or
m g = k (x – x0)
(eq. 2)
Fig. 1. Illustration of Hooke’s Law. As additional weights are added, there is a linear
increase in the length of the spring.
Equation 2 is in the form of the equation of a straight line (y = mx + b) so, we can see that if
we plot the force produced by different masses (mg) as a function of the displacement from
equilibrium the data should be linear. Hence, the slope of the line will be equal to the spring
constant k. The unit of k in standard metric units is N/m. If the mass is vertically pulled so
that the spring is stretched beyond its rest position, the restoring force of the spring causes an
acceleration back toward the equilibrium position, thus the mass oscillates in simple harmonic
motion.
One complete oscillation is the amount of time describing as the period of vibration called T,
and for the system described above is:
‫ ܂‬ൌ ૛ૈට
and
‫܍ܕ‬
(eq.3)
‫ܓ‬
me = m + me-spring
(eq.4)
where m is the mass which hangs from the spring and me -spring is the spring's equivalent
mass. Substituting equation 4 into equation 3 and squaring both sides of the equation yields:
‫܂‬૛ ൌ
‫܂‬૛ ൌ
૝ૈ૛
‫ܓ‬
૝ૈ૛
‫ܓ‬
൫‫ ܍ܕ‬൅ ‫ ܏ܖܑܚܘܛି܍ܕ‬൯ or,
‫ ܍ܕ‬൅
૝ૈ૛
‫ܓ‬
‫܏ܖܑܚܘܛି܍ܕ‬
(eq.5)
Comparing equation 5 to the equation for a straight line (y = mx + b), we see that the slope
and y-intercept, respectively, of the linear fit is:
‫ ܍ܘܗܔܛ‬ൌ
૝ૈ૛
‫ܓ‬
and
‫ ܡ‬െ ܑ‫ ܜܘ܍܋ܚ܍ܜܖ‬ൌ
૝ૈ૛
‫ܓ‬
‫܏ܖܑܚܘܛି܍ܕ‬
(eq.6)
Suggesstion : Extension of a spring is proportional to the applied force and a spring will
return to its rest length when the force is removed so long as the elastic limit has not been
exceeded. Beyond the elastic limit, springs exhibit plastic behavior where additional force
causes deformation of the spring such that the original or rest length is altered.
EXPERIMENT
Part I.
1. Using a scale, read the position of the last coil of the freely hanging spiral spring and record
it on your data sheet as x0 .
2. Hang an approximate 0.100 kg mass to the spring. Remember to include the mass of the
hanger and weigh the masses on a balance. Record this mass on your data sheet.
3. Calculate and record the weight of the mass using F = mg, where m is in kg and g is 9.8
m/s2 .
4. Read xi, the position of the same last coil of the spring as in step 1.
5. Calculate and record the total displacement of the last coil of the spring, Δx, by Δx = xi –
x0.
6. Repeat steps 2-5 for masses approximately equal to 0.200 kg, 0.300 kg, and 0.400 kg.
7. Make a graph of the force, F, versus displacement (Δx). You will have five data points for
this graph: the four data points for each of the four masses, and an additional data point at
(0,0). This data point is valid because when 0 kg hung on the spring, it was displaced 0 m
from its equilibrium position.
8. Fit the data with a linear function in the form of y = mx + b. Determine the value of the
spring constant from the slope of the best-fit line.
N
1
2
3
4
5
X0
m (kg)
F=mg (N)
xi (m)
Δx (m)
k
Part II.
We will determine the spring constant by means of period formula (eq.3)
1.
Setup the apparatus shown in Fig.1
2.
Using 4 different masses, used in the experiment, find the period of oscillation for 10 s
and record them as T 1, T2, T3, and T 4 to the table.
3.
Divide the period values by 10 to find the periods for 1 s.
4.
Using eq3. find the spring constant k.
5.
Plot the graph of m (kg) versus T 2 (s2).
6.
From eq3. and the slope of the graph, find the spring constant k.
N
m (kg)
10 T(s)
T(s)
T2(s2)
1
2
3
4
5
QUESTIONS
1. Calculate % error for k value between the calculated k value from the experiment and
k real value.
2. Using k values, calculate the mean standard deviation.
3. Write up the detailed error reasons you faced while conducting the experiment.
REFERENCES
1.Physics for Scientists and Engineers with Modern Physics (Serway Jewett)
2. Principles of Physics (Allyn and Bacon)
REPORT
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