+Physics – Measuring k for Springs
in Parallel and Series
Name: _________________
Lab Partner: _____________
In this lab, you will work with a variety of springs and investigate properties of their
spring/force constants when they are combined in parallel and series.
One third of your lab score is based on your ability to obtain the correct experimental
value for the force constants, so do you experiment with care! Another third is based on
your theoretical derivation for k effective in series and parallel combinations of springs.
The final third is based upon your discussion of the differences between your
experimental value for k effective and its theoretical value.
As always, record in your notebook your data for the lab.
Experimental Work #1
Measuring the Spring/Force Constant
You should be able to determine the spring/force constant for a spring. How do you do
it? Have a vertical spring hold a mass. The system will be in equilibrium once it no
longer oscillates. Equilibrium means that the downward force of gravity is equaled to the
upward force of the spring. When this happens, Fg=-ky, or k=mg/y. We’ve done a
few homework problems that find k this way. Although it is possible to determine a
value of k from a single measurement, it is better to use a variety of different masses and
to plot mg versus y. The slope of the line will be equal to –k. Should this line go
through the origin? Explain.
Describe spring being used (length, color, shape, etc): _________________
Spring Constant = ______________
(Attach your graph).
Theory for Working with Multiple Springs
Multiple springs can be used instead of a single spring. In order to work with these
systems, we need to first consider the system as one made of multiple springs. We then
work through the algebra to reduce the system to a single effective spring and to find the
corresponding single k effective.
Consider the system of two springs with different spring constants both being stretched
by the same mass m. We want to find a way (by understanding the physics) to describe
the two spring system as though it were actually a single spring. The illustration below is
often called a combination of springs in parallel (it only uses two springs, but one could
imagine a parallel combination with more than two springs).
k1
k2
keffective
m
m
m
m
Draw a free body diagram of what the block feels. On the left, draw it with two springs
(f1 and f2) and on the right draw it with only one effective spring.
Since both systems are experiencing the same downward force of gravity and they are in
equilibrium, write an equation relating f1, f2, and f effective.
(1.1)
Using Hooke’s Law, replace your f1 and f2 in the above with k1, y1, k2, and y2.
(1.2)
What must be true about y1 and y2?
Re-write you equation using the above fact and simply.
(1.3)
What is Hooke’s Law for the system on the right?
(1.4)
Combine these two expressions for Hooke’s Law to obtain an equation for what k
effective is equal to …
-2-
.
(1.5)
Another type of common combination of springs is a series connection. Here the springs
are linked end-to-end as illustrated below and combine to yield a single effective spring
force constant.
k1
keffective
k2
m
m
m
m
Draw a free body diagram of what the block feels. On the left, draw it with two springs
(f1 and f2) and on the right draw it with only one effective spring.
The total displacement that the block moves is given by y. Each spring contributes to
this displacement.
y  y1  y2
(2.1)
Imagine that the springs were magically made into the one effective spring and write
down Hooke’s Law with this same total displacement.
(2.2)
-3-
What force does each of the springs experience? (Note, we consider each spring to be
massless since their mass is much less than the attached weight).
(2.3)
Write Hooke’s Law for each displacement, now with this common effective force from
above.
(2.4)
Rewrite the expressions to give you y1 and y2 and add them together yielding the total
y. Simply the expression.
(2.5)
Rearrange the expression to match your original expression for Hooke’s Law for the
system in the right hand side box above.
(2.6)
What is k effective equal to?
(2.7)
Experimental Work #2
Parallel and Series Combinations
The rest of this lab requires you to work with another group. (Pick a group that has used
a different spring – you might want to check their data and their value for k). In this part
of the lab, you will need to find the k effective for one parallel combination and for one
series combination. Remember that you should use a variety of weights to determine the
spring constant and that you should graph the data to obtain the actual k. Do these
experiments with a great deal of care.
-4-
New lab partners: ___________________________________
Description of the springs and their force constants:
Drawing for the parallel combination.
Experimentally obtained value of k effective:
(attach graph)
How does this compare with your theoretical value?
Drawing for the series combination.
Experimentally obtained value of k effective:
(attach graph)
How does this compare with your theoretical value?
-5-