Objective: 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.
Hooke’s Law
Apparatus: A spiral spring, a set of weights, a weight hanger, a balance, a stop watch, and a twometer stick.
Purpose:
In this lab we will measure the spring constant of two springs using two different methods. First, we will
plot the force the spring exerts vs the distance the spring is stretched and second, we will measure the period
Theory:
for small oscillations.
The restoring force, F, of a stretched spring is proportional to its elongation, x, if the
Introduction: deformation is not too great. This relationship for elastic behavior is known as Hooke's law and is
Part I: If one described
end of a spring
is fixed and the other end is stretched, the spring will exert a force that
by
tends to pull it back to its unstretched position.
if a spring is compressed,
F = Likewise,
-kx
(eq. 1), the spring will exert a
force that tends to push it back to its uncompressed position. If one stays within the elastic limits of the
where kofis this
the constant
of proportionality
called
the spring
constant.
The is
spring’s
restoring
force acts
spring, the magnitude
force is directly
proportional
to the
distance
the spring
stretched.
These
in
the
opposite
direction
to
its
elongation,
denoted
by
the
negative
sign.
For
a
system
such
as
shown in
experimental results can be expressed in the form of a law known as Hooke’s Law.
figure 1, the spring's elongation, x – x0, is dependent upon the spring constant, k, and the weight of a
F If
=the
−kx
mass, mg, that hangs on the spring.
system of forces is in equilibrium (i.e., it has no (1)
relative
acceleration),
then
the
sum
of
the
forces
down
(the
weight)
is
equal
and
opposite
to
the
sum
where F is the force, x is the displacement of the free end of the spring and k is a constant that depends onof the
forces acting
the spring),
or constant and is always a positive
the internal properties
of theupward
spring. (the
Therestoring
constantforce
k is of
called
the spring
number. The minus sign in the above equation keeps track of the direction of the force, the force exerted
m gthe
= kdirection
(x – x0) the spring is stretched.
(eq. 2). Consider the diagram
by the spring is always opposite in direction to
below with the fixed end of the spring on the top. Let x = 0 (xo ) be the equilibrium position of the free end
of the spring and let positive x values be downward.
With this setupComparing
it is clear equation
that when
the displacement
x isequation
positiveofthe
spring is
stretched
thesee that if
2 with
the form for the
a straight
line
(y = mxand
+ b),when
we can
displacement x we
is negative
then produced
the springbyis different
compressed.
Note
that
this toof
bethe
true
the spring from
is not
plot the force
masses
(mg)
as aforfunction
displacement
compressed at its
equilibrium
Now
consider
Equation
When
is positive,
the minus
equilibrium
(x-xposition.
should
be linear
and the(1).
slope
of thex line
will be equal
to the sign
spring
0), the data
makes F negative.
This means
thatstandard
the spring
exerts
a force
in the negative direction, that is, it pulls back.
constant,
k, whose
metric
units
are N/m.
When x is negative then
minusissign
makes
F positive.
means that
the spring
exerts a force
in the
If the
the mass
pulled
so that
the springThis
is stretched
beyond
its equilibrium
(resting)
position, the
positive x direction,
thatforce
is, itofpushes.
In this
weanwill
measure the
spring
constant
of two springs
byof the
restoring
the spring
will lab
cause
acceleration
back
toward
the equilibrium
position
plotting the force each spring exerts versus the displacement the spring is stretched. We will suspend the
spring in a vertical position and attach a mass holder. Here, x will be the displacement of the mass holder
in the vertical direction and is measured from the equilibrium position
of the holder.
66
Part II: As we will learn later in the course, the spring constant also determines the period of oscillation
for a spring mass system. If a mass m is attached to an ideal spring and is released, it is found that the
spring will oscillate with a period of oscillation given by
r
m
T = 2π
(2)
k
where k is the spring constant for the spring.
1
Hooke’s Law
Laboratory Procedure:
Remember to do all work in your Lab Notebook. This means copy tables into your
notebook. Use correct heading and write in pen.
Part I - Taking Direct Measurements
1. Begin by hanging one end
of a spring from the ring stand. Hang a mass holder from the other end of the
Aim:
spring. The spring mustRobert
be stretched
untilone
theofadjacent
turns
of the
spring arebetween
no longer
Hooke was
the first to
notice
a relationship
thecompressed
force applied to an
against each other in order
for
the
elongation
of
the
spring
to
vary
linearly
with
the
applied
force, Law
in which
elastic object and its extension. This lab is designed to test and verify Hooke’s
accordance with Hooke’sstates
Law.that
For“the
theextension
brass spring,
mass
hangeris is
sufficient
to do this.
Forapplied
the force so
of an the
elastic
material
directly
proportional
to the
steel spring, add 500 g mass
to the
theelastic
mass hanger.
long as
limit is not exceeded”.
Diagram:
Apparatus:
Clamp and stand
Spring
Metre Rule
Mass holder
Slotted
masses
Force
2. Hold the meter stick alongside the spring and measure the height of the bottom of the mass holder.
Method:position (x = 0). Record this position in your lab notebook.
This position is the equilibrium
Set up the apparatus as shown in the diagram above. Apply various forces to the spring and
3. Add masses in increments
of 20.0
g for the (remember
brass spring
and
g for =the
steel
spring.
record
its extension
that
the100.0
extension
final
length
í original length).
4. Record the displacementData
fromCollection:
the equilibrium position, x.
x the
Record
pairs of data
forthe
force
and extension
in a suitable
table.
5. Record the uncertainty in
displacement
from
equilibrium
position,
δx, based
on the precision
x
Your
results
table
and
the
presentation
of
data
should
include
any uncertainties associated
of the meterstick.
with the apparatus that you have used.
6. Calculate the fractional uncertainty (δx/x) for this measurement and record this in your data table.
Data Processing and presentation:
7. Record the mass, m that was required to stretch the spring from the equilibrium position, i.e., the
x Use your results to plot a suitable graph which will allow you to accurately calculate the
mass added beyond that in step one.
spring constant, k, for your spring.
x in
SLyour
& HL,
include
bars subtracting
for each datasmall
point.amounts of mass to see
8. Determine the uncertainty
masses
δmuncertainty
by adding and
calculate
the uncertainty
in the
yoursystem
final value
for the spring constant by drawing
how much can be addedxor HL,
subtracted
while
still keeping
in equilibrium.
maximum and minimum slope on your graph.
9. Calculate the force (in Newtons) from your mass values. Note that there are 1000 grams
kg , and that
g = 9.81 sm2 .
Conclusion and Evaluation:
x Your
evaluation
should of
include
a final
value
for the spring
constant of your
spring
10. Repeat steps 3-9 for a total
of ten
measurements
xi and
mi , the
distance
and corresponding
mass
together
with
an
estimate
of
the
uncertainty.
measured from the equilibrium point, for one spring.
x Does your graph verify Hooke’s Law?
11. Repeat steps 1-10 for the
spring.
x other
Evaluate
the procedure and result including limitations, weaknesses or errors.
12. Draw a full-page graph of the force vs. displacement. The force, F , is to be plotted along the vertical
axis and displacement, x, is to be plotted along the horizontal axis. Include the data for both springs
Mike Dickinson
on the same graph or separate
graphs labeling them steel or brass.
13. Fit a line to your data and determine the slopes of your lines. The slope corresponds to the spring
constant, k of the spring.
2
Hooke's Law
Part II - Taking Indirect Measurements
1. Hang a small known mass, mk from the brass spring (including the mass of the holder, mh ).
2. Measure the period of small oscillations of the system by measuring the time, t, it takes for the spring
to make 20 oscillations (N=20).
3. Record the time and calculate the period, T , (T = t/N). As a check for blunders (miscounting, for
example), repeat twice more and average.
4. Determine δT by first determining the uncertainty in your reaction time. Find your reaction time by
starting and stopping the stopwatch quickly, then divide that time by your twenty revolutions.
δT =
reaction time
N
5. Determine the fractional uncertainty (δT /T ) for this measurement and record this in your data table.
6. As it turns out, the mass of the spring itself does affect the motion of the system, thus we must add 13
the mass of the spring to account for this. In order to calculate the mass, m, used in equation 2 then,
add 31 the mass of the spring (ms ), plus the mass on the hanger (mk ), plus the mass of the hanger
(mh ).
1
m=
∗ ms + mk + mh
3
7. Determine δm from the precision of the electronic balance.
8. Determine the fractional uncertainty (δm/m) for this measurement and record this in your data table.
9. Repeat steps 1-8 with the steel spring but use more mass (1Kg) on the mass holder.
10. Calculate the spring constant, k, for both springs in part II using the equation for the period of
oscillation for a spring mass system, equation 2.
Part III - Determining Uncertainties in Your Final Values
In the results section of your notebook, state the results of both parts of your experiment in the form
k±δk. Note, δk in part I should be equal to the largest fractional uncertainty from your values of mass
or displacement fractional uncertainties multiplied by your value of k from Part I. For Part II, δk should
be equal to the largest fractional uncertainty from your values of period or mass fractional uncertainties
multiplied by your value of k from Part II. For example,
δm δx
,
δk = k ∗ max
m x
You should also address the following question:
1. Do your results for k in the two parts for both springs agree within their uncertainties? Be sure to clearly
state the quantitative values you are comparing. If there are any large discrepancies, quantitatively
comment on their possible origin.
3