SPH 4UI
Conservation of Energy Lab Preparation
revised 2009
THE COIL SPRING
Some springs are made of spring steel bent into a tight coil so that, even when fully retracted with coils
touching, there is tension pulling the coils together. This makes the spring stiff. We could measure this “extra”
tension by loading the spring with just enough force to begin to separate the coils.
F (N)
F (N)
Rest length, l0
of the unloaded
spring
There is no gap
between the coils
so the spring
cannot relax
Same
spring with
fixed
weight on it
l (m)
l (m)
Rest length, l0
of the unloaded spring
FIGURE 1: A TIGHTLY COILED SPRING
FIGURE 2: A STRETCHED SPRING
For a new spring with coils which spring back tightly, the
“zero” of s is a value less than the actual length of the
spring. The coils are actually pulled even when the
spring is completely compressed. This makes the spring
stiff when it is not loaded.
For an older spring which sags to the point where the
coils have some space between them unloaded, the
“zero” of s is the l intercept. It is the length at which we
will assume the spring neither pushes nor pulls.
OSCILLATING MOTION (A MECHANICAL PERSPECTIVE)
Hooke’s law tells us that the force exerted by the spring is Fe = -ks. This means that the force changes
(increases) as the spring is extended. However, the force of gravity remains constant. When the mass is hung
on the spring at rest, the forces are in equilibrium or balanced (when Fe = Fg at l2 on the next page). The
oscillation can be started by slightly over or under extending the spring so that the vertical forces (gravity F g
and spring Fe) are not equal. When released the mass will accelerate in the direction of the unbalanced force
and its inertia will carry it past the point of equilibrium. The mass is subjected to a changing acceleration
opposing the motion because the net force (sum of spring and gravity forces) is what we call a restoring force.
See: http://www.walter-fendt.de/ph11e/springpendulum.htm and click on the force radio button.
GETTING STARTED
Get YOUR spring from the bin. You need the s=0 (l-intercept) and k (slope) values from your F vs. l graph
from the Spring Mini-Lab before you can analyze the energies. Think about the uncertainties in the masses and
lengths to produce a new graph with uncertainties. Be sure to use the SAME spring when measuring energies.
Do not deform the spring by overstretching or you will change its k and s=0 values and you will need to
begin again (and so will others who used your spring!!!).
To make it possible to directly measure an s=0 value of a new tight spring, a fixed mass can be added for the
duration of the lab to load the spring sufficiently to extend the spring to move the s=0 location (as the second
SPH 4UI
Conservation of Energy Lab Preparation
revised 2009
line on Figure 1 shows). New springs come with a small mass in the form of a fish weight that can be hung on
the spring to extend it beyond the non-linear behaviour shown above. Note that this does not affect the k value
(force per unit length extension). So as long as we do not change the mass on the spring during the
investigation, our results will not be affected. BUT THIS IS NOT NECESSARY IN THIS LAB. We will
carefully suspend a 1 kg mass on the spring and give it a 10 cm amplitude for the oscillation.
l0 intercept from graph
l1
s = 0 Fe = 0
l2
s1
l3
s2
s3
unloaded spring
s=0 length may
be shorter than
the spring itself
(where spring
neither pushes
nor pulls)
highest point
Fe < Fg
mass stopped
and accelerating
downward
middle point
Fe = Fg
mass not
accelerating
GATHERING DATA USING THE MOTION PROBE
lowest point
Fe > Fg
mass stopped and
accelerating upward
Since this lab is about studying the energy(ies) of the oscillating mass and conservation of mechanical energy,
determining the vertical positions and speed of the mass at the same time in real time is necessary to calculate
gravitational potential energy, spring potential energy and kinetic energy.
The motion probe senses only position at discrete equal time intervals. It must calculate the speed based on the
change in positions it has recorded (not continuous). So, it displays position and calculates speed as we have
done when we produced x-t and v-t graphs from a ticker tape.
time (s)
can be
displayed
0.0
0.1
position x
(cm)
can be
displayed
0
change in position
x
during time interval
(cm)
not displayed
average speed
vav (cm/s)
can be
displayed
2.5  0.0  0.0
2.5
 25
0.1  0.0
25  50
 37.5
2
2.5
7.5-2.5=5.0
0.2
5.0
 50
0.2  0.1
50  45
 47.5
2
7.5
12.0-7.5
0.3
12.0
adjusted average speed
vadj (cm/s)
not displayed
must be calculated
4.5
 45
0.3  0.2
SPH 4UI
Conservation of Energy Lab Preparation
revised 2009
To determine the total energy, we need the speed to determine EK and position x for EG and Ee at the SAME
moment. The probe will not provide us with that information. As in the chart above, the speed displayed is the
average speed for that time interval which, if the time interval is short enough, can be considered the speed at
the average time.
If we average these two average speeds, we get a good approximation to the speed at close to the average time.
Calculating each adjusted speed in this way will match position and speed to about the same time.
OBSERVATIONS AND ANALYSIS (YOUR INDIVIDUAL PART)
Following good lab procedures, collect and save your own set of data and cut and paste it into a spreadsheet for
manipulation. You should have several hundred data points for this lab.
TIPS FOR A GOOD RESULT:
o Measure carefully considering uncertainties
o Collect at least 10 cycles of the motion to analyze per person
o Many measurements (change sampling options) made in each complete cycle of the motion.
SETUP
Unloaded
length
l
s
m
h
motion
sensor
SPH 4UI
Conservation of Energy Lab Preparation
revised 2009
THEORY
Derivation of the mechanical energy equation of a vertical oscillating spring:
The total mechanical energy of the spring mass system is:
1
1
2
ET  mv 2  k  x0  x   mgx
2
2
In this equation, the first term is kinetic energy, the second term is the elastic potential energy of the
spring and the last term is the gravitational potential energy.
1
(m is the mass of weight hanger in kg and k is the spring constant in N/m, xo is the distance the spring
is stretched when it is in the equilibrium position measured in meters.)
We will choose the distance x such that:
 x is measured from the equilibrium position and x > 0 when the mass is above the equilibrium point
 x = 0 when the spring is in the equilibrium position
 x < 0 when the mass is below the equilibrium point.
Note: that when x = xo both the elastic and gravitational potential energy terms are 0.
Since kx0  mg then if the spring has a mass of m hanging on it, the elongation, xo, will be
Rewriting equation (1) substituting x0 
mg
k
mg
:
k
2
1
1  mg

ET  mv 2  k 
 x   mgx
2
2  k

 2
Combining the last two potential energy terms we obtain:
 3
1
1
 mg 
2
ET  mv 2  k  x  
2
2
2k
2
This equation is algebraically the same as the previous one, but it is
easier to use. Since the detector is not at x = 0, but at a distance ho
below the equilibrium point (see FIG.1), we define h, the position
measured by the detector as:
h = x + ho
Then: x  h  h0
This gives us a final equation to work with:
 4
1
1
 mg 
2
ET  mv 2  k  h  h0  
2
2
2k
2
Since the last term on the right hand side is a constant, the sum of the
first two terms on the right hand side must be a constant for mechanical
energy to be conserved.
FIG 1.
Note: If you add 1/3 the mass of the spring to the mass of the weight (m = mass of weight + 1/3
mass of spring) you will obtain a slightly more constant Total energy value. Why?