Spring Lab
Christina Manxhari
Lab Partners: Aarushi Pendharkar and Philip Economou
Section C
December 18, 2015
Introduction
Purpose: The purpose of this experiment is to investigate the conservation of
energy in an inclined spring system.
Researchable question: How does increasing the mass (kg) attached to the
end of a spring laid along an incline of a constant angle affect the maximum
length the spring stretches (m) when the mass is released from the top?
Hypothesis: If the mass is increased, then the length the spring stretches along
an incline will increase.
Methodology
Collection of Preliminary Data: We calibrated the spring constant for the spring
we used in our lab. We did this by using sensors for LoggerPro. We stretched the
spring to different lengths (which was recorded as its compression distances)
and recorded the average force required to stretch the string that distance for five
seconds.
We measured the mass of the frictionless cart itself using an electronic scale.
Collection of Experimental Data: We leaned a plank on a chair. We attached
the spring onto a hook on the back of the cart and also attached the spring onto a
nail protruding out of the plank. We then set up the initial compression of the
spring along the incline to what appeared to be no compression, so the spring
maintained its uncompressed length. We then let go of the currently maintained
cart. We recorded, by sight, the endpoint of the cart’s path along the incline. After
conducting ten trials for the cart without any additional mass, we added on a one
hundred gram mass on top of the cart. We conducted ten trials for each different
total mass setting, increasing by increments of one hundred grams.
Diagram
In this diagram, the viewer must suppose that the initial compression, which is
supposed to maintain displacement at 0 m, with no force applied, is located right
against the block that the spring is attached to.
Experimentation
Here is a picture of Philip about to release the cart, Aarushi paying close attention to the measurement marks on the ramp, and me recording the data of each trial. Constants
Equations
∑ Ei = ∑ E f
θ = 6.63346 °
2
g= 9.8 m/s
hi= 0.231034 m
k=3.0029 kg/s2
PEgi = PEg f + PEsf
PEg = mgh
1
PEs = kx 2
2
2mg(sin θ )
=∆ x
k
2m(9.8)(sin 6.63346)
=∆ x
3.00634
(See appendix A for derivation).
Calibrating the spring constant, k
∆x
FAvg
(m)
(N)
0.100
0.302
0.150
0.453
0.200
0.601
0.250
0.737
0.300
0.911
Average Force, FAvg (N) Force vs. Compression Distance 1 0.8 Using the slope
of the equation
shown to the
right, we found
our spring
constant, k.
k=3.0029 N/m
0.6 0.4 FAvg(m)= 3.0029m + 0.0003 R² = 0.99872 0.2 0 0 0.05 0.1 0.15 0.2 0.25 Displacement, ∆x (m) 0.3 0.35 Summarized Data
m
∆xAvg
STDEV
|%RSD|
∆xT
|% Error|
ΣEi
ΣEf
%
(kg)
(m)
(m)
of xAvg
(m)
of ∆x
(J)
(J)
Ei Lost
IV1
0.5009
0.316
0.00148
0.467
0.377
16.3
1.13
1.10
2.57
IV2
0.6009
0.390
0.00237
0.607
0.453
13.9
1.36
1.32
2.71
IV3
0.7009
0.456
0.00141
0.310
0.528
13.6
1.59
1.54
3.10
IV4
0.8009
0.524
0.000966
0.185
0.603
13.2
1.81
1.75
3.45
IV5
0.9009
0.594
0.00290
0.488
0.678
12.4
2.04
1.96
3.69
Avg.
0.411
13.9
Avg.
See Appendix B for all data collected.
Graph
Stretch of Spring vs. Mass 0.800 Displacement, ∆x (m) 0.700 ∆xT[m] = 0.7531m -­‐ (2*10-­‐15) R² = 1 0.600 0.500 0.400 y = 0.6908m -­‐ 0.0283 R² = 0.99972 0.300 0.200 0.100 0.000 0.000 0.200 0.400 0.600 Mass, m (kg) 0.800 1.000 Measured Data Theoretical Data Analysis
The low average percent residual was 0.411%, signifying very precise data
collection per setting of the manipulated variable. The percent error of 13.9% was
moderate (or arguably high) indicating that our data collection was not too similar
to the expected results.
The average spring stretch distances per independent variable setting differs
about .05 to .08 meters less than the expected spring stretch distances. The two
functions between mass and displacement (one theoretical, one experimental)
were best represented by a linear function (maintaining an r2 value of almost 1),
which is the type of relationship that these two variables, ∆x and m, should have
in this situation. Their slope should have been the same, but was not too far off.
The function that represents theoretical data should begin at the origin, but has
an extremely small, yet existent, y-intercept of 2 *10-15.
The percent of initial energy lost in the experiment was less than 4% for each
setting of the manipulated variable. Although this experiment was supposed to
show energy conservation, this low percent loss is not alarming to us as
experimenters in showing validity of our experimental procedure.
Conclusion
If the mass of an object connected to a spring following the incline of a ramp
is increased, then the change in the compression of the spring will also increase.
Our hypothesis was proven right.
However, the energy was not conserved probably because although the cart
was supposedly frictionless, it actually wasn’t. If we knew the friction, it would
maximize the accuracy of the data collected. This explains the consistent
difference of about 0.05 to 0.08 meters between the average spring stretch
distance and the theoretical spring stretch distance. Friction, which was
unaccounted for, did not allow the cart to move as far as it theoretically should
have.
If this experiment were to be run again, it would be best to test if friction is
present, even when something is claimed to be “frictionless” (as seen in this
experiment). Additionally, it would be best to know whether or not our initial
spring compression (which we determined was 0 m solely by eye) was in fact so.
Furthermore, human error with the release of the cart might have swayed results,
in which case it is recommended to create a frictionless release system with a
magnet.
Appendix A
∑ Ei = ∑ E f
PEgi = PEg f + PEsf
1
mghi = mgh f + k(∆ x)2
2
1
mghi = mg(hi − (sin θ )∆ x) + k(∆ x)2
2
1
mghi = mghi − mg(sin θ )∆ x + k(∆ x)2
2
1
mg(sin θ )∆ x = k(∆ x)2
2
1
mg(sin θ )∆ x = k(∆ x)2
2
∆x
1
mg(sin θ ) = k(∆ x)
2
1
k
2
2mg(sin θ )
=∆ x
k
2m(9.8)(sin 6.63346)
=∆ x
3.00634
0.753121m = ∆ x
Appendix B