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Mathematics
Date
Feb 20, 2024
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LAB # 2: Oscillations of a Spring
By Afra Tabassum Akand, 1807768
TA: Elenea Temnikova
Date: December 5th, 2023
RESULTS:
The experiments aimed to assess spring constants and ascertain a spring's mass. This entailed
creating an oscillating system by shi±ing different masses that were attached to a spring.
Equation 1 was linearized using the data in Tables 1 and 2.
Table 1: Variation in a spring's displacement when suspended at varying masses
:
Mass (Kg)
Change in
Displacement
Change in
Mass (kg)
Change in
Displacement
Force (change in
mass*gravity)
0.250
0
0
0
0
0.275
0.0241
0.025
-0.0241
0.24525
0.300
0.0485
0.05
-0.0485
0.4905
0.325
0.0724
0.075
-0.0724
0.737575
0.350
0.0971
0.1
-0.0971
0.981
0.375
0.1211
0.125
-0.1211
1.22625
0.400
0.1453
0.15
-0.1453
1.4715
0.425
0.1696
0.175
0.1696
1.71675
0.450
0.1640
0.2
-0.194
1.962
Equation 1:
ΔF= -kΔx
or,
ΔF=Δ mg
In the experiment, data obtained and recorded in Table 1 was used to create a visual
representation by graphing it. The y-axis of the graph represented the values calculated using
the formula ∆F = ∆mg, which was derived from Equation 1. On the other hand, the x-axis
depicted the horizontal change in displacement, ∆x.
GRAPH 1:
The graph plotted data points, revealing a slope of 10.113, identified as the spring constant (k) in
the static method. This parameter characterizes the spring's stiffness, revealing its behavior
under applied forces and displacements.
In the alternative method employed for determining the spring constant, various masses
were subjected to harmonic motion by oscillating. The Logger Pro so±ware was utilized to
measure and record the angular frequency (
ω
), as presented in Table 2. It's worth noting that the
angular frequency (
ω
) and the mass attached to the spring exhibit an inverse relationship. In
simpler terms, when a heavier mass is hung from the spring, the angular frequency (
ω
)
decreases. The second spring constant is determined using Equation 2, which relates angular
frequency, mass, and spring constant based on experimental data from mass oscillation.
Equation 2: T^2 = (4π^2)/k m + (4π^2)/3k m_s
k_2 = (4π^2)/a m
Table 2: calculated “T” ( T = 2π⁄
ω
) using given omega and mass
Mass (kg)
T^2
Omega (
ω
)
0.25
1.421
5.273
0.275
1.518
5.102
0.3
1.613
4.948
0.325
1.711
4.805
0.35
1.806
4.675
0.375
1.907
4.551
0.4
2.002
4.441
0.425
2.097
4.338
0.45
2.202
4.233
Plotting the data from Table 2 onto Graph 2 yields a linearized equation, which is then used to
find variable "a" in Equation 2. The 'x' axis on the plotted graph below denotes
mass (kg), and
the y axis denotes time squared.
> k
2
= 10.15
y= ax + b
GRAPH 2:
Since the spring constant is now solved, one can rearrange Equation 3 to solve for the mass of
the spring.
M
s
= 3b/a= = 3(0.4471)/3.89 = 0.345 kg
DISCUSSION:
The static method's spring constant, k=10.11±0.0106 N/m, was calculated using
Excel's LINEST function, while the oscillation method's spring constant, 10.15, was determined
through error propagation, as per the Lab Manual's Table 6.3.
δ
k=
)
2
4π
2
?
|
|
|
|
|
|
(
δ?
?
δ
k=
)
2
4π
2
3.89
|
|
|
|
|
|
(
0.013016
3.89
δ
k =
±0.0340 N/m, where
δ
a was obtained using the LINEST function in Excel.
The spring constants obtained through both methods exhibit relatively similar values, yet a
notable distinction arises when comparing their uncertainties. The static method provides a
significantly more precise spring constant, as indicated by its lower uncertainty. Several factors
contribute to the error in the harmonic motion oscillation method. One primary factor is the
deviation from a purely vertical motion when the mass is suspended and released. Instead of a
complete vertical oscillation, the mass exhibits a slight pendulum motion. This pendulum
motion introduces inaccuracies in the measured time (T) since the mass takes a longer time to
return to the equilibrium position. The impact of this deviation is reflected in the error
propagation of the time period, further affecting the overall precision of the spring constant
determination.
δ
m
s
=
3?
?
|
|
|
|
(
δ?
?
)
2
+ (
δ?
?
)
2
δ
m
s
=
3(0.4471)
3.89
|
|
|
|
(
0.013016
3.89
)
2
+ (
0.004632
0.447056
)
2
δ
m
s
= ±0.000375 kg, where
δ
ms was obtained using the LINEST function of Graph 2 in Excel.
The friction created by the spring contacting the surrounding plastic was brought about by the
pendulum motion. It was observed during data recording that the oscillation was damped. One
possible way to solve this and conduct an experiment with greater accuracy is to hang the spring
from a rod. Furthermore, by positioning a second motion sensor horizontally, it is possible to
guarantee that the mass oscillates just vertically, removing the pendulum motion and reducing
friction. The goal of this modification is to set up the necessary parameters for a more accurate
and regulated measurement of the spring constant.
CONCLUSION
This laboratory experiment aimed to assess and contrast the spring constant values derived
through static measurement of the initial displacement caused by the mass versus oscillating the
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spring with various masses. The overarching objective was to unveil the unknown mass
associated with the spring.
REFEREENCES
[1] PHYS 130, LAB#2: Oscillations of a Spring, Lab Manual, University of Alberta, Department
of Physics.
ACKNOWLEDGEMENT
[1] Elena Temnikova ( TA )
[2] Adiba Akand (Lab Partner)
[3] HA-MIM Rahman Khan (Lab Partner)