Internal Assessment Lab Report
Introduction
This investigation explores the relationship between tension and period of a massspring system. For the purpose of this investigation, linked rubber bands were used
instead of a spring. Using the periods obtained from the experiment, a theoretical
spring constant was determined for every band.
The independent variable for this experiment is the tension of the linked rubber
bands. This varied due to the width of the rubber bands. Three different widths
were used: 0.15 cm, 0.25 cm, and 0.6 cm. A standard ruler measured this, and its
instrumental uncertainty was 0.05 cm, or half of its lowest increment.
The dependent variable was the time taken to complete one full oscillation, or
period. The time was calculated using a stopwatch with an uncertainty of 0.3
seconds: this rounded value takes into account human reaction time (0.3 s) and the
instrumental uncertainty, or its lowest increment of measurement (0.01 s).
The controlled variables included the length of the rubber bands, the height of the
table at which the rubber bands rested, the mass used, thickness of the bands, and
the length of extension. Due to varying elasticity, the narrower band of 0.15 cm
stretched farther than the wider band of 0.6 cm, hence a different number of rubber
bands were linked for each width as to assure the same length of 61.4 cm. The
height of the table was 103.5 cm. The mass used was a 200g weight. The length of
extension of the spring was 42.1 cm.
The materials used were:
 Rubber bands
o Widths of 0.15, 0.25, and 0.6 cm
 Stopwatch
 200g weight
 Pencil
 Duct tape
 Table
First, a pencil was attached horizontally to the end of the table using duct tape. For
the first trial, the thinnest rubber bands (0.15 cm) were linked together and the
200g mass was attached to the end. Partner 1 extended the rubber band 42.1 cm.
Partner 2 started the stopwatch. As to eliminate as much uncertainty as possible,
Partner 1 watched the stopwatch and released the mass when the stopwatch
reached two seconds. Partner 1 then counted 5 oscillations and said, “stop” when
the five oscillations were completed. Partner 2 stopped the stopwatch as soon as
Partner 1 said “stop.” These steps were repeated for five trials. Five trials were then
repeated using the 0.25 cm and 0.6 cm rubber bands, respectively.
In order to calculate the period length of one oscillation, the times were then
divided by 5. Below are the period lengths (for a single oscillation) of the three
bands:
Period of Oscillations (s) (±0.01 s)
Band Width (cm) (± 0.05 cm) 0.15
0.25
Trial 1
1.126
1.036
Trial
2
1.106
1.044
Average
Period of Oscillations
Trial
3
1.126
1.074
Band Width (cm) (± 0.05 cm)
0.15
0.25
Trial 4
1.086
1.066
Period Length (s)
1.11 ± 0.04
1.06 ± 0.04
Trial 4
1.100
1.062
0.60
0.842
0.858
0.814 0.6
0.816
0.84 ± 0.04
0.850
Band Width vs. Period
1.4
1.2
Period (s)
1
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Band Width (cm)
To find a theoretical k value (spring constant) for each rubber band, we used the
𝑚
formula: = 2𝜋√ 𝑘 . To isolate k, the formula was then transformed to: =
4𝜋 2 𝑚
𝑇2
.
Uncertainty was taken into account when finding these values: the uncertainty of
the period (time) was multiplied by 2, as it is squared in the denominator the
second equation. The theoretical k values are as follows:
Theoretical K Values
0.15
0.25
6.4 ± 0.5
7.0 ± 0.6
Band Width (cm) (± 0.05 cm)
K Value (kg/s2)
0.6
11.2 ± 0.9
Spring Constant vs. Period
1.4
1.2
Period (s)
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
12
Spring Constant (kg/s2)
Through our data we were able to affirm the relationships between tension and
time period (greater tension leads to a longer period), as well as between tension
and spring constant (a greater tension results in a greater spring constant).
Theoretical spring constants were found for every band, and these values align with
our data.
There were, though, limitations in our experiment: a systematic error in our
experiment was failure to account for all the directions in which the mass could
move. With the mass and tension of spring that we used, there was no way to make
sure that the mass would move straight up and down. The mass did move up and
down in the experiment, but it also moved side to side which must have made all of
the period lengths collected shorter than their actual value. Using a greater mass
would fix this issue, as its increase in inertia would cause it to deviate less in
direction. However, in doing so, the tensions of the rubber bands must be adjusted
accordingly. Simple harmonic motion in springs is only executed when the mass is
not extended too far from the original position (as to not manipulate the spring
constant k) - if the mass was too great for the rubber bands, the tension of the spring
would be damaged and simple harmonic motion would not occur.
This experiment could also be improved by the usage of a video camera to record
the length of 5 periods of oscillation. In this experiment, we judged by eye when 5
oscillations had occurred - yet, with a video camera, these measurements could be
made even more accurate because then we could freeze-frame the recording and
note the times of exactly when the mass returned to its original position. The use of
a camera would eliminate not only the uncertainty of human reaction time but also
the possible random error of parallax, as we made our observations from above the
mass-spring system instead of at eye-level.