Determining the relationship between Radius, Satellite Mass, and Attractive
Force during Circular Motion
05/23/2023
Mariam Abugoukh
Partners: Ryan C & Ashish R
Introduction
Rotational dynamics are used by engineers and scientists in many ways, such as when
constructing roller coaster loops. As the cart travels around the loop, its force normal is
perpendicular to its surface, causing it to point radially inwards at all times. The cart’s direction
is forwards, its velocity tangential to its travel. As a result, the cart moves in a circular motion,
with the centripetal force keeping it on the track and pushing it around the center of rotation at
a constant speed. Although designing a roller coaster loop may seem simple, many different
factors are taken into account to ensure that the speed of the cart doesn’t exceed a certain
amount and endanger the passenger. Some of the factors that vary per roller coaster loop are the
radius of the loop, the mass of the cart, and the amount of centripetal force.
Therefore, the following research question was formed to find what affects its velocity,
“What affects the motion of a satellite?”. There were 3 hypotheses for the research question, the
first was about how the radius affects the satellite’s motion, the second was about the mass of
the satellite, and the third was about attractive force. It was predicted that the velocity of the
object would increase if the radius was increased, this might be because less force is needed to
return the satellite to its original point and to change its direction. For the mass of the satellite,
as the mass increases or decreases, the velocity would not change because it’s weight becomes
negligible as it travels around the circle. The third hypothesis is that changing the amount of
attractive force would increase the satellite’s velocity. The rationalization for this hypothesis was
that the attractive force is a component in the satellites motion, therefore an increase in this
component would increase the satellite’s velocity as well. To answer the research question, an
experiment was run to find whether the radius, mass of the satellite, and attractive force affect
the motion of the satellite.
Experimental Method
The independent variables for the three sets of experiments is the radius/length of the
string, the mass of the satellite, and amount of attractive force.
Figure 1: Top view labeled diagram of satellite in circular motion, including radius and attractive force
Instead of using designated masses for the satellite, the satellite is measured in stoppers,
and increased when a larger mass must be tested. The attractive force is measured in washers,
and turned into a force using gravitational force, which is then used as a tension. The force is
increased by increasing the amount of washers. The dependent variable for all three
experiments is the velocity of the satellite/stoppers. The controls for the experiment were the
other two variables not being tested. Meaning that the “base” or default for the three variables
were 0.3 meters of string, 20 washers, and 3 stoppers. When testing the radius, 20 washers were
used for the attractive force and 3 stoppers were used for the satellite. The radii tested were 0.3
meters, 0.35 meters, and 0.4 meters. When testing the attractive force, the radius was set to 0.3
meters and the satellite was set to 3 stoppers. The masses that were used for the attractive force
were 20 washers, 25 washers, and 30 washers. When testing the mass of the satellite, the radius
was set to 0.3 meters and the mass of the attractive force set to 20 washers. The satellite masses
tested were 3 stoppers, 2 stoppers and 1 stopper. The constant in the experiments was
gravitational acceleration.
The materials needed to conduct the experiments were 30 washers, 3 stoppers, at least
0.45 meters of string, a stopwatch, a plastic tube, a camera, a meter stick, and a marker. First,
the amount of stoppers tested or kept controlled was tied to the end of the string and 3 different
measurements were marked starting at the stopper(s), 0.3m, 0.35m, 0.4m. The end of the string
was threaded through the tube, and the amount of washers being tested or controlled was tied to
the end.
Figure 2: Side view labeled diagram of satellite in circular motion, with components used
Then, the stoppers/satellite is spun for 3 revolutions while its travel is recorded and
timed. This process is repeated for all three independent variables, only changing one at a time
and keeping the other two at the controlled value, each for three trials.
The main safety precaution taken during this experiment was to stay away from the
orbiting satellite, as it could hurt the researchers. It was ensured that all researchers stayed at
least a meter away from the spinning projectile.
Data
Satellite Time and Angle during 3 Revolutions for each Radius
Radius(m)
Time for 3 Rev.(s)
Angle of Ft(°)
Trial 1
2.15
63
Trial 2
2.1
63
0.3 Trial 3
2.02
69
Trial 1
2.21
58
Trial 2
2.16
57
0.35 Trial 3
1.68
58
Trial 1
2.46
58
Trial 2
2.45
57
0.4 Trial 3
2.51
55
Table 1: Raw data collected for the Satellite’s Time and angle for three revolutions for 3 trials when spun with
varying radii
Satellite Time and Angle during 3 Revolutions for each Attractive Force Mass
Attractive Force Mass(# of
washers)
Time for 3 Rev.(s)
Angle of Ft(°)
Trial 1
2.15
63
Trial 2
2.1
63
20 Trial 3
2.02
69
Trial 1
2.25
47
Trial 2
2.27
41
25 Trial 3
2.24
60
Trial 1
2.03
61
Trial 2
2.12
47
30 Trial 3
2.2
59
Table 2: : Raw data collected for the Satellite’s Time and angle for three revolutions for 3 trials when spun with
varying Attractive Force masses
Satellite Time and Angle during 3 Revolutions for each Satellite Mass
Satellite Mass(# of stoppers)
Time for 3 Rev.(s)
Angle of Ft(°)
Trial 1
2.15
63
Trial 2
2.1
63
3 Trial 3
2.02
69
Trial 1
1.91
65
Trial 2
1.81
67
2 Trial 3
1.76
67
Trial 1
1.43
73
Trial 2
1.47
66
1 Trial 3
1.31
73
Table 3: Raw data collected for the Satellite’s Time and angle for three revolutions for 3 trials when spun with
varying Satellite masses
Results
Averaged Satellite Time and Angle during 3 Revolutions for each Radius
Radius(m)
Time for 3 Rev.(s)
Angle of Ft(°)
0.3
2.09
65
0.35
2.016666667
57.66666667
0.4
2.473333333
56.66666667
Table 3: Averaged data collected for the Satellite’s Time and angle for three revolutions when spun with varying
radii
Averaged Satellite Time and Angle during 3 Revolutions for each Attractive Force Mass
Attractive Force Mass(# of washers) Time for 3 Rev.(s)
Angle of Ft(°)
20
2.09
65
25
2.253333333
49.33333333
30
2.116666667
55.66666667
Table 4: Averaged data collected for the Satellite’s Time and angle for three revolution when spun with varying
Attractive Forces
Averaged Satellite Time and Angle during 3 Revolutions for each Satellite Mass
Satellite Mass(# of stoppers)
Time for 3 Rev.(s)
Angle of Ft(°)
3
2.09
65
2
1.826666667
66.33333333
1
1.403333333
70.66666667
Table 5: Averaged data collected for the Satellite’s Time and angle for three revolution when spun with varying
Satellite Masses
The velocity was calculated by finding the quotient of the time per revolution and the
circumference, which was calculated using the angle and the length of the string. Because the
string wasn’t perpendicular to the plastic tube, the radius of the satellite’s circular travel
decreased. To account for the smaller radius, trigonometry was used to find the horizontal
distance from the pole to the satellite. The formula used, where S is the length of the string, a of
the angle created between the tube and string, and R of the radius of the satellite’s travel, was:
R=Stan(a)
Then, the new radius was used to find the circumference of the satellite’s travel. The
formula used, where R is the radius and C is the circumference, was:
C=2πR
Additionally , the time per revolution of the satellite was found using the following
formula, where P was the time per revolution and L the time it took to complete 3 revolutions,
was:
P=L/3
Lastly, the quotient of the circumference and time per revolution is found to calculate the
velocity. The formula used, where C is circumference, P the time per revolution, and V for
velocity, is:
V=C/P
A sample calculation for finding the velocity for the satellites travel with 1 stopper
attached to it during the 3rd trial is shown below:
0.3mtan(73)
0.276m
0.276m is the radius of the satellite. The rest of the equation for finding the
circumference is shown:
2π0.276m
1.733m
The circumference of the satellite’s circular travel is 1.733m. Then the period is found
and used with the circumference to find the satellite’s speed:
1.31s/3
0.437s/rev
0.437s/rev is the satellite’s period, and is then used to find the velocity:
1.733m/0.437s
3.966m/s
3.966m/s is the velocity of the satellite as it travels around the circular path.
Circumference, Period, and Velocity of Satellite for each Radius
Radius(m)
Circumference(m)
Period(s/rev)
Velocity(m/s)
0.3
1.708349931
0.6966666667
2.452176935
0.35
1.858143895
0.6722222233
2.764180997
0.4
2.099809897
0.6466666667
3.247128707
Table 6: Circumference, Period, and Velocity of the satellite and its travel varying by Radius
Circumference, Period, and Velocity of Satellite for each Attractive Force Mass
Attractive Force Mass(# of washers)
Circumference(m)
Period(s/rev)
Velocity(m/s)
20
1.708349931
0.6966666667
2.452176935
25
1.429764349
0.7511111111
1.903532418
30
1.556540362
0.7055555556
2.206120197
Table 7: Circumference, Period, and Velocity of the satellite and its travel varying by Attractive Force
Circumference, Period, and Velocity of Satellite for each Satellite Mass
Satellite Mass(# of stoppers)
Circumference(m)
Period(s/rev)
Velocity(m/s)
3
1.708349931
0.6966666667
2.452176935
2
1.726423775
0.6088888889
2.835367513
1
1.778660131
0.4677777778
3.802361326
Table 8: Circumference, Period, and Velocity of the satellite and its travel varying by Satellite Mass
Figure 3: Graph of the satellite’s velocity varying by radius
Figure 4: Graph of the satellite’s velocity varying by Attractive Force
Figure 5: Graph of the satellite’s velocity varying by Satellite Mass
Discussion
One of the assumptions for this experiment was that the length of the string stayed at the
same value. This was important because the experiment couldn’t be conducted for testing the
radius’s effect on the velocity otherwise. Another assumption was that the circular travel of the
satellite was perpendicular to the ground, which was important as speed could be lost
counteracting gravity.
One of the reasons that could’ve caused error in the experiment was the assumption that
the radius stayed the same. This could’ve caused errors and made testing the radius not effective.
Another source of error was that the angle measured wasn’t constant throughout the satellite’s
travel, therefore the angle used to find the new radius might not be accurate.
The first hypothesis, that velocity increases with radius, was supported by the data
collected. According to Figure 3, with the tested radius 0.3m, the velocity was about 2.5m/s, and
with the radius 0.4m, it was at around 3.25m/s. Meaning the relationship between velocity and
the radius is directly proportional, with a high R^2 of 0.985.
The second hypothesis was that the attractive force would increase the velocity of the
satellite, and it was not supported by the data due to a low R^2 value of 0.2 as seen in Figure 4.
This result might’ve occurred because of the low angle between the tube and the string, creating
friction and causing it to slow down.
The final hypothesis was about how satellite mass doesn’t affect its velocity. This wasn’t
proven by the data collected, as in Figure 5, the trendline has a high R^2 value with an inverse
relationship between mass of the satellite and velocity. One reason this might’ve happened was
that more force was used to spin the satellite when it was lighter, causing it to spin at a higher
velocity.