Atwood's Machine Report Template
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School
Carleton University *
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Course
1007
Subject
Industrial Engineering
Date
Dec 6, 2023
Type
docx
Pages
7
Uploaded by MajorSparrow2743
Carleton University
Laboratory Report
Course #: 1007A
Experiment #: 4
Atwood’s Machine Report
Ali Khelil
(101258560)
Date Performed:
11-06-2023.
Date Submitted:
11-13-2023.
Lab Period:
L1
Partner:
Goodness
Station #:
16
TA:
Teresa
Purpose
The purpose is to use an Atwood's machine to measure a time interval for the acceleration due to gravity
with the motion affected by weights. The found values will be used to find gravity and torque.
Observations/Graphs
Figure 1: A graph of the linearized form of the values received from the Atwood’s machine test.
Table 2: The weighed and measured values for the masses of the 2 separate weights, washers, height, and diameter of the pully.
The uncertainty values were calculated and written bellow as well.
Masses
Lengths
Iron core + screw
Other weight +
screw
One washer
(mass of all/10)
Distance travelled by
m
1
Diameter of the
pulley
m
1
m
2
m
w
h
d
Units
g
g
g
cm
mm
Trial 1
254.44
254.82
0.997
103.1
121.12
Trial 2
254.39
254.84
0.997
103.3
121.97
Trial 3
254.38
254.83
0.997
103.3
121.79
Average
254.40
254.83
0.997
103.2
121.63
Instrumental
Uncertainty,
σ
IU
0.01
0.01
0.01
0.05
0.5
Standard Deviation,
σ
SD
0.00105
0.0001
0
0.015
0.20065
Standard Deviation
of the mean,
σ
mean
0.0006
0.00005
0
0.009
0.12
Chosen Uncertainty
(
σ
IU
∨
σ
mean
)
0.01
0.01
0.01
0.05
0.5
Final Measurement
(…±…) units
254.40±
0.01
254.83±
0.01
0.997±
0.01
103.2±
0.05
121.63
±
0.5
Calculations
Calculating the total mass undergoing linear motion and its uncertainty:
M
=
m
1
+
m
2
+
10
m
w
M
=
254.40
+
254.83
+
10
(
0.997
)
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M
=
254.40
+
254.83
+
10
(
0.997
)
M
=
519.2
g
Uncertainty calculation:
σ
M
=
√
σ
m
1
2
+
σ
m
2
2
+
100
σ
m
w
2
σ
M
=
√
0.01
+
0.01
+
100
(
0.01
)
σ
M
=
0.346
g
Therefore M
=
519.2
g±
0.346
g
Rearranging to find acceleration due to gravity:
Δm
=
g
2
h
(
M
+
1
2
M
p
)
2
mh
(
M
+
1
2
M
p
)
=
g
g
=
2
(
0.01558
)(
103.2
)
(
519.2
+
1
2
(
165
)
)
g
=
1934.89
cm
/
s
2
g
=
1934.89
c m
/
s
2
×
1
m
100
cm
g
=
19.349
m
/
s
2
Calculating uncertainty:
σ
g
=
√
(
2
h
(
M
+
1
2
M
p
)
)
2
σ
m
2
+
(
2
m
(
M
+
1
2
M
p
)
)
2
σ
h
2
+
(
2
mh
)
2
σ
M
2
+
(
mh
)
2
σ
M
p
2
1
¿
2
0.346
¿
2
+
(
(
0.43
) (
121.63
)
)
2
¿
0.0006
¿
2
+
(
2
(
0.43
)
(
519.2
+
1
2
(
165
)
)
)
2
(
0.05
2
)+
(
2
(
0.43
) (
121.63
)
)
2
¿
(
2
(
103.2
)
(
519.2
+
1
2
(
165
)
)
)
2
¿
σ
g
=
√
¿
σ
g
=
√
5552.415
+
669.417
+
1309.877
+
4955.739
σ
g
=
111.747
cm
/
s
2
σ
g
=
1.117
m
/
s
2
therefore g
=
19.349
±
1.117
m
/
s
2
Doing a t test to compare the experimental value of
?
you just found with the accepted value of (9.81
±0.01) m/s
2
:
¿
x
1
−
x
2
∨
¿
√
σ x
2
1
+
σ x
2
2
t
=
¿
¿
19.349
−
9.81
∨
¿
√
1.117
+
0.01
t
=
¿
t
=
8.985
Since t is greater than 2 that means x1 and x2 are consistent.
Calculating the value of
𝛤:
Convert d from mm to cm:
121.63
mm×
1
cm
100
mm
¿
1.2163
cm
σ
=
0.05
Γ
=
hd
(
M
+
1
2
M
p
)
Γ
=(
103.2
)(
1
.
2163
)
(
519.2
+
1
2
(
165
)
)
Γ
=
75526.684
T
Calculating uncertainty for
𝛤:
σ
Γ
=
√
(
hd
(
M
+
1
2
M
p
)
)
2
σ
b
2
+
(
bd
(
M
+
1
2
M
p
)
)
2
σ
h
2
+
(
bh
(
M
+
1
2
M
p
)
)
2
σ
d
2
+
(
bhd
)
2
σ
M
2
+
(
bhd
2
)
2
σ
M
p
2
1
¿
2
0.346
¿
2
+
(
(
0.08828
) (
103.2
) (
1.2163
)
2
)
2
¿
0.5
¿
2
+
(
(
0.08828
) (
103.2
) (
1.2163
)
)
2
¿
0.05
¿
2
+
(
(
0.08828
) (
103.2
)
(
519.2
+
1
2
(
165
)
)
)
2
¿
0.0002
¿
2
+
(
(
0.08828
) (
1.2163
)
(
519.2
+
1
2
(
165
)
)
)
2
¿
(
(
103.2
) (
1.2163
)
(
519.2
+
1
2
(
165
)
)
)
2
¿
σ
Γ
=
√
¿
σ
Γ
=
√
(
228.171
)
+
(
10.435
)
+
(
7512492.911
)
+
(
14.7
)
+(
30.698
)
σ
Γ
=
2740.945
thereforethevalue of Γ
=
75526.684
±
2740.945
T
Discussion
The quantities that were being found in this experiment is gravity (g) and torque (T). the values
that were found for gravity is
19.349
±
1.117
m
/
s
2
and the value found for torque is
75526.684
±
2740.945
T
. In the lab itself the setup was very easy and straight forward. Once the
masses were weighted and the uncertainties were written down. You would then set up your experiment
and start collecting your values which has no incidents of falling or disconnecting itself from the
Atwood’s machine. Using the Atwood’s machine is different compared to dropping an object to calculate
the gravity. This is because it’s a fixed system that minimizes sources of external errors like having an
incorrect time due to having a human use a stopwatch. The arrangement of weights is also another
difference as the weights minimizes the effects of random errors to happen so you can get a better value
for gravity. With the addition of using an electronic timer the results will be more accurate. An estimate
error for calculating gravity would be around 2-5m/s
2
since there’s many sources of errors involved in a
straightforward drop. The value of
𝜞 was calculated with a positive value. Since the Atwood’s machine was
spinning counterclockwise you can use the right-hand rule to see if the value is positive or negative. Since
the hands curl in a counterclockwise pattern, it shows that the value should be positive.
Friction plays a small part of error for the calculation of g. This is because the pully system isn’t frictionless.
That means the string is slightly slowed down from the pully and that means the results aren’t 100%
accurate. Air resistance is not a significant source of error because the use of weights is involved in the
Atwood’s machine. The weights pull speeds up the system and any air resistance that is there is minor. The
limitations of the experiment is that you are bound to the weight that the machine can handle. You are also
required to have a weight limit so that the mass on one side could click at the top of the system.
The reason the diameter of the pully was measured from between the strings instead of measuring it directly
from the pulley is because the string isn’t attached to the pully and the string is included in the system. The
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tiny gap between the pully and the string is decently big and could alter the accuracy of the calculations.
Some improvements that could be done in the machine is to have a built-in ruler for the diameter because
when measuring the diameter of the pully if you accidently bumped into the system the whole system would
shake and could potentially make the string fall of the pully system. The shake would also mean you’d have
to take the time to stop the shaking and make sure that the system is motionless. By dividing the dominant
source of uncertainty by a factor of 2 the uncertainty value would be half the value or potentially more. The
smaller the dominant source of uncertainty the smaller the total uncertainty would be.
The calculated value of gravitational acceleration is 19.349 m/s
2
, it is significantly larger than the
variations that geologists would be looking for in oil exploration. This value is more consistent
with the normal gravitational acceleration at the Earth's surface, which is approximately 9.8
m/s
2
. In summary, the precision required for measuring variations in gravitational acceleration
for oil exploration is much higher than the provided value of 19.349 m/s
2
. Instruments used for
such measurements are sensitive enough to detect much smaller changes.