workandenergy-1201 Lab final
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1201A
Subject
Mechanical Engineering
Date
Dec 6, 2023
Type
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8
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Work and Energy - 9
Data and Work Sheets
- Print or bring it on an electronic device
Work and Energy - Physics 1201A
2022-2023
Please circle the appropriate values
Course
1101A
1201A
1401A
1501A
Lab Section
002
003
004
005
006
007
008
009
010
013
014
Lab Subsection
A
B
C
D
Name
First:
Last:
Student #
Lab Partner
First:
Last:
Lab Station #
Date
Demonstrator
Disclaimer: Please note that some but not all questions in this lab writeup will be graded.
EXPERIMENT 1: PRELIMINARY ADJUSTMENT OF THE APPARATUS
DATA AND CALCULATIONS
Levelling the air track
Place the car at one end of the track and send it through both photogates. If the track is hori-
zontal, the timers should read the same time to within 5%. If necessary, adjust the leg(s) at one
end of the track in order to level it.
(Adjust at the end that is not connected to the air hose.)
Timer reading:
t
1
=
t
2
=
Percent difference between the two readings: (Should be less than 5% for better accuracy.)
What conclusion can you draw from the percent difference test?
Senitha
Kumarapeli
2
5
I
2
9
3
0
3
0
clarence
18
Nov
22
2022
Adrien
t
Annika
0.164
0.165
diff
É
Hoot
01
4
100
0.6
257.1
Since
the
percentage
is
less
than
St
it
means
the
track
is
level
Work and Energy - 10
EXPERIMENT 2: CONSERVATION OF MECHANICAL ENERGY
Units in data:
Measure
x
1
,
x
2
,
`
,
D
and
H
in centimeters (cm) and record in cm. Convert data into
SI units in the first step in your calculation. All calculations must be done in SI units.
Uncertainty in x
1
, x
2
: These are ‘positions’ of the photogates, and the uncertainty is
±
0.05 cm.
Uncertainty in
`
,
D
,
H
and
L
:
These are length measurements. The uncertainty is
±
0.1 cm.
However, estimate
d
D
to be higher since
D
is difficult to measure.
Uncertainty in
m
:
Add the uncertainties in masses and the pan balance used.
Record absolute uncertainties in data to one significant figure. Follow the rules for subtraction,
addition, multiplication and division. Keep the correct number of significant figures in derived
quantities.
DATA
Table 1:
Dimensions of the Apparatus
Item
Measurement & Uncertainty
1st Photogate
x
1
±
d
x
1
=
2nd Photogate
x
2
±
d
x
2
=
Length of flag on air car
`
±
d
`
=
Distance between track legs
D
±
d
D
=
Mass of air car
m
±
d
m
=
Height of aluminum tube
H
±
d
H
=
=
=
CALCULATIONS
Convert all units into SI units in your calculations.
d
H
avg
= meter ruler reading uncertainty +
1
/
2
(
H
max
-
H
min
)
d
L
=
d
x
1
+
d
x
2
Distance between photogates
L
±
d
L
=
L
=
x
2
-
x
1
Average H
H
avg
±
d
H
avg
=
0.56m
I
0.0005
1
268m
I
0.0005
0.12
MI
0.001
0.1015m
I
0.001
0.2187kg
I
0.001
0.0099m
I
0.0005
0.01
M
I
0.0005
O
011m
I
0.0005
0.588m
I
0.001
0.01
I
0.0005
Work and Energy - 11
Calculate the vertical height
h
between the points
A
and
B
. Find the absolute uncertainty
d
h. Write
your answer in the form
h
±
d
h
units. Since
h
is an intermediate value, keep
d
h
to two significant
figures (Notes to Students: Section 3.4).
h
=
L
sin
q
=
LH
avg
D
(14)
d
h
h
=
d
L
L
+
d
H
avg
H
avg
+
d
D
D
Record
h
±
d
h
as an intermediate value:
h
±
d
h
(Intermediate value of
d
h
has two significant figures.)
Calculate the car’s initial gravitational potential energy at
A
,
U
A
=
mgh
. Use
g
= 9.804
±
0.001
m/s
2
. Then find the fractional (
d
U
A
/
U
A
) and absolute uncertainties (
d
U
A
) in
U
A
. Record
U
A
in
the form
U
A
±
d
U
A
units.
U
A
=
m g h
(15)
d
U
A
U
A
=
d
m
m
+
d
g
g
+
d
h
h
Intermediate value:
U
A
±
d
U
A
4
0
58844018,01m
0.005793m
8k958m
t
m
t
0.0440m
Sh
0.00291
0.005793
I
0
00291
0
004587
0
000102
0.218kg
9
804m15
0.005793
Vat
0.0124J
Tig
9
4
É
0.0124
80A
0
00628
0.0124
I
0.00628J
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Work and Energy - 12
The values in the last column
E
A
-
E
B
must be close to zero. If they are not, your data may have a
problem. Verify your data and calculations, and contact the demonstrator if necessary.
Table 2: Energy of the Air Car
(initial)
t
A
(s)
±
(final)
t
B
(s)
±
v
A
(m/s)
v
B
(m/s)
K
A
(Joules)
(total initial
energy)
E
A
=
U
A
+
K
A
(Joules)
(total final
energy)
E
B
=
K
B
(Joules)
E
A
-
E
B
(Joules)
Sample calculation of E
A
- E
B
Select one trial from Table 2, and starting from the first column, show how you will calculate
v
A
,
v
B
,
K
A
,
K
B
,
E
A
and
E
A
-
E
B
. Show the equation and the first step of substituting numerical
values into the equation and units.
Each
t
value in the first two columns has three significant
figures. Carefully think of how many significant figures you should keep in subsequent columns.
v
A
=
l
/
t
A
E
A
=
U
A
+
K
A
v
B
=
l
/
t
B
E
B
=
K
B
K
A
=
1
/
2
mv
2
A
E
A
-
E
B
K
B
=
1
/
2
mv
2
B
In an ideal experiment,
E
A
(initial total mechanical energy) should be equal to
E
B
(final total
mechanical energy) because the total mechanical energy in the system is conserved, and therefore
the values in the last column in Table 2 should be zero. However, you will probably find that the
two values,
E
A
and
E
B
, are not exactly equal because of the experimental uncertainties involved.
Next, perform the overlap test to check whether the two values
E
A
and
E
B
are equal within their
experimental uncertainties
(Notes to Students: Section 5).
0.237
0.2060
5060.583
0.0279
0.0403
0.037
0.0033
0.285
0.225
0.421
0.533 0.0193
0.0317
0.0310
0.0007
0.288
0.237
0.4180
5060.0189
0.0313
0.0279
0.0034
0.286
0.229
0.419 0.5200.0191
0.03153
0
02950.0015
0.5060ms
0.01245
0.02795
70.040352
HE
0.5830ms
420.211kg
0.506ms
0.0270g
0.037
0.04035
0.037
J
0.0033
Y2
0.211kg
0.58345
0.03585
Work and Energy - 13
How to compare
E
A
and
E
B
using the overlap test:
Select one trial to perform sample cal-
culations and the overlap test. First we must calculate the absolute uncertainties,
d
E
A
and
d
E
B
.
Absolute uncertainty,
d
E
A
Since
E
A
=
U
A
+
K
A
, you can write,
d
E
A
=
d
U
A
+
d
K
A
. Use the
d
U
A
value found in the last
part. Calculate
d
K
A
using the following equation. Use
t
A
,
v
A
and
E
A
values that you selected from
Table 2.
K
A
=
1
2
m v
2
A
=
1
2
m
✓
l
t
A
◆
2
,
d
K
A
=
✓
d
m
m
+
2
d
l
l
+
2
d
t
A
t
A
◆
K
A
(16)
d
E
A
=
d
U
A
+
d
K
A
The total initial mechanical energy =
E
A
±
d
E
A
Record the above values in
intermediate value
format.
Absolute uncertainty,
d
E
B
E
B
=
U
B
+
K
B
=
K
B
and therefore,
d
E
B
=
d
K
B
.
We can use the Equation (17) to calculate
d
K
B
. However, we have to use
t
B
and
v
B
instead of
t
A
and
v
A
. Use
t
B
,
v
B
, and
E
B
values obtained for the trial you selected.
K
B
=
1
2
m v
2
B
=
1
2
m
✓
l
t
B
◆
2
,
d
K
B
=
✓
d
m
m
+
2
d
l
l
+
2
d
t
B
t
B
◆
K
B
(17)
d
E
B
=
d
U
B
+
d
K
B
=
d
K
B
The total initial energy =
E
B
±
d
E
B
Record the above values in
intermediate value
format.
Ka
0.02795
gpa
4
8
4884
2
8
55
0.02795
0.0047
0.0034
0.0084
0
0229
0
0004615
0.00628
0.000461
0.0067415
0.0403 I
0.006741J
143
0.037
gap
18
1
1
2
8
14
26
60.037
SUB
40.00045
0.0034 0.0048
0
037
0.000325
0.03710
00032J
Work and Energy - 14
OVERLAP TEST:
Finally, compare
E
A
and
E
B
using the overlap test. An example of overlap test
is given in Notes to Students: Section 5.1. Use the intermediate values of
d
E
A
and
d
E
B
(i.e., 2
significant figures).
|
E
A
-
E
B
|
(
d
E
A
+
d
E
B
)
(18)
Is the test successful? If
E
A
=
E
B
, what does this mean? Write your answers in the space above.
EXPERIMENT 3: WORK DONE BY AN EXTERNALLY APPLIED FORCE
DATA AND CALCULATIONS
Table 3: Work Done by an External Force
(initial)
t
A
(s)
(final)
t
B
(s)
v
A
=
`
/
t
A
(m/s)
v
B
=
`
/
t
B
(m/s)
E
A
=
1
/
2
mv
2
A
(Initial energy)
(Joules)
E
B
=
1
/
2
mv
2
B
+
mgh
(Final energy)
(Joules)
W
ext
=
E
B
-
E
A
(Joules)
Experimental
W
ext
=
F L
(Joules)
Theoretical
Average
W
ext
:
Theoretical work done, W
ext
:
The force
F
is the tension in the string attached to the car. An analysis of the system used in this
experiment (Figure 5) shows that the magnitude of
F
is given by Equation (19) where: sin
q
=
H
avg
/
D
,
m
= mass of the air car,
m
1
= 30 g = mass attached to the string. Calculate the theoretical value
of
F
using Equation (19), and then
W
ext
using Equation (20).
F
is an intermediate value. Think
about how many significant figures you should keep in the answer.
F
=
m m
1
m
+
m
1
g
(
1
+
sin
q
)
(19)
10.0403
0.0371
0.00674
0.00032
0.0038
0.007020
yes
testgefessted
This
means
thatthe
mechanical
energy
was
conserved
0.168
0.0990.7143
1.212
0.0558
0.1725
0
11720.1546
0.167
0.9099
0.7185
1.212
0.05627
0
1725
0.1162
0.1546
0.167
0.09
0.7185
4.212 0.05627
0.1725
0.1162 0.1546
0.1165
0.03
5488
1.0169
9.804
0.01
É
fan
0
017
0
0.01701
sina.oizj
qqF
o.am
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Work and Energy - 15
W
ext
=
F L
(20)
Compare the average
W
ext
from Table 3 to the theoretical
W
ext
above using the percent difference
test. Is the test successful? If
W
ext
,
experimental
=
W
ext
,
theoretical
what does that mean? (See the
example in Section 5.2, Notes to Students).
Percent difference test:
W
ext
,
experimental
-
W
ext
,
theoretical
(
W
ext
,
experimental
+
W
ext
,
theoretical
)
/
2
⇥
100
<
10%
(21)
If
W
ext
,
experimental
=
W
ext
,
theoretical
within 10% error, what does that mean?
Post-Lab Question:
Answer the following questions without help from your demonstrator. You
may discuss with your lab partner.
1) Suppose you throw a ball of mass
m
straight up with an initial vertical velocity
v
(m/s) and catch
it later. Answer the following questions assuming there is no air friction or no non-conservative
forces acting on the ball. Define potential energy as zero at the level where the ball leaves your
hand.
Your answers
must contain only
m
and
v
and no other variables like height. You can find the
answers easily if you think about the conservation of mechanical energy of the system.
(a) What is the kinetic energy of the ball when it was leav-
ing your hand?
(b) What is the mechanical energy of the ball when it was
leaving your hand?
(c) What is the mechanical energy of the ball at the maximum height?
(d) What is the mechanical energy of the ball when you catch it later?
2) How much work is done by gravity on the hanging mass in Experiment 3 as it drops a height
L
?
01844
5
4
100
0
027
lot
oozy
lot
08
gg
z
This
proves
that
the
work
energy
theorem
is
true
K
Yz
my
2
K
Yzma
energy
put
into
E
Yzma
the
throw
Ee
Yz
my
2
we
É
d
so
led
and
É
mg
so
the
work
done
by
gravity
can
be
calculated
by
taking
the
multiplication
ofLama
9.804
Work and Energy - 16
CONCLUSIONS AND DISCUSSION
Summarize all your conclusions drawn from your results. Separate your conclusions into different
sections rather than writing an essay. Discuss your results. Are they successful? If not, explain
why? Your results must support your explanation. Do you have any suggestions to improve any
part of your experiment?
Final Mark
In
experiment
1
the
air
track
was
tested
If
the
percentage
difference
test
between
the
two
timer
readings
was
less
than
st
the
track
was
level
In
experiment
2
the
energy
of
a
car
travelling
down
an
inclined
ramp
was
observed
After
calculating
the
intial
and
final
energies
of
the
car
the
conservation
of
total
mechanical
energy
was
proven
to
be
true
using
the
overlap
test
In
experiment
3
the
work
done
by
an
external
applied
force
to
the
car
The
theoretical
work
was
compared
tothe
experimental
work
done
using
the
percentage
difference
test
The
percentage
difference
found
was
below
the
10
uncertainty
range
so
it
proved
that
the
work
energy
theorem
is
true
the
work
done
by
an
external
force
is
equal
to
the
difference
between
the
final
and
initial
mechanical
energies
ofthe
system
since
all
results
were
successful
no
changes
one
needed
tobe
made
to
the
test
An
improvement
could
be
to
repeat
the
exprivent
at
different
heights
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