PHYS 164
LAB #15 Resistors in Series and Parallel
ANALYSIS
Part 1: Resistors in Series - Table 1
I4 (mA)
V4 (V)
Req (W)
Experimental
Result
3.28
±
0.01
4.0
±
0.1
1200
±
12
Predicted
Value
3.6
±
0.1
4.0
±
0.1
1100
±
11
% Diff
10
1
9
Agree?
Yes
Yes
No
Part 2: Resistors in Parallel - Table 2
I4 (mA)
V4 (V)
Req (W)
Experimental
Result
76.00
±
0.01
4.0
±
0.1
48
±
0.48
Predicted
Value
73
±
3
4.0
±
0.1
54.5
±
0.5
% Diff
3.64
0
12
Agree?
Yes
Yes
Yes
PHYS 164
LAB #15 Resistors in Series and Parallel
DATA
NOTE: All resistors have a tolerance of ± 1%
Part 1: Resistors in Series
R1 = 200 W; R2 = 400 W; R3 = 600 W; V = 4 V
I1
I2
I3
I4
I (mA)
3.28
3.28
3.28
3.28
±
0.01
0.01
0.01
0.01
Req (W)
1200
±
V1
V2
V3
V4
V (V)
0.66
1.30
1.97
3.96
±
0.1
0.1
0.1
0.1
Ohm's Law
Prediction
IR (V)
±
0.66
0.01
1.31
0.02
1.97
0.03
12
Part 2: Resistors in Parallel
R1 = 100 W; R2 = 200 W; R3 = 300 W; V = 4 V
I1
I2
I3
I4
I (mA)
39.8
13.2
14.2
76.0
±
0.01
0.01
0.01
0.01
Req (W)
48
±
V1
V2
V3
V4
0.5
V (V)
4.0
4.0
4.0
4.0
±
0.1
0.1
0.1
0.1
Ohm's Law
Prediction
IR (V)
±
3.98
0.04
2.64
0.03
4.26
0.05
Professor Line
PHYS 163 A
Albert Square
April 1, 2024
Experiment #0
Free Fall
My Partners: Emmy Noether
Billy Cobham
Activity Goals:
The goal of this activity is to determine if free-fall motion correctly describes the motion of a
500-g steel ball falling vertically through air.
Procedure:
The procedure used for this activity can be found on pp. 300-305 of the Phys 163 Fall 2024 Lab
Manual.
1
Data:
A 500-g steel ball was dropped from four different measured heights, and the time to reach the ground
was measured.
Ball mass: 500 ± 1 g
Initial
Height 1
y (m)
3.5
±
0.1
Measure
±
1
2
3
Fall
time t
(s)
0.82
0.88
0.86
Avg t
Fall Time
0.85
0.85
0.02
0.03
Initial
Height 3
y (m)
10.5
±
0.1
Measure
±
1
2
3
Fall
time t
(s)
1.48
1.50
1.45
Avg t
Fall Time
1.48
1.48
0.02
0.03
Trial
Trial
0.02
0.02
0.02
0.02
0.02
0.02
Initial
Height 2
y (m)
7
Deviation
from
average
0.03
0.03
0.01
Initial
Height 4
Deviation
from
average
0.00
0.02
0.03
±
0.1
Measure
±
1
2
3
Fall
time t
(s)
1.19
1.20
1.19
Avg t
Fall Time
1.19
1.19
0.02
0.02
Measure
±
1
2
3
Fall
time t
(s)
1.64
1.66
1.63
Avg t
Fall Time
1.64
1.64
0.02
0.02
Trial
y (m)
13.5
0.02
0.02
0.02
Deviation
from
average
0.00
0.01
0.00
±
0.1
Trial
0.02
0.02
0.02
Deviation
from
average
0.00
0.02
0.01
Measurement Uncertainties:
The following quantities were measured:
• Initial height, H (m)
• Time of fall, t (s)
2
Uncertainty in initial height: 10 cm
The ball was dropped by hand from an open window located on each floor of Kirkbride Hall
and its roof. The distance from the location of release to the ground was determined from the
length of a string, weighted at one end and lowered until it reached the ground. Using a 3meter ruler, with reading uncertainty of 0.001 m, the reading uncertainty associated with the
largest distance was estimated at 0.005 m. However, because the ball was dropped by hand
reaching out the open window, an initial height uncertainty of 0.1 m was chosen to account for
a lack of precision in locating the hand at the same height as the bottom of the window.
Uncertainty in fall time: 0.02 s
A digital stopwatch, which displayed time to 0.01 seconds, was used. It was assumed that the
stopwatch worked correctly, so the reading uncertainty was taken to be 0.005 s. However, to
account for reaction times (starting and stopping the stop watch just at the beginning and end of
the fall) an uncertainty of 0.02 s was used. This was determined by making several trials of
timing a second hand on the lab’s wall clock as it ticked-off 2 seconds.
Analysis of Data:
Sample Calculations:
The following calculations were made:
•
Initial height, using a 3-meter ruler. As an example, the following are the calculations used in
finding height 4, along with its uncertainty:
H4 = (3±.001) + (3±.001) + (3±.001) + (3±.001) + (1.5±.001) =13.5± 0.005 m
δH4 = 0.001+ 0.001+ 0.001+ 0.001+ 0.001= 0.005 m
•
Absolute difference from the average – the absolute deviation
For example, the deviation associated with gcalc for the height 4 (H4) data:
10.0 −9.7 = 0.3 ms 2
•
Average value
3
The average value of N measured or calculated quantities, as well as the average uncertainty
for the average value, is determined using the formula
xavg
1 N xi
For example, to determine the average measured value of t (for Initial Height 1 data):
tavg = (0.82 + 0.88+ 0.86) s = 0.85 s
3
For repeated measurements or calculations, the reported uncertainty for the average is
determined by the larger of (the average of the individual uncertainties) and (the largest
absolute difference between the individual quantities and the average of those quantities). The
absolute difference from the average is given by xi − xavg .
•
Uncertainty for repeated measurements
The largest of the deviation values was compared to the average reading uncertainty. The
larger of these two was the reported uncertainty:
For the Initial Height 1 data, 0.03 > 0.02, and the reported uncertainty is 0.03 s.
•
δ tavg = 0.03 s
Calculation of the value for g
The value of g was calculated using the constant acceleration ∆y = vo,y∆t − g(∆t)2
For example, using the height 4 data:
(
)
∆
g =−(2∆ t)y2 =− 2×(1 .−6413s.)52m =10.0ms2
δg = g
δ∆(∆yy)+δ(∆∆tt)+δ(∆∆tt)
10..6402
=10.0
130..15 + 10..6402 +
=10.0(0.0074 + 0.012 + 0.012)
4
=10.0(0.031)= 0.3 ms 2
• Comparison of gcalc to gtheory by % difference
9.7 −9.8
×100% = 0.8%
9.8
•
Comparison of gcalc to gtheory by agreement within uncertainty
For Initial Height 3 data, g = 9.6 ± 0.4 m/s2. This means the experimental value falls
somewhere within the range
9.2ms 2 ≤ gexperimental ≤10.2ms 2
The accepted value for g is 9.81 ± 0.01 m/s2, which falls within the experimentally-determined
range.
Results:
•
2H
Calculation of g, assuming free-fall analysis: g = t 2
Table R1 – Calculated values of g
•
H (m)
±H
t (s)
±t
3.5
0.1
0.85
0.03
9.6
1.0
7.0
10.5
13.5
0.1
0.1
0.1
1.19
1.48
1.64
0.02
0.03
0.02
9.8
9.6
10.0
0.5
0.4
0.3
gcalc (m/s2)
± gcalc
Summary of calculations of g from data: gtheory given in lab manual
Table R2 – Comparison of gexperimental to gtheoretical
5
9.6
±
gexp
1.0
9.8
9.6
10.0
0.5
0.4
0.3
gexp (m/s2)
9.81
±
gtheory
0.01
9.81
9.81
9.81
0.01
0.01
0.01
gtheory (m/s2)
% Diff
Agree?
2.0
Yes
0.2
1.8
1.9
Yes
Yes
Yes
Conclusions and Discussion
When a steel ball is dropped and falls through the air, both the gravitational and air drag forces will act
on the ball.
For free fall motion with constant gravitational force, the acceleration of the ball should be the
same value for all initial heights, and equal to 9.81 m/s2. If air resistance had a significant effect,
the acceleration should decrease with increasing initial height, and be less than 9.81 m/s2.
The results support the conclusion that free-fall motion describes the motion of the falling steel
ball over a range of heights up to 13.5 m.
This conclusion is supported by both graphical and numerical analysis.
For free fall motion with constant gravitational force, the value of g near Earth's surface should be
9.81 ± 0.01 m/s2 ( source: Lab Manual, p. 302 ). Tables R1 and R2 in the Results section provide a
summary of the experimentally determined values of g for different heights.
Table R2 shows that the experimentally-determined acceleration of the steel ball is consistent with 9.81
m/s2 for all four heights investigated – each experimental value agrees with the accepted value of
9.81 ± 0.01 m/s2 within their uncertainties, with percentage differences between 0.2% and 2%.
In addition, Graph 1: Calculated values of g for different fall distances, gives results consistent with a
value for g which is independent of the initial height. This is seen from the fact that it is possible to
draw a horizontal line through the data + error bars; a horizontal line on a graph indicates that the
"dependent variable" is independent of the "independent variable."
It can also be noted that, if air drag was significant, its effect would be to reduce the apparent
value of g with increasing height. On the graph Calculated values of g for different fall distances it
can be seen that a straight line of negative slope could also fit the results, which would be consistent
with a conclusion that air resistance plays a part in the motion. However, the data points themselves do
not appear to have any obvious negative slope trend, so a conclusion of significant air resistance
effects is not very well supported by the data. (See answer to Question below.)
In summary, the results of this activity supports the assumption that a 500-g steel ball, falling from a
height of 13.5 m or less, can be considered as having free fall motion.
6
Question:
1) How would the presence of air resistance show up in the data?
If air resistance were present, the net force acting on the falling ball would be decreased (Fg down,
but FR up, opposite the direction of travel), so the acceleration of the ball would be reduced from
its free fall value. Because the effects of air resistance increase with speed, the effects should also
increase with initial height (the greater the fall distance, the longer the fall time, and the longer the
time for the ball to accelerate). Therefore, the calculated value of g should decrease with
increasing initial height. A possible line that would support the presence of air resistance is shown
and labeled in Graph 1.
7
6
#15 Resistors in Series and Parallel
Objective
The object of this experiment is to provide a simple experimental
study of the characteristics of resistors in series and parallel configura-
tions.
Introduction and Theory
There are two basic ways of connecting circuit elements: in series and
in parallel. To illustrate, suppose we are given three resistors to connect
by means of some wires such that the resistors form a network (a network
is simply a portion of a circuit). If we connect them so that there is only
one conducting path across the network (Fig. 15.1) they are said to be
connected in series. If there is more than one conducting path (Fig. 15.2),
they are said to be connected in parallel.
R
R2
W
R;
w
Fig. 15.1. Three resistors connected in series.
The following statements and equations apply to a network of series-
connected resistors:
1. The current in every part of the series network is the same.
I = I1 = 12 =13
(15.1)
2. The voltage across the network of series-connected resistors is equal to
the sum of the voltages across the individual resistors.
V = V1 + V2 + V3
(15.2)
3. The total resistance of the series network is equal to the sum of the in-
dividual resistances.
R = R1 + R2 + R3
(15.3)
Statement #1 for the series case is a statement of the conservation of
charge, since an electric current is the rate of flow of charge.
Statement #2 for the series case is merely another way of stating that
the energy per unit charge expended in the whole circuit is the sum of the
energy per unit charge expended in the various resistors. This must follow
from the law of the conservation of energy.
Statement #3 is a direct consequence of statements 1, 2, and Ohm's
law, applied to the total resistance R and the individual resistances Ri, R2,
and R3. Using Ohm's law and statements 1 and 2:
V = IR = IR1 + IR 2 + IR 3
(15.4)
whence
R = R1 + R2 + R3
(15.3)
R
R
R3
Fig. 15.2 Three resistors connected in parallel.
When the conductors are connected in parallel (see Fig. 15.2), the re-
spective ends of the resistors are connected to common points and the total
current divides among the resistors. For a network of parallel-connected
resistors the following statements and equations apply:
4. The total current is the sum of the individual currents.
I= 1 + I2 + I3
(15.5)
5. The voltage across any one resistor is the same as that across any other
resistor. This voltage is also the same as that across the entire network.
V = V1 = V2 = V3
(15.6)
6. The reciprocal of the effective resistance of the network is equal to the
sum of the reciprocals of the individual resistances.
1 1
1
1
=
+
+
(15.7)
R R
R₂
R
Statement #4 follows from the fact that charge does not pile up at a
junction. The currents in each of the resistors may be different, being in-
versely proportional to the respective resistances, since the voltage across
each resistor is the same.
Statement #5 follows from the fact that the terminals of each resistor
are connected to common points, so that the difference in potential be-
tween these points is the same no matter through which individual resistor
the path is chosen through.
8
Statement #6 follows from the application of Ohm's law and state-
ments 4 and 5 (the student should verify this).
It is important to observe that the connection of additional resistors in
series increases the total resistance, while inserting additional resistors in
parallel decreases the total resistance.
Apparatus
Check that you have the following items:
low-voltage power supply (LVPS)
three resistance boxes (+1% resistors)
two multimeters (one set as an ammeter, the other as a voltmeter)
assorted wires
-
-
-
Experimental Procedure
Set up the circuit as shown in the various wiring diagrams as directed.
Unless otherwise specified, circuit elements may be inserted with either
terminal connected toward the positive side of the power supply. A low-
voltage power supply (LVPS) will be used in this experiment. Be sure that
all switches on the LVPS are in the off position before plugging in the
power supply. Set the HI/LO switch on the LVPS to the LO position and
adjust the CURRENT control to about half way between the fully CCW
and fully CW positions. The power supply should be plugged in last. In
all cases have the instructor check your circuit before turning on the
power switch!
1. Connect the LVPS (turned off) and three resistance boxes in series as in
Fig. 15.3. Set the resistances to 200, 400, and 600 ohms. Have the in-
structor check your circuit.
LVPS
V
Mw w
R;
R2
R3
Fig. 15.3 Circuit with three series-connected resistors.
2. Turn on the LVPS and set the voltage to 4.0 volts as measured with the
digital voltmeter. Do not adjust the LVPS or the resistances of the re-
sistance boxes while performing the next two steps.
9
3. Measure and record the currents 11, 12, and 13 through each of the three
resistors, as shown in Fig. 15.4. In addition, measure and record the
current entering the network of series-connected resistors, 14.
LVPS
+
V
+
А4
A
+
+
w
AL
A2
w
R2
w
R3
RI
Fig. 15.4 Measurement of current through resistors connected in series.
4. With the ammeter in the circuit at A4 measure and record the voltage
across each resistor and then across the entire network of resistors by
placing the voltmeter in turn at the four positions indicated in Fig.
15.5.
LVPS
+
V
+
A4
+
+
V
V2
V3
w
R1
w
R2
R3
+
V
Fig. 15.5. Measurement of the voltage drop across resistors connected in series.
5. Remove two of the resistance boxes from the circuit. Adjust the re-
sistance of the remaining resistor until the current is equal to 14 in step
3 above. Record the value of this resistance as Req:
6. Connect the LVPS (turned off) to three parallel-connected resistance
boxes as in Fig. 15.6. Set the resistances to 100, 200, and 300 ohms.
Have the instructor check your circuit.
10
LVPS
V
w
R;
w
R
WW
R
Fig. 15.6. Measurement of the current through resistors connected in parallel.
7. Set the voltage on the LVPS to 4.0 volts as measured with the digital
voltmeter. Measure the current I, by inserting the ammeter into the lo-
cation A, shown in Fig. 15.7. In the same manner measure and record
the currents 12 and 13 in resistors R2 and R3, respectively. Finally,
measure and record the current entering the network of parallel-
connected resistors, 14.
LVPS
ur
R
A
W
R
B
A
w
w
R;
Fig. 15.7. Measurement of the current through resistors connected in parallel
8. With the ammeter in position As, measure the voltages across each of
the three resistors, and then measure the voltage V. across points A and
B
11
9. Remove two of the resistance boxes. Adjust the resistance of the re-
maining box until a current is obtained which is equal to the total cur-
rent originally observed by A4 in step 7 above. Record the value of this
resistance as Req.
Analysis of Data
1. From the data obtained in the series combination, calculate values of 14,
V4, and Req using Eqs. 15.1, 15.2, and 15.3.
2. From the data obtained in the parallel combination, calculate values of
14, V4, and Req using Eqs. 15.5, 15.6, and 15.7.
Conclusion
Don't forget to include a table summarizing your final results.
Discussions of results include claims supported by evidence.
When discussing agreement between experimental and expected val-
ues, be sure to explicitly discuss the overlap of the range of values
indicated by their uncertainties. Percentage difference does not
necessarily indicate agreement between two numbers.
Comment on whether or not the results from the series and parallel
circuits agree with the predictions, within the uncertainties. What are some
of the possible sources of error in this experiment?
Questions
1. A rheostat is simply a resistor with adjustable resistance. Design a cir-
cuit containing a lamp and a rheostat such that adjustment of the rheo-
stat will result in the dimming or brightening of the lamp. Explain
clearly why your circuit works correctly.
2. In a house, multiple electrical outlets are parallel-connected in a single
115-V “circuit,” which also contains a 15-A circuit breaker. Draw a
circuit which shows three such outlets connected to a circuit breaker and
115-V power source. What is the advantage of having outlets parallel-
conn ed, and why is a circuit breaker a necessary safety item in the
circuit?
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