P185 Lab Manual (Drum) Revised 4/16/14
1
Orange Coast College
Physics 185 Lab Manual
Contents
Subject
Page
Lab 1
Galileo and the Pendulum
2
Lab 2
Free-fall
5
Lab 3
Understanding Motion
8
Lab 4
Force Vectors
11
Lab 5
Newton’s 2nd Law
12
Lab 6
Forces and Collisions
14
Lab 7
Energy Conservation
15
Lab 8
Momentum and Collisions
19
Lab 9
Buoyancy
21
Lab 10
Heat and Temperature
24
Lab 11
Simple Harmonic Motion
27
Lab 12
Resonance
31
Lab 13
Statistics
34
Lab 14
Velocity of Sound
33
Lab 15
Cannonball Range
37
Lab 16
Fluid Flow
41
Lab 17
Moment of Inertia
45
Appendix
48
P185 Lab Manual (Drum) Revised 4/16/14
Lab 1: Galileo’s Pendulum
2
Name: _________________________
1. Period and amplitude
Find the period of a pendulum for various amplitudes.
Stamp
Table 1.1 Period vs. Amplitude
 ()
# of Osc.
Total t
Total t
trial 1
trial 2
T
2
4
6
8
10
15
20
30
40
50
60
1.2
Graph T() with the y-axis starting at 0.
Graph T() with the y-axis covering just the range of your data.
Each graph should have a linear fit line.
In your report, discuss the validity of Galileo’s hypothesis based on your
evidence.
P185 Lab Manual (Drum) Revised 4/16/14
3
2. Finding g with a pendulum
2.1
For one small (< 5) amplitude, find T for two trials of 100 oscillations each.
()
N
Total t
Total t
Trial 1
Trial 2
T
100
2.2
Re-write the equation T  2 d / g to find g.
Length of string:
__________
Diameter of ball:
Total length d:
__________
Period T:
Experimental g:
__________
Theoretical g:
__________
__________
__________
P185 Lab Manual (Drum) Revised 4/16/14
4
Error Analysis: Determining basic errors
List all measured quantities and the measuring instrument. Include units.
List the errors on each quantity. Briefly indicate how you determined the errors.
Quantity
Errors
& Instrument
t
Read.
Cal.
Other

Read.
Cal.
Other
L
Read.
Cal.
Other
D
Read.
Cal.
Other
How determined
P185 Lab Manual (Drum) Revised 4/16/14
Lab 2: Free-fall
5
Name: _________________________
Position and speed vs. time
Find the time for the ball to drop various distances.
Stamp
For h = 5 to 35 cm, take a data point every 5 cm.
For h = 40 to 70 cm, take a data point every 10 cm.
For h = 80 to 120 cm, take a data point every 20 cm.
Table 2.1 Position vs. time
x
t1
t2
tAVG
x
2. Analyzing the data
2.1
Make four graphs:
(1)
(2)
(3)
(4)
Graph v(t) using Excel and a linear fit line
Graph x(t2) using Excel and a linear fit line
Graph x(t) using Excel and a power fit line
Graph x(t) using Data Studio and a power fit line
t
v
t1/2
P185 Lab Manual (Drum) Revised 4/16/14
6
Print each graph. For each graph,




Write the equation you expect from the theory of kinematics.
Write the equation of the computer-generated fit line to your data.
Circle each number or variable in the kinematic equation and draw a line to the
corresponding quantity in the computer equation.
Calculate the value of “a”. Find your average “a” and compare to the book value.
Kinematic
Equation
Value
for a
V(t)
Computer
Equation
Kinematic
Equation
Value
for a
x(t2)
Computer
Equation
Kinematic
Equation
Value
for a
x(t)
Computer
Equation
Kinematic
Equation
x(t)
Computer
Equation
Questions
1.
2.
Is the acceleration constant? How do you know?
Is the object in “free fall?” How do you know?
Value
for a
P185 Lab Manual (Drum) Revised 4/16/14
7
Error Analysis: Determining basic errors
List all measured quantities and the measuring instrument. Include units.
List the errors on each quantity. Briefly indicate how you determined the errors.
Quantity
Errors
& Instrument
t
Read.
Cal.
Other
x
Read.
Cal.
Other
How determined
P185 Lab Manual (Drum) Revised 4/16/14
Lab 3: Understanding Motion
8
Name: _________________________
1. Position graph
Get the motion sensor set up and working properly. Pull up the
position-match graph.
Stamp
DO NOT PRACTICE FOR THE FIRST TRY.




Make ONE trial where you match your movement to the graph.
Print out this trial.
Delete the graph.
Let the next person try.
When everyone in the group has printed their trial, sit down and analyze your
graph. Look at the graph and circle regions where your graph doesn’t match the
red line. Explain why the two graphs don’t match. Be as specific as possible.
1.2
When everyone is finished with part 1.1, make more trials to match the graph.





Make several trials to you match your movement to the graph. Keep going until
you have one you think is a good fit.
Delete all the graphs but your best.
Print out this trial.
Delete the graph.
Let the next person try.
2. Velocity graph
2.1
This procedure is exactly the same, but using the velocity-match graph.
Questions
1.
Which graph, x(t) or v(t), was smoother. Why?
2.
Which one was easier to match? Why?
P185 Lab Manual (Drum) Revised 4/16/14
Lab 4: Force Vectors
9
Name: _________________________
1. Theory


For an object in equilibrium, FNET  0
Stamp
2. Two forces
2.1
Put two hangers on the table: 100 g for m1 at 0 and an empty
hanger for m2 at 180.
Find the range for m2 at equilibrium. MMIN = _____ g
mMAX = _____ g
3. Three forces
3.1
Put three hangers on the table: 100 g for m1 at 0, 200 g for m1 at 160 and an
empty hanger for m3.
What mass and angle on the third hanger do you predict will create equilibrium?
Draw a free-body diagram and find the vector components; do not use any other
method. Check your prediction with the instructor before hanging any weights.
PREDICTED = _______
MEASURED = _______
% difference _______
mPREDICTED = _______
mMEASURED = _______
% difference _______
P185 Lab Manual (Drum) Revised 4/16/14
10
4. Three forces
4.1
Put three hangers on the table: 100 g for m1 at 0, 150 g for m2 at 140 and an
empty hanger for m3. Find the mass and angle for m3 by trial and error.
M3 = _______
3 = _______
What is the net force? Draw a free-body diagram and show all your work.
P185 Lab Manual (Drum) Revised 4/16/14
11
Error Analysis: Absolute and % Errors
Pick one data point from part 1 and one point from part 2 and estimate the uncertainty
on each quantity.
Express each uncertainty as both an absolute number and a relative number.
To change between %  to absolute  for a measurement of “x” use:
% 
Quantity
READING
Abs.
m

m

abs 
100
x
abs  
%
x
100
CALIBRATION
%
Abs.
%
OTHER
Abs.
%
P185 Lab Manual (Drum) Revised 4/16/14
Lab 5: Newton’s 2nd Law
12
Name: _________________________
1. Predicting acceleration
1.1
Stamp
A cart on a frictionless track is attached to a weight. The
cart’s mass is M and the weight’s mass is m.
Draw two free-body diagrams: one for the cart and
one for the weight. Don’t include friction or air
resistance.
M
Write FNET = ma for the cart in terms of M, m, g, and
the tension T.
m
Write FNET = ma for the hanging weight in terms of M,
m, g, and T.
Solve for a.
2. Measuring F and a
2.1
You instructor will explain how you will measure a.
For each of the weights in the table below, find your expected acceleration and
then measure a.
Table 2.1 Acceleration for Different Weights
m (kg)
aEXPECTED
aMEASURED
% Difference
.050
.100
.150
.200
% Diff 
aMEASURED  aEXPECTED
100
aEXPECTED
P185 Lab Manual (Drum) Revised 4/16/14
3.
13
Inclined plane
Set up an inclined plane. Measure the angle:  = _____
Draw a free-body diagram and predict your cart’s acceleration. Include friction.
Measure a both up the ramp and down the ramp. Use this data to estimate the
force of friction on the cart.
Error Analysis: Absolute and % Errors
Pick one data point from part 1 and one point from part 2 and estimate the uncertainty
on each quantity.
Express each uncertainty as both an absolute number and a relative number.
To change between %  to absolute  for a measurement of “x” use:
% 
Quantity
abs 
100
x
READING
Abs.
abs  
%
x
100
CALIBRATION
%
Abs.
%
OTHER
Abs.
%
P185 Lab Manual (Drum) Revised 4/16/14
Lab 6: Forces and Collisions
14
Name: _________________________
1. Collisions
Cart A collides with cart
B. Sensors measure
the forces during the
collision.
B
A
Stamp
In the table, various combinations of bumper, mass, and motions are
given. Cart A always moves to the right initially. Fill in the
“Predicted” column with a >, =, or < symbol.
Clay
Rubber
Magnetic
Bumper
Table 1.1 Forces in Collisions
Mass
B
FA ? FB
Maximum F
Move
s
Predicted
FA
A=B
Left
A=B
Stop
A>B
Left
A>B
Overtake
A=B
Left
A=B
Stop
A>B
Left
A>B
Overtak
A=B
Left
A=B
Stop
A>B
Left
A>B
Overtak
FB
Measured
>, <. =
Impulse
IA
IB
Measured
>, <. =
P185 Lab Manual (Drum) Revised 4/16/14
Lab 7: Energy
15
Name: _________________________
1. Definitions

The formulas for kinetic energy (KE) and potential energy
(PE) are:
KE  (1/ 2)mv2
Stamp
PE  mgy
1.1
Open C:\Program Files\DataStudio\Library\Physics\P01
1.2
Write a formula expressing PE in terms of m, g, and x:
2. Energy
2.1
On the left-hand graph below, draw a plot of what you expect a graph of
the PE to look like as the cart goes up and down the track.
On the right-hand graph below, draw a plot of what you expect a graph of
the KE to look like as the cart goes up and down the track.
0.5
0.5
PE (J)
KE (J)
0
0
-0.5
-0.5
0
1
2
2
3
t (s)
4
5
0
1
2
2
Graph 2.1: PE and KE of a Coasting Cart
3
t (s)
4
5
P185 Lab Manual (Drum) Revised 4/16/14
16
Find : ________ 
2.2
Weigh your cart. m: ________ (kg)
2.3
Get a good run for position vs. time. Check with your instructor. Erase all but the
best run.
2.4
Your instructor will show you how to use the calculator function.
2.3

To graph PE, click on the calculator icon and enter the formula for PE.

To graph KE, click on the calculator icon and enter the formula for KE.

To graph E, click on the calculator icon and enter the formula for E.
Draw graphs of PE and KE on graph 2.1 using solid lines. How are the two
graphs different? Explain any differences in your lab report.
Print out your graphs of PE, KE, and total E.
2.4
Find the total energy for the beginning and end of the run.
EINITIAL _________ J
What % of the original energy was lost?
% lost 
EFINAL _________ J
Where did it go?
E FINAL  E INITIAL
 100
E INITIAL
P185 Lab Manual (Drum) Revised 4/16/14
3.
Falling Weight
3.1
Set up a cart with a pulley and a hanging weight. Obtain a graph of x(t) for the
cart as the weight falls.
Create a graph of kinetic energy vs. time.
Create a graph of GPE for the falling weight vs. time.
Create a graph of total energy vs. time.
17
P185 Lab Manual (Drum) Revised 4/16/14
18
Error Analysis: Combining Errors
Find the uncertainties on “m” and “v” for two different trials. Express each uncertainty
as both an absolute number and a relative number.
To change between % to absolute  for a measurement of “x” use:
% 
abs 
100
x
abs  
%
x
100
Find the total error on each measurement.
To find the total error use:    READING   CALIBRATION   OTHER
2
Quantity
READ
Abs
2
CAL
%
Abs
2
OTHER
%
Abs

%
Abs
%
P185 Lab Manual (Drum) Revised 4/16/14
Lab 8: Conservation of Momentum
19
Name: _________________________
1. Collisions
In this lab we will measure the momentum of colliding carts.
Stamp
Magnetic bumpers: 1. Equal mass cars, one car at rest.
2. Equal mass cars, both cars in motion.
3. Unequal mass cars, both cars in motion.
Clay bumpers:
4. Equal mass cars, one car at rest.
5. Equal mass cars, both cars in motion.
6. Unequal mass cars, smaller car at rest.
“Explosion:”
7. Equal mass cars.
8. Unequal mass cars.
In your lab report, you will want to include the following:



Trial 1
Initial
Final
m
Final
Final
p
KE
Totals
Initial
Cart B
Final
Cart A
%Lost
p
KE
p
KE
p
KE
Cart B
m
v
p
KE
Totals
Cart A
Initial
Cart B
Final
Cart A
%Lost
Cart B
Trial 3
Initial
v
Cart A
Trial 2
Initial
Was momentum conserved in all cases, according to your data?
When was kinetic energy conserved, according to your data?
Was the percent difference always a reliable measure? If not, what else would
you use to compare initial and final states?
m
v
p
KE
Totals
Cart A
Initial
Cart B
Final
Cart A
%Lost
Cart B
P185 Lab Manual (Drum) Revised 4/16/14
Trial 4
Initial
Final
Anaylsis
Final
Final
Final
Final
Totals
Final
Cart A
%Lost
p
KE
p
KE
p
KE
p
KE
p
KE
Cart B
m
v
p
KE
Totals
Cart A
Initial
Cart B
Final
Cart A
%Lost
Cart B
m
v
p
KE
Totals
Cart A
Initial
Cart B
Final
Cart A
%Lost
Cart B
m
v
p
KE
Totals
Cart A
Initial
Cart B
Final
Cart A
%Lost
Cart B
Trial 8
Initial
KE
Cart B
Trial 7
Initial
p
Initial
Trial 6
Initial
v
Cart A
Trial 5
Initial
m
20
m
v
p
KE
Totals
Cart A
Initial
Cart B
Final
Cart A
%Lost
Cart B
P185 Lab Manual (Drum) Revised 4/16/14
21
Analysis: Using the percent difference formula
If two momentums are opposite and almost equal, or both zero, you will get a total
momentum of nearly zero. Using the percent error formula will give you a meaningless
number in this case. Consider comparing the absolute values or using some other
method to decide if momentum is being conserved.
Analysis: Kinetic Energy
Under what conditions do you expect KE to be conserved? Partially conserved?
Error Analysis: Combining Errors
Find the uncertainties on “m” and “v” for two different trials. Express each uncertainty
as both an absolute number and a relative number.
To change between % to absolute  for a measurement of “x” use:
% 
abs 
100
x
abs  
%
x
100
Find the total error on each measurement.
To find the total error use:    READING   CALIBRATION   OTHER
2
Quantity
READ
Abs
m
v
m
v
2
CAL
%
Abs
2
OTHER
%
Abs

%
Abs
%
P185 Lab Manual (Drum) Revised 4/16/14
Lab 9: Buoyancy
22
Name: _________________________
1. Theory
1.1
The theoretical buoyant force is given by FB  gV



Stamp
ρ = 1000 kg/m3 for water
g = 9.8 m/s2
V is the volume of the object in m3
To measure the buoyant force, compare the weight of an object in and out of the
water: FB  WOUT  WIN
The volume for various shapes is
4
V ( sphere )  R 3
3
V (cylinder )  R 2 h
V (block )  LWH
For this lab, use meters, kilograms, and newtons.
2. Predicting Buoyancy
2.1
For each object, measure the dimensions and calculate V and FB.
Table 2.1 Theoretical Buoyant Force
Object
Dimensions (m)
V (m3)
FB (Theory) (N)
P185 Lab Manual (Drum) Revised 4/16/14
23
3. Measuring Buoyancy
3.1
Calibrate your force sensor. Your instructor will explain this. Find the buoyant
force of various objects. Compare to the predictions.
% Difference 
Measured  Theoretical
 100%
Theoretical
Table 3.1 Measured Buoyant Force
Object
WIN (N)
WOUT (N)
FB (Measured)
Table 3.2 Summary
Object
FB (Theory)
FB (Measured)
% Diff
4. Capacity of a boat
4.1
Find the maximum buoyant force the water could exert on your “boat” (really it’s a
tuna can). Show your work on a separate sheet.
Use this to predict the maximum load of your boat.
4.3
Load up your boat until it sinks. How much could it hold?
Predicted Capacity: __________
Measured Capacity: __________
P185 Lab Manual (Drum) Revised 4/16/14
24
Error Analysis: Propagation by Substitution; Comparing Numbers Using Errors
For a sphere, find the error on r. For a cylinder, find the errors on r and h.
If you have a relative error, convert it to an absolute error.
Quantity and Value
READ
CAL
OTHER

rS=
rC=
hC=
Find the largest and smallest possible value for each quantity.
Quantity
Max Value
Min Value
rS=
rC=
hC=
Find the largest and smallest possible value for each quantity.
Quantity
Max Value
Min Value
FB,S =
FB,C =
Express FB in “” notation
FB,S = ____________ ± ____________
FB,C = ____________ ± ____________
Compare this value of F to the theoretical calculation. Are the two values comparable
within the stated uncertainty?
P185 Lab Manual (Drum) Revised 4/16/14
Lab 10: Specific Heat & Abs. Zero
25
Name: _________________________
1. Theory


Heat energy (Q) is related to temperature by Q  mcT
The ideal gas law is PV  nRT , T in kelvins.
Stamp
2. Specific Heat of a Metal
2.1
Weigh out a metal sample in a cup. Add about 200 cc of hot
water. Find c.
Metal
Aluminum
Copper
Iron
Lead
Amount
200 g
500 g
500 g
800 g
Metal
Metal
mMETAL
c
MWATER
TINITIAL
cBOOK
TFINAL
% Diff
3. Constant-volume Thermometer
3.1
Graph P vs. T. Graph your data. Draw a fit line and extend it back to P = 0. At
what T does P = 0? What is the significance of this?
Table 3.1 Pressure of air at different temperatures
T (C)
P (kPa)
P185 Lab Manual (Drum) Revised 4/16/14
26
Error Analysis: Propagation by Substitution; Comparing Numbers Using Errors
For a sphere, find the error on r. For a cylinder, find the errors on r and h.
If you have a relative error, convert it to an absolute error.
Quantity and Value
READ
CAL
OTHER

Find the largest and smallest possible value for each quantity.
Quantity
Max Value
Min Value
Find the largest and smallest possible value for each quantity.
Quantity
Max Value
Min Value
Express FB in “” notation
FB,S = ____________ ± ____________
FB,C = ____________ ± ____________
Compare this value of I to the theoretical calculation. Are the two values comparable
within the stated uncertainty?
P185 Lab Manual (Drum) Revised 4/16/14
Lab 11: Simple Harmonic Motion
27
Name: _________________________
1. Static stretching and Hooke’s Law
Find the stretching distance as a function of force.
Stamp
Table 8.1 Stretching vs. Force
m
x
x
F
0
2. Period of oscillation
2.1
Find the period of oscillation for various masses. Leave “M” blank for now.
Table 8.2 Period vs. m
m (g)
100
150
200
250
300
350
400
450
500
# of Osc.
Total t
trial 1 trial 2
T
M
P185 Lab Manual (Drum) Revised 4/16/14
28
3. Analysis
3.1
The period “T” is related to m and k by T  2 (m  m) / k where m is the moving
mass and m’ is the effective mass of the spring.; m’ is not the same as the
spring’s actual mass. Re-write this to get m(T2):
Plot F(x). Write the theoretical relationship in table 8.3 and find k.
Make a plot of m(T2). Write the theoretical relationship in table 8.3 and find k.
The effective oscillating mass is M = m + m’. Fill in the last column of table 8.2.
Make a graph of T(M). Write the theoretical relationship in table 8.3 and find k.
Table 8.3 Fit-line Analysis
Theoretical
Equation
Value
for k
F(x)
Computer
Equation
Theoretical
Equation
Value
for k
m(T2)
Computer
Equation
Theoretical
Equation
Value
for k
T(M)
Computer
Equation
What fraction of the spring’s mass is its effective mass? m´/mSPRING = ________
P185 Lab Manual (Drum) Revised 4/16/14
29
4. Effective mass of a vibrating rod
4.1
Find the period of oscillation for various masses at the end of a rod.
Table 8.4 Period vs. m
m
# of Osc
Total t
trial 1
trial 2
T
Make a plot of m(T2). Use a fit line to find the slope and intercept. Include units.
Slope: ________
kDYNAMIC: ________
Intercept: ________
m’: ________
What fraction of the rod’s mass is its effective mass? m´/mROD = ________
P185 Lab Manual (Drum) Revised 4/16/14
30
Error Analysis: Propagation by Substitution; Comparing Numbers Using Errors
Write the formula for “k” in terms of “T,” “m,” and “m´”
Pick one data point and find the errors on T,” “m,” and “m´.” If you have a relative error,
convert it to an absolute error.
Quantity and Value
READ
CAL
OTHER

T=
m=
m´ =
Find the largest and smallest possible value for each quantity.
Quantity
Max Value
Min Value
T
m
m´
Express “k” in “” notation k = ____________ ± ____________
Compare kSTATIC to kDYNAMIC using this uncertainty. Are the two values comparable
within the stated uncertainty?
P185 Lab Manual (Drum) Revised 4/16/14
Lab 12: String Resonance
31
Name: _________________________
1. Theory
Everything has one or more natural frequencies of vibration and
will resonate at these frequencies.
Stamp
2. Resonances and nodes
2.1
Set L to 1 m. Put a mass of 200 g on your string. Find the first five resonant
frequencies. For each resonance measure  and calculate the wave speed.
Make a graph with n on the x-axis and f on the y-axis. Print the graph.
n
f (Hz)
 (m)
v (m/s)
1
2
3
4
5
Table 2.1 Resonances
3. Resonance and length
3.1
Set m = 200 g. For lengths from 20 cm to 1 m, find the resonant frequency f.
Make a graph with L on the x-axis and f on the y-axis. Print the graph.
L (cm)
20
40
60
80
f (Hz)
Table 3.1 Resonant Frequency vs. Length
4. Resonance and tension
4.1
Set L = 1 m. For masses below find the resonant frequency f.
m (g)
50
200
800
f (Hz)
Table 4.1 Resonant Frequency vs. Mass
100
P185 Lab Manual (Drum) Revised 4/16/14
Lab 14: Speed of Sound
32
Name: _________________________
1. Finding resonances
Find the resonances for a variety of frequencies. The practical
range for f is 800 Hz to 3 kHz.
Stamp
Table 10.1 Resonance Frequencies
f
f
Nominal Actual
Position of nth Resonance
1
2
3
4
5
6
7
8

v
800
1000
1200
1400
1700
2000
2500
3000
1.2
The speed of sound depends on temperature. Find the average of your
measured speeds, the book value for v, and the book value corrected to room
temperature. Use the formula
v1
T
 1 .
v2
T2
T (ºC)
Experimental average
Book value
Book value at room T
T (K)
v (m/s)
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33
Error Analysis: Propagating Errors by Formula
Write the formula for v: _________________________________
Find the errors on each quantity used in your formula. If you have a relative error,
convert it to an absolute error.
READ
Quantity
CAL
OTHER

f
Find the error on v. To find this you will need to use derivatives:
 v, 
dv
 
d
 v, f 
dv
 f
df
Add the errors to get the total error on v:
 v   v , 2   v , f 2
Compare your v to the book value of v using this uncertainty.

P185 Lab Manual (Drum) Revised 4/16/14
Lab 13: Statistics
34
Name _________________________
1. Trials and Randomness
Stamp
1.1 Put 8 pennies in a cup. Shake the cup, dump the pennies out,
and count the number of heads. This is a “trial.”
Make a mark in the appropriate column on table 1. For example,
if you get 5 heads, put an “X” in column labeled “5.”
2002
Do 20 trials per person, making an “X” for each one. Every
person should keep his or her own record.
2002
2002
After this, combine all the data from the entire group.
This is called a histogram or a bar graph.
2002
Column
0
1
2
3
4
5
6
7
8
# Heads
0
1
2
3
4
5
6
7
8
Table 1.1: Tossing 8 Pennies Per Trial
P185 Lab Manual (Drum) Revised 4/16/14
35
2. Larger Numbers
2.1
Repeat exercise 1, but use 32 pennies for each toss instead of 8.
Column
0
1
2
3
4
5
6
7
8
#Heads
0-2
3-6
7-10
11-14
15-18
19-22
23-26
27-30
31-32
Table 2.1: Tossing 32 Pennies Per Trial
3. Averages
3.1
Your instructor will tell you how to find the various different kinds of averages.
“Heads”
Expected
Mode
Mean
Median
8 pennies
32 Pennies
P185 Lab Manual (Drum) Revised 4/16/14
36
3.2
The mode is the column with the most X’s. There may be more than one.
3.3
The mean is the total number of heads for all trials ÷ by the number of trials.
8 Pennies
Column
3.4
# of X’s
32 Pennies
# of Heads
Column
0
1
1
4.5
2
8.5
3
12.5
4
16.5
5
20.5
6
24.5
7
28.5
8
31.5
Total
Total
# of X’s
# of Heads
Use the following worksheet to find the median:
Start by adding all the X’s in each column.
8¢
32 ¢
(a) Which column did you get to without exceeding 30?
______
______
(b) What was the total number of X’s up to this point?
______
______
(c) How many more X’s would you need to equal 30?
______
______
(d) How many X’s are in the next higher column?
______
______
(e) Calculate the median: a  (c / d )  (1 / 2)
______
______

In what situations would the mode be the most useful average?

In what situations would the mean be the most useful average?

In what situations would the median be the most useful average?
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37
4. Comparing the Two Graphs.
4.1
Use Excel to graph your results for parts 1 and 2 on the same graph. Use a
graph that connects the data points.
4.2
In what way are the two graphs different? The same?
4.3
Which experiment (8 pennies or 32) is more likely to give an “unusual” result?
4.4
In which case is an unusual result more significant, when the group being tested
is large or small? (Hint: what do I mean by “significant”?)
5. Proof
The 5-year survival rate for leukemia is about 50%, meaning about half of all
people diagnosed with leukemia will be alive after five years.
Suppose you have an experimental drug which you give to a group of mice with
leukemia. You start with eight mice and observe that, after the mouse equivalent
of five years, 6 are still alive (that’s 75% of the mice).
What are the odds that this would happen by random chance?
_______
What are the odds that the increased survival rate is from the drug? _______
If you work for the FDA (Food and Drug Administration), would you approve this
drug for use in leukemia patients? Would you fund more experiments?
You decide to do a larger trial. You test 32 mice and find that 24 survive (again,
this is 75% of the mice).
What are the odds that this would happen by random chance?
_______
What are the odds that the increased survival rate is from the drug? _______
If you work for the FDA (Food and Drug Administration), would you approve this
drug for use in leukemia patients? Would you fund more experiments?
P185 Lab Manual (Drum) Revised 4/16/14
Lab 15: Cannonball Range
38
Name: _________________________
1. Theory

The range of a cannonball depends on the angle of launch.
Stamp
2. Range vs. Angle
2.1
Take shots to cover the range of 10 to 75.
Table 3.1 Range vs. Angle
Angle ()
Range
Range
(1 click)
(2 clicks)
10
15
20
25
30
35
40
45
50
55
60
65
70
75
Graph your data on the computer. Put both sets of data on one graph.
Print the graph. By hand, draw a smooth line through your data points.
P185 Lab Manual (Drum) Revised 4/16/14
2.2
39
Place a target at a distance you know you can hit.
Using your graph, predict the angle needed to hit this target.
Distance to target: ________ cm
Predicted angle: ________ 
Now try to hit the target. How close were you? D: __________ cm
3. Accuracy
3.1
Set your angle to 45 and find the range for 24 shots (1 click). Measure the
range as accurately as possible.
Table 3.1 Variation of Range
Find the average of all 24 shots. Average range: _________ cm
Cross off the lowest 4 ranges and the highest 4 ranges. Write down the lowest
and highest ranges remaining.
Low: __________ cm
High: __________ cm
Take half of the difference between the low and the high. This is the uncertainty
in the range.
Express your range in “plus-or-minus” notation.
Range = __________  __________ cm
P185 Lab Manual (Drum) Revised 4/16/14
40
Error Analysis: Absolute and % Errors
Pick one data point from part 1 and one point from part 2 and estimate the uncertainty
on each quantity.
Express each uncertainty as both an absolute number and a relative number.
To change between %  to absolute  for a measurement of “x” use:
% 
Quantity
READING
Abs.
R

R

abs 
100
x
abs  
%
x
100
CALIBRATION
%
Abs.
%
OTHER
Abs.
%
P185 Lab Manual (Drum) Revised 4/16/14
Lab 16: Fluid Flow
41
Name: _________________________
5. Flow rate in a fixed column
The flow rate of a fluid is f  m / t .
Stamp
If the flow rate “f” is proportional to the height of water in a tube
and the density of water is ρ = 1 g/cm3, then
f (h)  kh
;
h(t )  h0e  ( k / A)t
Find the flow rate in g/s for various column heights. Remember,




Write down the number on the column as “x,” then find the height “h.”
Keep the level of fluid constant while measuring f.
Keep your collection times reasonable.
Vary the collection height from 20 cm 100 cm.
Dia. Of tube: ________
Area of tube: ________
Table 4.1 Flow Rate vs. Mass
x
h of H2O
in tube
m of H2O
in cup
Collection Flow rate
time
1
2
3
.4
5
6
7
8
9
10
Make a graph of f(h) with a linear fit. Is our model of fluid flow justified?
From this graph, kFIXED = ________
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42
2. Flow rate in a decaying column
Table 7.1 Find the flow in g/s as the column drains. You’ll need to use teamwork.
Table 4.2 Height vs. Time
x
h of
H2O in
tube
t
x
1
13
2
14
3
15
4
16
5
17
6
18
7
19
8
20
9
21
10
22
11
23
12
24
h of
H2O in
tube
t
Use Data Studio to graph h(t). Estimate “kDECAY” (show your work):
Use this estimate of k to create an exponential fit.
Make a semilog plot of h(t) and fit a line to it.
P185 Lab Manual (Drum) Revised 4/16/14
43
Table 4.3 Fit-line Analysis
Value
for k
Theoretical
Equation
f(h)
Computer
Equation
Value
for k
Theoretical
Equation
h(t)
Computer
Equation
Value
for k
Theoretical
Equation
lnh(t)
Computer
Equation
kFIXED______
kDECAY = _____
% Diff = ________
KDECAY,1 = ________
kDECAY,2 = ________
% Diff = ________
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44
Error Analysis: Absolute and % Errors
Pick one data point from part 1 and one point from part 2 and estimate the uncertainty
on each quantity.
Express each uncertainty as both an absolute number and a relative number.
To change between %  to absolute  for a measurement of “x” use:
% 
Quantity
READING
Abs.
x
t
x
t
abs 
100
x
abs  
%
x
100
CALIBRATION
%
Abs.
%
OTHER
Abs.
%
P185 Lab Manual (Drum) Revised 4/16/14
Lab 17: Moment of Inertia
45
Name: _________________________
1. Finding I
Draw a diagram of the apparatus here:
Stamp
Falling mass m = ________
Rotating weights M = ________
Dia. Of spindle d = ________
Width. Of weights W = ________
Write an expression for:
a) a in terms of h and t:
____________________
b)  in terms of m, g, and a:
____________________
c) α in terms of a and R:
____________________
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46
For the weights in various radial positions, find the moment of inertia I.
Table 7.1 I for various r’s
No M
Position of
masses
r
Height of
falling weight
h
Time for
weight to fall
t
Accel. Of
falling weight
a
Torque

Angular
acceleration

Moment of
inertia
I
I of masses
I – I0
1
2
3
4
5
0
Make three graphs: I(r), I(r + d/2), and I(r – d/2).
Write down the computer fit equations for each graph. Which works best? Why?
Equation
Value for
M
Value for n
Theory I®
Computer Fit I®
Computer Fit I(r + d/2)
Computer Fit I(r – d/2)
Find the moment of inertia for a disk and a ring and compare to theory.
P185 Lab Manual (Drum) Revised 4/16/14
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Error Analysis: Propagation by Substitution; Comparing Numbers Using Errors
Write the formula for “I” in terms of “h,” “t,” “m,” and “r.”
Pick one data point and find the errors on “h,” “t,” “m,” and “R.” If you have a relative
error, convert it to an absolute error.
Quantity and Value
READ
CAL
OTHER

h=
t=
m=
r=
Find the largest and smallest possible value for each quantity.
Quantity
Max Value
Min Value
h
t
m
R
Express “I” in “” notation I = ____________ ± ____________
Compare this value of I to the theoretical calculation. Are the two values comparable
within the stated uncertainty?
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48
Error Analysis for Physics 185
Overall Goals:
1.
Distinguish between systematic errors, random errors, and blunders.
Determine basic uncertainties: reading error, calibration error, and other errors.
2.
Understand absolute and % errors.
3.
Combine errors into a total and compare numbers using uncertainties.
4.
Propagate errors by direct substitution.
5.
Propagate errors by formulas.
Lab
Error Analysis Goal
1
Pendulum
Basic uncertainties
2
Free-Fall
Basic uncertainties
3
Vectors
Absolute and % errors
4
Motion
None
5
Newton’s 2nd Law
Absolute and % errors
6
Newton’s 3rd Law
None
7
Energy
Combine errors
8
Collisions
Combine errors
9
Buoyancy
Propagation by substitution; comparison
10
Heat and Temp
Propagation by substitution; comparison
11
Oscillators
Propagation by substitution; comparison
12
Resonance
None
13
Velocity of Sound
Propagation by derivatives
14
Statistics
None
P185 Lab Manual (Drum) Revised 4/16/14
49
Appendix: Lab Protocol
At the beginning of each lab I will lay out cards with each student’s name and place the
cards on the tables. You will have a different lab group for each set of labs.
It is very important to be on time. The roll will be taken at the beginning of each lab. If
you are late you will lose points.
Lab reports are due the following week at the beginning of your lab session.
If you miss a lab call and see if you can attend another lab session. This will be done
only if you have a good reason for missing lab, such as a serious illness. If the lab
cannot be made up an alternate assignment will be given. If you don’t have a good
reason the lab will be scored as a zero.
I will only answer questions when you are in your group.
Your entire data sheet must be filled out before leaving class. Don’t say “I’ll get the rest
of it later.” All graphs should be printed out before leaving class.
The Lab Report
A report should have the following, stapled together:
1.
2.
3.
4.
You lab report.
The original lab handout with my stamp on it.
Your notes from the pre-lab lecture.
All graphs and data sheets.
The report should be typed, double-spaced. The write-up must be done in the standard
five-section format: Theory, procedure, results, conclusions, error analysis.
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50
How to write a lab report
Let’s take as an example a free-fall experiment.
You drop a small steel ball from various heights and use an
electronic timer to measure how long it takes the ball to hit
the ground. From this you calculate the final speed of the
ball using v = 2x/t.
You believe that the ball will have a constant
acceleration of “g,” 9.8 m/s2. This will be seen if you graph
velocity vs. time and get a straight line with a slope of 9.8.
You end up with a table of data giving distances and fall times and a graph of v(t).
x (m)
t (s)
v (m/s)
.10
0.14
1.4
.20
0.21
1.9
.30
0.26
2.3
.40
0.27
3.0
Velocity in Free-fall
v (m/s)
3
2
y = 7.7058x + 0.3239
1
0
0
0.1
0.2
0.3
0.4
t (s)
Before you start writing, you have to know what audience you’re writing for. You
are writing for a fellow student who has not done this lab. You will assume he has
about the same knowledge of physics as you do. You need to give him enough
information to do the following:




Understand what you are trying to accomplish and how.
Evaluate how accurate and reliable your measurements are.
Evaluate the results of the experiment.
Reproduce the experiment himself.
Now you have to write the report. The report will always have the same format with five
sections. Each section should be labeled exactly as shown below.
Section 1: Theory
Describe the purpose of the lab. This may be one or more of three things:


You are trying to prove a theory. In our case we’re trying to show that the
acceleration of a body in free-fall is constant.
You are examining a relationship. This is what you do if you don’t have a
theory. For example if you measure the time it takes a pendulum to make
P185 Lab Manual (Drum) Revised 4/16/14

51
one swing as you vary the size of the swing, but without having a theory or
formula that allows you to make a prediction in advance.
You are measuring a quantity, for example the acceleration of gravity.
Describe any simplifying assumptions you are making, such as no air resistance.
Also give the equations you are using to analyze the data. For our experiment, the
theory is that a = g, but what we are actually measuring directly are x and t, not a. From
kinematics we derive the equation 𝑣𝐴𝑉𝐺 = (1⁄2)𝑎𝑡, from which we will get a.
This section will usually be brief.
Section 2: Procedure
You will describe three things in this section:
 Any equipment you used to make measurements (meter sticks,
stopwatches, etc.). This is important so the reader can get an idea of how
accurate your experiment is. For our experiment we used an electronic
timer and a meter stick.
 The procedures you used, but don’t go into too much detail. This section
should be brief. A drawing may be useful here.
 The range of any independent variables. These are quantities you select
yourself. For example, for our experiment, you might say “The height
ranged from 10 cm to 40 cm.” Don’t put any values for the time or speed
here, since these are quantities you measured experimentally: you didn’t
know them in advance.
Section 3: Results
There are three things that are commonly found in this section:



The range of measured values. From our example of dropping a ball, you
would list the range of times speeds you measured in this section: “The
fall time ranged from 0.14s to 0.27s. The calculated speeds ranged from
1.4 m/s to 3.0 m/s.”
Describe any trends in the data. Did the data fit a straight line, or some
other kind of curve? Give the equations for any computer fit lines. If the
data is supposed to be linear, use your eye to judge whether it really fits a
straight line or if it curves. (Note: If the data fits a straight line and the line
passes near the origin, you can say the quantities being graphed appear
to be directly proportional.)
Compare measured values with expectations or theoretical values. For
example “Our measured value for “a” was 7.7 m/s2, compared with the
book value g = 9.8 m/s2, a 22% difference.”
There shouldn’t be anything controversial in this section. Anything that involves
an interpretation or speculation should go in the next section.
P185 Lab Manual (Drum) Revised 4/16/14
52
Section 4: Conclusions
If you were trying to prove something, did you? How well does your data support
the theory? There are three common answers. If your data matched the theory, the
answer is yes. This means that you results matched the expected results within the
limits of uncertainty of the experiment. It means that any trends you observed were as
expected.
Sometimes the data does not support the theory. If this is the case, be clear
about how. For example, “The data showed a direct proportion between speed and
time, but the acceleration value we obtained was 22% below the theoretical value.”
Finally, you may get data that supports your theory within a certain range of
values but deviates from it outside this range. For example, “The graph of v vs. t was a
straight line up to a speed of 250 cm/s but curved downwards for higher speeds.”
If your theory is not supported by your data, you may speculate on why not.
Keep in mind, though, that “human error” is usually a bad explanation unless you know
specifically of something you did incorrectly that you couldn’t fix.
Discuss any weaknesses in the experiment and how they might be improved.
Section 5: Error Analysis
In this section you discuss the accuracy and validity of your experiment. You will
include the handout, which will be different for each set of labs.
You need to list any significant sources of uncertainty in the values you
measured directly (the raw data).
You need to give uncertainty values on the final results.
You need to discuss any potential other sources of error that were not directly
accounted for.
You need to discuss how you might reduce your uncertainties or improve the
experiment.
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53
Grading
Each lab is graded from 0 to 100. Grading will be based on the following:
Absolute essentials




The lab is stamped or attendance was taken.
The lab is handed in on time.
You were ready for the lab and participated actively in the lab.
All the necessary data was taken. The data is clear and neat
Good lab practices:






All data has units.
Raw data is taken in pen.
The number of digits is correct (not too many or too few).
Graphs:
o The axes are labeled and units are shown.
o The graph has a title at the top.
o The data points are NOT connected.
o An appropriate fit line is there if required, with equation.
Error analysis is done correctly.
The report is in the correct format. Your descriptions are clear, your conclusions
are based on the data and are logical, and your error analysis is complete and
reasonable.
P185 Lab Manual (Drum) Revised 4/16/14
Calibration Errors for Commonly Used Instruments
Instrument
Error
Meter Stick
0.2%
Protractor
0.2%
Force table
Stopwatch
0.05 s
Free-fall timer
0.05 s
Photogate timer
0.05 s
Vernier Calipers
0.1 mm
Pasco Motion Sensor
Pasco Force Sensor
Balance
Function Generator
Oscilloscope
0.2 %
54