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Instruction Manual
Manual No. 012-08581A
Physical Pendulum
Accessory
Model No. ME-9858
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Physical Pendulum Accessory Model No. ME-9858
Table of Contents
Equipment List........................................................... 3
Introduction ............................................................. 4
Equipment Setup ..................................................... 4-5
Taking Measurements and Collecting Data ............................ 6
Part I: Measure and Adjust the Mass of the Car ................................................................................ 6
Part II: Record the Pendulum’s Mass, Length, and Radius .................................................................6
Part III: Collecting Velocity Data .........................................................................................................7
Option 1: Measuring the Car’s Velocity with Photogates and a Computer Interface.............7
Option 2: Measuring the Car’s Velocity without a Photogate ................................................7
Theory ................................................................ 8-9
Experiment .............................................................. 9
Calculations .........................................................10-11
Appendix A: DataStudio Setup Instructions .........................11
Appendix B: Technical Support ....................................... 12
Appendix C: Copyright and Warranty Information .................. 12
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Model No. ME-9858 Physical Pendulum Accessory
Physical Pendulum Accessory
Model No. ME-9858
Equipment List
6
5
2
4
3
1
Included Equipment Replacement
Model Number*
1. Pendulum rod with pivot pin (1)
2. Pivot Rod with Ball Bearings (1)
3. Mass, 1 gram (2)
4. Mass, 2 grams (2)
5. Mass, 5 grams (1)
6. Mass, 10 grams (1)
003-08577
616-101
648-06505
648-06506
648-06507
648-06508
*Use Replacement Model Numbers to expedite replacement orders.
Additional Equipment Required Replacement
Model Number*
A PASCO computer interface (PASPORT™ or
Roller Coaster
Photogate and Smart Timer or computer system with interface
ScienceWorkshop
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)
ME-8930 and ME-9498A
Various
(See PASCO catalog.)
ME-9812
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Physical Pendulum Accessory Model No. ME-9858
Introduction
The Physical Pendulum Accessory (ME-9858) is used with the PASCO Roller Coaster
(ME-9812). The Mini-Car provided with the Roller Coaster Apparatus has a rubber band
bumper, which collides elastically with the Physical Pendulum Accessory. The Physical
Pendulum Accessory attaches to a PASCO Roller Coaster. The collision allows detailed
studies of angular momentum, linear and rotational kinetic energy, and gravitational potential
energy. The release point of the Mini-Car can be adjusted until the pendulum just makes it to
the top of its 180-degree rotation.
A detailed theoretical explanation is provided in Appendix A of this manual.
Setup
1. a) Assemble the track to start on the left board at the middle height position (Figure 1). The
track should extend past the edge of the board on the right.
b) You can place a catcher at the end of the track as a back-up, but it is not usually needed.
Figure 1: Track Configuration for Pendulum
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Model No. ME-9858 Physical Pendulum Accessory
2. a) Attach the pendulum bar by sliding the pivot pin into the middle of the pendulum
bearings. Insert the pivot rod into the exact hole shown on the peg board (Figure 2a).
support peg
peg
pivot rod
pivot rod
bumper
Figure 2a: Front view of
pendulum with pivot pin
Figure 2b: Side view of
pendulum with pivot pin
b) Place a normal length support peg in the hole as shown at the top of the swing (Figure
2a). This acts as a stop to prevent the pendulum rotating more than 180o.
3.a) Put the bumper on the car, as shown in Figure 3a. Use two rubber bands. b) Release the
car "backwards" with the rubber band part of the bumper in front.
Note: The bumper pin on the end of the pendulum must hit the middle of the rubber band
(Figure 3b). Adjust the position of the track side to side to achieve the correct position. If the
bumper is not the correct height to hit the pin, you have the pivot support rod in the wrong
hole.
bumper
Figure 3a: Bumper on Car
Figure 3b: Bumper against Pin on
Pendulum
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Physical Pendulum Accessory Model No. ME-9858
Taking Measurements and Collecting Data
Part I: Measure and Adjust the Mass of the Car
1. Release the car from halfway up the hill and carefully watch the collision. If the car has no
added masses, it will rebound off of the pendulum, and thus be moving "backwards" after the
collision. If the ballast mass was added, the car would continue in the forward direction after
the collision. (Here is a good reason to use the catcher at the end of the track as a safety stop.)
CAUTION: Catch the pendulum before it swings back down! Don't let it swing down
and hit the car!
2. With the small screw provided with the car, secure the
small masses in the tray on top of the bumper (Figure 4.)
3. Using the small masses provided, adjust the mass of the
car so that after the collision, the car is neither moving
forwards or backwards. The car should stop completely,
so that its velocity after the collision is zero. The mass
needed to do this is independent of the speed of the car,
but if the car is moving too slowly, friction will stop the
car any way. Start the car at least half way up the hill
during this part of the experiment.
mass
Figure 4: Mass in Car
screw
4. After adding the proper amount of mass, weigh the mass of the entire car (m
Part II: Record the Pendulum’s Mass,
Length, and Radius
1. Find the pendulum's mass (M). (Note: This
does not include the pivot support rod.)
pivot
2. Measure the length (l) of the brass part of the
pendulum. Do not include the bumper pin.
3. Measure the distance (d) from the center of the
bumper
pivot pin to the top end of the pendulum (the
non-pin end) (See Figure 5).
4. With the pendulum on the support pivot, and
the car's bumper against bumper pin, measure
the distance (r) from the center pivot pin out to
Figure 5
the position of the rubber bands on the car
bumper. This is the radius at which the car applies a force (and thus a torque) to the
pendulum, causing it to rotate. Note: Because of the bumper pin, l=d+r.
).
o
rubber
band
d
l
r
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Model No. ME-9858 Physical Pendulum Accessory
Part III: Collecting Velocity Data
Option 1: Measuring the Car’s Velocity with Photogates and a Computer Interface
1. Place a Photogate on the last peg of the Roller
Coaster track. This will allow you to measure
the speed of the car just before it hits the
flag
pendulum (Figure 6).
2. Connect the Photogate to a Smart Timer or to
a computer interface (ScienceWorkshop or
PAS PO RT).
Note: If you are using a PASPORT interface,
first plug the photogate into a PASPORT
Figure 6: Car with flag
entering photogate
Photogate Port; then plug the port into the
computer interface.
3. Insert the flag into the car (Figure 6). When the car passes through the photogate, the flag
must block the photogate beam. If not, adjust the photogate up and down, so that the flag
blocks the photogate beam. The nuts on the photogate support peg allow the sideways
position of the photogate to also be changed. (For more information, see the Roller
Coaster Manual).
4. For the Smart Timer, choose speed, one gate, and the velocity will be calculated and
displayed. To use a computer, you will have to use the calculator function and the spacing
on the flag (For instructions, see Appendix A in this manual).
Option 2: Measuring the Car’s Velocity without a Photogate
If you do not have a photogate available, you can estimate the car’s speed using conservation
of energy. By measuring the height of the car above the table both at the top of the hill and at
the bottom, the theoretical speed can be calculated, as follows:
mgh=½mv
2
Note: The theoretical speed does not depend on the mass of the car.
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Physical Pendulum Accessory Model No. ME-9858
Theory
mo = mass of car (including bumper and masses)
M= mass of pendulum
vo = speed of the car just before collision
l = length of physical pendulum (ignoring bumper pin)
x= distance from center of mass (c.m.) to pivot (calculated)
d= distance from pivot to (non-bumper) end of pendulum
(directly measured by student)
pivot pin
center of
mass
d
x
x
l
r = distance from pivot to rubber band on bumper (directly
measured by student)
Figure 7
I = rotational inertia of pendulum about pivot (calculated)
1. Center of Mass
The center of mass of the physical pendulum can experimentally be found by balancing it on
a knife’s edge, but since the pins have negligible mass, you can assume its center of mass is
at the center (l/2). Thus the distance, x (Figure 7), the distance from the center of mass to
1
x=
the pivot point is (1).
l-d
2
2.Rotational Inertia
The rotational inertia of a thin rod (total length l ) about its center of mass is
1
Icm=Ml
12
2
(2).
If the axis of rotation is not about the center of mass, but is instead about a parallel axis a
distance x from the center of mass, the Parallel Axis Theorem states that the new rotational
inertia is
IIcmMx
2
+=
(3).
Thus, combining these two gives the rotational inertia of the physical pendulum as
8
1
I= Ml2 + Mx
12
2
(4).
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Model No. ME-9858 Physical Pendulum Accessory
3.Angular Momentum
The linear momentum (p) of the car is movo, but in a collision that causes rotation, we have to
look at the angular momentum , where r is the distance from the car (rubber band) up
Lrxp=
to the pivot, and the magnitude of the initial angular momentum of the car about the pivot is
Lo=movor
(5).
After the collision, the car is stopped, and the pendulum is rotating with some initial angular
speedω. The pendulum has angular momentum
L=I
ω
(6).
4. Kinetic Energy
The initial kinetic energy (KEo) of the car is
KEo=½mov
2
o
(7).
After the collision, the car is stopped, and the pendulum is rotating with some initial angular
speed ω. The pendulum rotational kinetic energy is
KE=½I
2
ω
(8).
5.Potential Energy
When a point mass increases its height by an
amount h, it increases its gravitational potential
energy (PE) by an amount mgh. For a non-point
mass like the physical pendulum, the distance h is
the amount the center of mass moves upward.
For a complete 180o rotation (Figure 8), that
distance h=2x, and thus the gain in potential
pivot
x
energy is
h=2x
PE = 2Mgx
Experiment
1. Adjusting the release height on the track.
Release the car from half the way up the hill,
so that the pendulum does not hit the top
support peg. Repeat, releasing the car from a higher elevation each time, until the
pendulum "just" makes it to the top. You will need to use the marker on the white board to
keep track of your position. If you do it just right, you can get the pendulum to stop upside
down. If the pendulum hits the top peg and bounces off, your release point was too high.
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(9).
center of
mass
x
After
Figure 8
Before
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Physical Pendulum Accessory Model No. ME-9858
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Also, recheck what the car does after the collision. If the car is moving at all, you don't
have the mass correct; go back and adjust the mass.
2.Measure the speed of the car.
If using photogates, start measuring the speed for each trial, so that when you do get the
pendulum to stop at the top, you can use the value as the speed (vo) of the car. Otherwise,
calculate the theoretical speed using the change in height.
Calculations
1
x=
1.Using equation (1), calculate the distance .
2.Calculate the rotational inertia of the pendulum (I) using equation #4.
l-d
2
3.Initial angular speed ( ) of the pendulum:
ω
Because there is very little friction, we can assume conservation of energy as the pendulum
swings up to its stopped position at the top. Thus, the pendulum's total energy at the bottom
(only rotational KE) is equal to its total energy at the top (only PE).
2
Using equations #8 and #9, and solving for the angular speed,
=
4Mgx
---------------
I
.
ω
Using your values, calculate the initial angular speed, .
1
---
Iω
2
2Mgx=
ω
4. Angular momentum before and after the collision
Using equation #5, calculate the angular momentum before the collision.
Lo=movor
Using equation #6, calculate the resulting angular momentum of the pendulum.
LIω=
How do these two values compare? Calculate the percent difference between them. What
are the sources of error? Did the momentum of the system really change?
5. Kinetic energy before and after the collision:
Using equation #7, calculate the initial KE
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KE
of the car.
o
2
1
---
mov
=
o
o
2
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Model No. ME-9858 Physical Pendulum Accessory
Using equation #8, calculate the resulting KE of the pendulum.
1
2
---
KE
Iω
=
2
How do these two values compare? Calculate the percent difference between them. How
"elastic" was the rubber band collision? What are the sources of error? Did you lose or
gain energy? What would you expect? Explain why.
Appendix A: DataStudio Setup Instructions
Procedure Steps
PART I: Plug the
photogate into the
interface:
PART II: Set up your
experiment
PAS PO RT : a) Plug the Photogate Port into either port on a PASPORT
Photogate Port. b) Plug the Photogate Port into a PASPORT interface.
c) Connect the PASPORT interface (i.e. USB Link, PowerLink, etc.) to a
USB port on your USB-compatible computer. d) When the PASPORTAL
window opens, select “Launch DataStudio.”
OR
ScienceWorkshop: a) Plug the stereo plug on the Photogate into a
ScienceWorkshop
interface. b) Launch DataStudio.
PAS P ORT: a) In the PASPORT Setup window, click on the Add Timer
button and select Photogate Timing from the available choices. b) At
the prompt, enter the flag length and photogate spacing and click OK. c)
In the PASPORT Setup window, scroll to the Photogate Timing box, and
select “time in gate” and “velocity in gate.”
OR
ScienceWorkshop: a) In the Welcome to DataStudio window, double
click “Create Experiment.” b) In the Sensors list, click and drag a
Photogate icon to the picture of the interface (i.e. to the same digital
channel in which you have the photogate connected.) c) In the
Experiment Setup window, double click on the Photogate icon. d) In the
Measurement tab, select “time in gate” and “velocity in gate.” e) In the
Constant tab, enter the flag length in meters and click OK. f) In the main
toolbar, click the Timers button. g) Under Label, type a name for your
timing sequence. h) Click on the down arrow next to channel 1 and
select “blocked.” i) Click on the arrow again and select “unblocked.”
j) Click the Done button to save your timing scheme.
PART III: Create the
velocity equation
PART IV: Collect Data a) From the Data list, drag the velocity equation over a y-axis in a
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a) Click the New button to create a new equation. In the equation box,
type in the velocity equation: v=d/t, where v= velocity in m/s, d=distance
in meters, and t=time in seconds. Click the Accept button. b) Under
Variables, next to variable “d”, click on the down arrow, select constant,
and type in the flag distance from leading edge to leading edge (about
.001 m). c) For variable “t”, click on the down arrow, select “data
measurement,” “time in gate” and click OK. (For more instructions on
building equations, see the DataStudio online help.)]
separate Table display. b) Click the Start button on the main toolbar.
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Physical Pendulum Accessory Model No. ME-9858
Appendix B: Technical Support
For assistance with the ME-9858 or any other PASCO products, contact PASCO as follows:
Address: PASCO scientific
10101 Foothills Blvd.
Roseville, CA 95747-7100
Phone: (916) 786-3800
FAX: (916) 786-3292
Web: www.pasco.com
Email: techsupp@pasco.com
Appendix C: Copyright Notice and Warranty
Information
Copyright Notice
The PASCO scientific 012-08581A Physical Pendulum Accessory Manual is copyrighted and
all rights reserved. However, permission is granted to non-profit educational institutions for
reproduction of any part of the 012-08581A Physical Pendulum Accessory Manual,
providing the reproductions are used only for their laboratories and are not sold for profit.
Reproduction under any other circumstances, without the written consent of PASCO
scientific, is prohibited.
Warranty Information
For information about the product warranty, see the PASCO catalog.
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