PHYSICS 5 & 7

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Updated 9/04




PHYSICS 5 & 7

Laboratory Manual

Foundations of Mechanics &

Analytical Mechanics






























2



TABLE OF CONTENTS




ORIENTATION










3


INTRODUCTORY LAB SESSION






11


UNIT A: PARTICLE MOTION IN ONE DIMENSION


12


UNIT B: PARTICLE MO
TION IN TWO DIMENSIONS


17


UNIT C: MOTION IN SIMPLE SYSTEMS




20





UNIT D: ROTATION OF A RIGID BODY




23


LABORATORY MEASUREMENTS





27


AIR TRACKS









37


MEASUREMENTS AND ERRORS





41






3


ORIENTATION



Our goal in this lab is to challenge
your understanding of physics by asking you to apply
it experimentally to real situations. To further enhance this experience, instead of
performing predetermined lab exercises, you will be required to create your own
experiments.


Following the Introduct
ory Lab Session, there will be eight two
-
hour lab sessions divided
into four two
-
week “Units”. Each Unit is dedicated to a specific topic. You will
propose, create and carryout an experiment to explore each topic. Suggestions for
experiments are offered

in each Unit but you are encouraged to come up with you’re your
own experiments based on your life experience, a homework problem, a lecture
demonstration, a TV commercial, etc.


Note:

Each student is required to submit a full independent report on each
of the four
experiments.


What To Do



This style of laboratory, featuring original investigation, requires that you do most of the
work outside of the lab sessions; there is no
t enough time for planning in the lab period.
Mostly, you will work in pairs, and the outside planning should be done in
collaboration with your partner. Start by reading the appropriate section of this Guide
and examining the referen
ces therein. Before the first lab session of the unit you must
settle on your objective, the physical system to be used, the general method by which you
will go after your objective, the step
-
by
-
step procedure you intend to follow, and a
complete list of
equipment and supplies which you will need. This information (your
“Proposal”) is to be submitted to the instructor at the lab session
preceding

the Unit to
which it applies; see “Research Proposals” section.
Partners submit a single Proposal
.



All the TAs will have office hours


you can discuss your ideas and questions with them
while you are formulating your Proposal.



A detail
ed calendar is furnished to you early in the semester and posted in the laboratory.
In the first of the two lab sessions devoted to each proposed experiment, you will start to
carry out your experiment as planned, analyzing your data between the two sessi
ons.
Very often you will discover that something was overlooked in your original planning


for example, some unanticipated physical effect, inappropriate equipment, or additional
information needed. Such oversights are to be dealt with between sessions,

and the
experiment completed successfully in the second period.


If the results of the first session are satisfactory, the second period should be used to
extend or improve on the original experiment, either by taking more data of the same
kind to improv
e accuracy or by devising new measurements with the same apparatus,
building on the first week's work.


4



If the first session reveals that the planned experiment is impossible, “standard
experiment” is to be done in the second period; in this case you will
be graded on the
planning of the first experiment
, i.e.
, your proposed experiment, and the execution of the
second, back
-
up, experiment.



All data are to be recorded as they ar
e obtained, in a bound or spiral notebook, together
with all relevant information about the conditions of the measurements, e.g., diagrams
and definitions of terms; loose
-
leaf is not satisfactory. These numbers are later
transcribed into your report, whic
h is a separate document, not written in your data
book. Each student must have his or her own notebook


never leave the lab without
your own current and complete record. Don't depend on your partner in this matter.




The first five sections of the lab report are covered in your Proposal, revised as necessary
to show what was actually done. Between the two lab sessions you should make at least
a trial analysis of your first

data, including error analysis, as this may reveal what needs
to be done in the second week. After the second session the report is to be completed by
transcribing the data, calculating results and writing the final discussion. The report is
due at the
first lab session after the end of the Unit.
Note:

Each student must turn in
their own independent report, with conclusions and discussion in their own words.



The first five

sections of the proposal will be identical with your partner's, except
possibly for the wording of revisions made after the first week. However the remainder
of the report will be your own, though you may collaborate in the analysis and discuss the
resul
ts with each other as much as you like.


How To Do It



Be selective in designing your experiment
. The notes in this Guide, under each unit,
suggest a variety of valid experimental objectives and of specific systems (things and
combinations of things) to which they can be applied. These are shopping lists for ideas,
and you should not try for broad
coverage among the items listed. You will do only one
experiment in each unit, and it should be limited enough that it can be done in a single lab
session. The allocation of two sessions is to provide for rectifying oversights, debugging
and surprises,
o
r for improvements and extensions
, and
you are required to work in the
lab both weeks
. Certainly you should work with only one system, and your objective (or
objectives) should be narrow enough to be perfectly definite and clear in your own mind.




Work at a level commensurate with your background. Each person should be working
somewhat beyond his/her previous experience, but not so
far as to be completely at sea.
As a rough guide the suggestions listed in later sections are annotated as follows.



CNV:

conventional system, which is the b
asis of a standard freshman lab
experiment; “lab notes” for such experiments can be found among the references;


STR:

straightforward system, for which there is no standard experiment, but
which is certain t
o work;


5


NOV:

novel system, which may or may not work. These entail exploratory
experiments, which may not go the way one expects, and therefore are more
challenging than the others.




There is no grade advantage or handicap attached to these categories: the more ambitious
the undertaking, the less that will be expected in the way of results. What counts is how
much you yoursel
f put into the experiment's planning and execution, and the strength of
understanding your report shows. The effort made in the
two weeks

of each unit will be
considered.



Eq
uipment suitable for specific purposes of broad applicability will be available in the
orientation unit periods. Presentation of these techniques is deliberately disassociated
from any individual experiment or unit. Part of your task is to select those t
hat are best
for your experiment.



References




Referenc
es are an important part of designing an experiment. Your own textbook can be
the primary source. References can provide basic theory, and/or describe the system you
propose to use, and may even outline detailed procedures suitable for your use or provid
e
the specific theory of your experiment for you. They should be examined to find useful
information before much planning is done.



Referen
ces designated “L
-
...” are mimeographed documents available in the laboratory;
you may have a personal copy of any of these on request, but you must ask for what you
want to see.


Almost all other references are single copies on our B&H 136 reference she
lves
(non
-
circulating), identified below. References designated {RH} refer to Resnick and
Halliday, in the edition listed below.




RH
-
Resnick, R. and Halliday, D.,
Physics, Part I
, Third Edition (Wiley, 1977)


Beers, Yardley,
Theory of Errors

(Addison Wesley, 1957)




Ford, Kenneth W., Classical and Modern Physics (Xerox College Pub, 1972),
Volumes 1, 2 and

3


Parratt, Lyman G.,
Probability and Experimental Errors in Science

(Wiley, 1961)



Sears, F.W., Zemansky, M.W. and Young, H.D.,
University Physics
, Fifth Edition
(Addison
-
We
sley 1976)



You may also use your own text. Copies of a few other texts are also available on the
shelves.
Note:

None of these books are to be removed from room 136. A few references
are available on reserve in the Science Library
.


6


Research Proposals



When the present lab format was tried for the first time, those disappointments that
occurred resulted in most cases from insufficient advance planning.
It is not acceptable
to attempt to improvise an experiment from scratch during the lab period. To counteract
anything of this sort, the following procedure has been adopted:





A written Research Proposal is required for each Unit, to be submitted at the second
session of the
preceding

Unit. It is to consist of the first five sections of the laboratory
repor
t (Objective, Method, Theory, Apparatus and Procedure; see below) written in full.
It will then be checked as quickly as possible for appropriateness and feasibility. The
Proposal will be returned to you within a few days, schedule to be announced, eithe
r
“approved” or “disapproved”. (The most likely reasons for disapproval would be that the
procedure is not planned in sufficient detail, the necessary equipment is not available to
us, or that for some reason the experiment will not yield useful results.)

There will be
some time available to correct a disapproved proposal before the next lab session. This is
a good time to make use of the lab instructors' office hours! You should make changes,
consult with the TA's, so that you will start the lab with a
n approved Proposal. If your
Proposal is disapproved and not likely to be corrected, you will be assigned a standard
experiment from the appropriate Unit to be done in the lab. A disapproved Proposal may
be submitted with the final Lab Report for credit a
s to the planning involved.



The requirement of a written Proposal to be submitted for review a week in advance does
not add to your work, sin
ce it will be returned and may be used as the first part of your
Report (except for changes that develop in the course of the experiment). What it does
do is require the
detailed planning

to be completed well ahead of time. This is an
accurate

reflection of the life of science, in which it is always necessary to do detailed
planning to design experiments to achieve specific unambiguous goals, as well as to
secure support for work to be done through the mechanism of research proposals
submitted
far in advance.



The Laboratory Report


This is the vital conclusion to the experiment, the primary form in which its value is
preserved and the means by which you show your unders
tanding of your experiment and
get credit for your effort. The report needs to be self
-
contained and complete. The main
way the instructor will know what you did, and why, is through your success in
communicating those facts in your report.




The report is a written, self
-
contained document and should be handed in as such, not
torn from a notebook. It should be on standard 8 ½ x 11 paper,

and if you are able to type
it, you should. It's obviously to your advantage that it be not merely legible but
easy

to
read.


Please make sure to print your name and that of your partner's on the report, to write the
title of the projec
t and the date.


7


Note:

You and your partner must submit separate reports. The first five sections may be
identical with the single joint proposal you and your partner submitted; however any
changes in apparatus and procedure that were made during the exper
iment itself should
be in your own words. As stated above, and reemphasized here,
each partner

must have
an independent notebook with a current log of the experiment progress and the data. But
the data numbers should be identical! Never leave the lab wi
thout a complete set of data
in your own notebook. Each partner must be prepared to complete the analysis
independently once the data
-
taking is finished.


However, you and your partner can collaborate on the calculations and analysis of the
data to any de
gree you both wish, and discussion with your partner of the results is
welcome, as an important part of the collaboration. But your presentation of the Results
and your DISCUSSION should finally be done independently in your own words


not a
Xerox or com
puter copy of your partner's report.



We ask you to follow the format below exactly, for our ease in evaluating the work and
your practice in organizing a scientific report. A
ll sections except the Discussion should
be
brief
. It should take a lot longer to decide what to say than to write it down, and you
should certainly make notes for each section before you write it.



Sections of the Report


Remember the first five sections (
a
-
e
) are submitted in advance, as the Proposal. The
Report will contain the same sections, with any revisions made during the experiment.




(a)

Objective:

This should be a single phrase, or at most a sentence, short and
specific.


(b)

Method:
One or two sentences at most, telling what system you worked with,
what things had to be measured to achieve the ob
jective.


(c)

Theory:
The purpose of this section is to tell what the theoretical background
of your experiment is, and what equations you plan to use to analyze your
data. Usually your objective will be to verify the relationship among variables
which an eq
uation expresses, or to find the value of one of the constant
quantities in the equation. You can shorten this section considerably by
making use of references; anything you find in a reference (listed in the last
section of your report) can be used witho
ut proof in your “theory” section, and
if you find a reference in which the theory of your experiment is presented,
you can simply refer to that by a numbered footnote and quote the final
equation(s). However, whenever you do use something from a reference

you
must justify it, by telling what special conditions apply to the thing quoted,
and why or how the system you are studying meets those conditions.
Note:

In
any case, your “
theory
” must contain clear definitions of all symbols in its
equations, relatin
g them exactly to your apparatus and your measured
numbers.


8


(d)

Apparatus:
This is a
complete

list of all pieces of equipment and supplies
needed. To make sure it's complete, imagine yourself going through the entire
experiment, and make a note of every item

you see yourself handling. We
need this list in the Proposal in order to provide what you need in the lab. A
diagram of your setup is essential.


(e)

Procedure:
Write a step
-
by
-
step list of things to be done, from start to finish.
You should bring a copy o
f this to the lab and work from it, modifying it as
you work so that the final form is a true list of what you did.


(f)

Data:
Transcribe your measurements from the lab notebook to this section of
the report. Use tables for clarity, and be sure that measured

numbers are
identified with the symbols defined under “
theory
.” Specify the units for each
number. You don't need to include in the report trial runs or mistakes, only
the measurements you intend to use. The notebook remains a complete record
of the va
rious operations and trials.
Note:

It may be necessary to turn in your
notebook to resolve questions arising in the evaluation of the report.


(g)

Calculations:

Here you make use of your
data

in conjunction with the
theory

to produce your
results
. You don'
t need to work out all the arithmetic steps in
the report, but you should include enough explanation to make clear exactly
how you arrive at the results quoted. A sample calculation will usually do
this.


(h)

Results:
State your final results, simply and cle
arly. These should obviously
refer back to the
objective
, and will usually be expressed in the form of
numbers or graphs obtained in the
calculations

section. Numerical results
must always be accompanied by estimates of experimental uncertainties


se
be
low (ERROR ANALYSIS).


(i)

Discussion and Conclusions:
This is the only section you need not try to
keep short, although there is no reason to pad it and a few sentences may
well be sufficient. It is your final review of the experiment, and three asp
ects
are to be considered:



(1)

Error analysis:

what is the reliability, or range of uncertainty,
associated with your results? (This is a numerical statement; see next
section of Guide.)

(2)

Significance of results
:

In the light of your error analysis, how wel
l
do your results correlate with the theory? If there is significant
disagreement, what are the most probable reasons? Your discussion
may, at your option, extend beyond this basic consideration into
subjective matters: what the experiment shows, why it
is interesting,
why you chose the objective and the system you did, why you did or
did not enjoy it.


9


(3)

Recommendations
:

unexpected problems you encountered, and how
you would suggest improving the experiment if it were to be done
again.





(j)

References:
List all references you found helpful in any way. Number them,
and use footnotes to refer to the specific sources of any material you have
quoted in your report.




Error Analysis


This means determining and expressing the reliability of an experimental result. A
thorough error analysis is sometime more complicated than the experiment itself, but a
simplified set of proced
ures is commonly used. These are described in Reference L
-
1,
which will be distributed in lab; relevant references in B&H 136 include a 65
-
page
paperback (Beers) and a full
-
length book on theory of errors (Parratt). A summary can be
found in
Wall, Levine
and Christensen, Physics Laboratory Manual
, pp. 1
-
26, on reserve
in the Science library. The working rules of error analysis, as presented in the handout
(L
-
1) are needed for the laboratory work, and should be studied before the first
experiment is done.

These techniques will be discussed during the Orientation Unit.



The expression of reliability involves two considerations: the accuracy of instruments
used, expressed by wri
ting only “significant figures” for any given number; and sources
of random error, expressed by attaching a “measure of error” to the number
(written with a

sign, after the number itself). Any number read from an ins
trument
(e.g., a pointer alongside a scale) contains one or more figures which are definitely
identified by the scale markings, plus one last figure which is estimated from the position
of the pointer between scale markings (estimated fraction of smallest
scale division).
These are the
significant figures

of the reading; the significant figures in a number
derived from one or more measurements are those determined by the significant figures
in the input data and by the rules of the
propagation of errors
.




Two
measures of error

may be used alternatively to express the uncertainty introduced by
random sources of error: average deviation (A.D.) and standard deviation (S.D.
). The
significance of the measures of error is as follows.
If

the errors are truly random
and if

the error measures are based on a very large number of readings,
then

there is a 67%
probability (odds of 2 to 1) that the true value lies within
S.D. of the expressed value;
there is a 57% probability that the true value lies within

A.D.; and there is a 50%
probability (equal odds) that the true value lies within
, where P = 0.67 S.D. = 0.
85
A.D. (The measure P is called the “probable error”. The advantage of S.D. is that the
probability statement just given holds for smaller numbers of readings than in the case of
A.D.).


A
s a checklist, the things you need to know about error analysis are the following.



The nature of systematic errors and random errors. (Only random errors are calculated;
systematic errors can only be d
iscussed speculatively.)


10


For a series of measurements of a single quantity:



What is gained by repeating the measurement several times;

how to calculate the best (most probable) value from

the measurements;


how to calculate standard deviation for individual measurements in the set;

how to calculate standard devia
tion to be attached to the best value.

For a quantity calculated from two or more others which have been measured separately:



ho
w to determine significant figures in the calculated quantity;


how to determine the measure for error to be attached to the calculated
quantity,
from those associated with the input quantities (“propagation of error”).



For any measure of error: the meaning of absolute error, and percen
t error, and how to
convert from one to another.


Grading

Proposals and reports submitted beyond the due date will not be accepted, except in
special circumstances which are verified by a written medical excu
se or a note from a
dean.


The unit proposal
, which consists of the first 5 sections of your final report,

will be
graded as a
first draft

of the final report,
using the standards described above
.


Point credit will be assigned as follows:





Points

Prop
osal


10

Final Report

Objective

1


Method

1


Theory

5


Apparatus

5


Procedure

5


Data

10


Calculations

20


Results

20


Discussions and Conclusions

2
0


References

3




Total


100





11


INTRODUCTORY LAB SESSION



The
Introductory L
ab Session

is the first lab session of this course. At it you will write
and hand in your proposal for Unit A (
the first lab on Unit A will meet the week
following the Introductory Lab Session).

Before attending the Introductory Lab Session
read the mate
rial on “Unit A” in this manual and think about experiments you might like
to do.
Also, try to find a partner to work with in the lab.


The introductory lab session will include:


Your TAs name, room number, phone number and office hours,

Procedures and g
oals of this laboratory,


A lecture on
error analysis, significant figures, etc.,

How to write a lab report,

A talk on safe lab practices and procedures,


A demonstration of basic lab equipment & common
measurement methods

A schedule of lab related deadlin
es,

How to write a lab proposal.



You will have the remainder of the time to check references, discuss ideas with TAs and
others, learn about the available equipment, and write
your proposal on Unit A.


Note
: You must submit your proposal on Unit A at the end of the Introductory Lab
Session.


After the TA and Lab Manager have evaluated your proposal, it will be made available to
you and your partner. Please pick it up promptly a
t the specified time and location so you
and your partner will have time to make any corrections or adjustments that may be
required for approval.


12


UNIT A: PARTICLE MOTION IN ONE DIMENSION





Material included: RH Chapters 3 and 5 and Sec. 6.2; Ford Chapter 7 and Secs. 12.1,
12.5.



A real object moves as a “particle” when there is no rotation or inte
rnal motion of any
kind, so that the object's location and orientation in space at any instant are entirely
determined by the position of
any

point in it. The object's motion is then wholly
described by the position of a chosen reference point in it (the
center, a corner, etc.) as a
function of time.



The object's motion will be controlled by one or more of these forces: gravity, the
supporting

force of a tabletop or track, tension in attached springs or cords, friction, and
air resistance or fluid drag. Most of these are constant forces (unchanging in time), the
exceptions being spring tension, which changes if the spring length changes, and a
ir
resistance and fluid drag, which are velocity dependent. The most natural objectives for
this unit would concern the law of force for one of those listed. A particular force can be
studied either by choosing a system in which it is the only force acti
ng, or by determining
the resultant force in a situation in which several forces act and subtracting those that are
known. The law of force can be investigated either by verifying the motion predicted by
it, or by testing its dependence on the appropriate

properties of the system (
e.g.
, showing
force of gravity proportional to mass, friction proportional to compressive force and
independent of contact area, fluid drag proportional to speed). Alternatively the objective
might be to measure one of the const
ants appearing in a known force law, such as the
acceleration due to gravity or the coefficient of friction. Yet again, the objective might
be to verify the Second Law of Motion itself by testing the asserted relationship among
force, mass and acceleratio
n as these are varied.



In most experiments the basic data will be a table of paired numbers giving the object's
position at various times
,
. To relate this information to the resultant force acting, the
basic procedure is to calculate values of the derivative of
, giving speed as a function
of time,
; and then to repeat th
e process, giving acceleration as a function of time,
. Multiplying
by the object's mass, gives force as a function of time,
. The
result is a table of correlated values of
x
,
v

and
F

(al
l for the same
t

values), from which
F

can be expressed as a function of either
x

or
v
, whichever is desired.



Determination of

may be
done either in a one
-
shot experiment, in which the object
goes through the motion
once

and its positions are recorded at a series of predetermined
times; or repetitively, the motion being executed many times with a measurement of the
time to reach a predet
ermined position being made in each run.

The latter may seem easier but requires care to insure that conditions are the same for all
runs, especially the relation between the start of the motion and the start of the timing
device. Various timin
g devices are available; yours should be chosen on the basis of how
fast a motion you plan to investigate.


13




There are three ways to deal with friction


unless of
course, it's the thing you plan to
investigate.



1)

Elimination.

It may be reduced to negligible magnitude, as by the use of air
track; or, more crudely, dry ice sliding over smooth metal; or simple
lubrication.


2)

Compensation
. The added force needed to overcome friction can be
determined in a preliminary trial and then left in place during the experiment
proper. This might be difficult.


3)

Calculation.

The coefficient of friction can be measured in a preliminary
experiment (s
ee A
-
8) and thereafter used to calculate and subtract the
frictional force in the main experiment.





Suggested Systems



STR (A
-
1
)
Two interacting masses
. This system is designed to measure the ratio of two
masses without measuring any forces. (The ratio of their
weights

can be measured
independently, with a pan balance, allowing the relation between weight and mass to be
studied.
) The interacting masses are two air
-
track gliders, loaded with additional mass as
desired, and connected by a spring. They are separated and released, and their mass ratio
can be determined from their accelerations, since the spring exerts the same forc
e on
each.

Caution: the accelerations are not constant, and so must be compared at identical instants
of time. Variation: if you can find a way to measure the spring's length at various times
during the motion, that can tell you the force acti
ng on each mass, in case you can think
of a way to use this information. Can you think of another way to make the masses act
on each other without interacting with anything else?




References:

RH Sec. 5
-
8, 16
-
4


Ford p. 215





CNV (A
-
2)
Acceleration by a constant force
. Designed

for the direct study of the
Second Law of Motion, this system consists of a single mass, an air track glider with
whatever mass is loaded into it, accelerated by a single force, the tension in a string
passing over a pulley to a hanging weight. This tens
ion is constant in time but is not
equal to the weight of the hanging mass. There are two ways to deal with this unknown
force. It can be eliminated from the theory of the experiment, using Newton's Law to
predict the system acceleration as a function of

the two masses involved which you then
verify as your objective; or you can measure the string tension by inserting a calibrated
spring in the cord. (Because the tension is constant the spring will take on a constant
length during the motion, unlike A
-
1.
)


14


Photocell
-
activated electronic timers are suitable for timing. If you have trouble working
with small acceleration weights, try two nearly equal weights hanging at opposite ends of
the track and pulling on the glider in opposite directions.
If you are limited in track range
by the vertical space available for the descending weight, run the cord
under

the end
pulley and then
over

a second pulley fastened overhead.




References:


L
-
3 “Uniformly accelerated motion”

L
-
3A “The relati
on between force and acceleration: study of Newton's
second law of motion”


RH p. 88 Ex. 6


Ford pp. 213, 217
-
218


Guide to Laboratory Measurements



CNV (A
-
3)
Motion under gravity alone.



Ex
periment 1 (NOT TO BE USED FOR
PROPOSALS IN ANY SIMPLE WAY.) The objective in a study of free fall may be an
accurate measurement of “g” or verifying the dependence of the force of gravity on mass
and its independence on density, speed, direction of motion

or any other variable you
consider worth checking out. Any of the timing devices can be applied. Air resistance is
present, but it is negligible for high
-
density objects and short drops, and is the same for
two objects of the same size and shape (such a
s a golf ball and a ping
-
pong ball). An
interesting variation would be study of an object that loses mass as it falls, if you can
think of a way to do it.


References:

“Motion of a free falling body”


RH Sec. 3

10



Guide to Laboratory Measurements























STR (A
-
4)
Motion under reduced gravity
(“Atwood machine”).

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are hung 潮 the en摳 潦 a⁣潲搠灡ssing 潶er a

fixe搠 灵lley⸠ ⁔he fall 潦 the larger mass is
灲潤oce搠批 gravity 扵t 摩mi湩s桥搠 批 the
tensi潮 in the c潲搻 this m潴i潮 is sl潷er than
free fallⰠ an搠timi湧 is sim灬er⸠ ⁉f the masses
are varie搠in such a⁷ay that their rati漠stays
c潮stantⰠthe net f潲c
e 潮 the larger mass will
扥⁡⁣潮stant fracti潮 潦 its true weight an搠
“free fall” can be studied with various values
潦=re摵ce搠gravity⸠ ⁁lternativelyⰠ any=潦=the=
潢oectives= suggeste搠 f潲=A
-
㌠Pre⁡灰pica扬e= t漠
this=摥vice⸠An=interesti湧= variation= is=th

“double Atwood machine,” with M1 = M2 +
j
㌻=灲e摩ct=an搠verify= the=three
=
accelerati潮o⸠ =
v潵=may=have=t漠fin搠 a⁷ay=t漠摥al=with=
灵lley= friction= in=any=versi潮= 潦=A
-
㐮††††††††††††††††††4
=

15






Reference:

RH p. 89 ex. 7








NOV (A
-
5)
Elevator forces
. In A
-
4 with single pulley, one mass is replaced by a box or
frame (elevator) containing a mass (passenger). The elevator can be acc
elerated up or
down by suitable choice of counterweight, and to measure the force exerted on the
passenger by the elevator, the passenger
-
mass is suspended within the elevator by a
calibrated spring whose extension measures that force. How does it depend
on the
passenger's mass and acceleration? The main experimental problem is to measure the
spring's length during the motion.




Reference:

RH p. 90, Ex. 8



CNV (A
-
6)
Inclined plane.

If the object slides down a straight frictionless track the
resultant force on it is a fixed fraction of its weight, det
ermined by the inclination, and the
objectives of A
-
3 and A
-
4 are applicable. Variation: a controllable force can be added by
running a cord from the object up the plane to a pulley and on to a hanging weight, to
develop a variation of A
-
2.




Reference:
L
-
5 “Acceleration of a body down an inclined plane”




STR (A
-
7)

Friction.

In A
-
2 and A
-
6 friction was eliminated, but if instead the contact
surfaces have significant friction, its magnitude can be determined from the object's
acceleration, by subtracting known forces from the resultant force. A special case of thi
s
procedure is to determine the applied force needed to produce a resultant of zero, shown
by constant speed. Alternatively, the frictional force can be found in a situation in which
it is the only force acting by measuring the object's stopping distance
for a known initial
speed.

In studying the law of friction, possible objectives include its relation to the nature of the
interacting surfaces (including lubrication effects), contact area, compressive force and
speed.




References:

RH Sec. 6
-
2


Ford Sec. 12.1





NOV (A
-
8)
Fluid drag
. A steel ball or similar object falls through a vertical transparent
tube filled with water or other liquid. The drag force exerted by the fluid can be found by
subtracting the force of gravity from the resultant force on the falling objec
t, and can be

16


plotted as a function of the object's speed (see third paragraph of general notes on this
unit). This relationship would be the basic objective; an advanced extension would be
investigation of how the “drag coefficient” depends
on the object's size and shape, or on
the nature of the fluid. Variation: if the fluid can be made to flow smoothly upward it
might be possible to find the flow speed necessary to balance out the object's weight and
hold it motionless, which would be a di
rect measurement of the object's terminal speed.
Finding the terminal speeds of objects of different weight but the same size and shape
(
e.g.
, a ping
-
pong ball filled with different amounts of lead shot) would give the
relationship between speed and drag
force. In this approach the practical problem would
be measuring the volume of fluid flowing through the cylinder per second.





References:
RH p. 95, Prob. 33


Ford p. 517




NOV (A
-
9
) Ai
r resistance.

It might be possible to use the methods of A
-
8 with air as the
resisting medium, if a light object such as a ping
-
pong ball were used. Alternatively, the
study might be done with a heavier object moving horizontally on a frictionless surfac
e,
with an air stream providing the driving force. Either way, the air
-
track blower, a
vacuum
-
cleaner output or a hair
-
dryer might be used to provide the air stream.


Reference:

Ford pp.

513
-
516




STR (A
-
10)
Sliding chain.

If a flexible chain lies on a horizontal surface with part
hanging over the edge, it will slide of
f if the friction is low enough. The driving force is
the weight of chain hanging over, so the chain's acceleration varies with time. Objective:
predict and then verify either the chain's acceleration as a function of the hanging length,
or total slip
-
of
f time as a function of initial overhang. (For the former, prediction is easy
but the measurements are elaborate; for the latter, measurements are simple but
prediction requires solving the chain's equation of motion.) Friction can't be eliminated
in this

setup, but it can be compensated (cancelled out) by setting the plane at an
angle of inclination. With the plane set at this angle, there will be no net force on
whatever portion of chain is resting on it at different times in the main run even

though
that portion is variable. (Your theory should explain this.)




Reference:

RH p.
185, Prob. 41 (IS THE GIVEN SOLUTION CORRECT?)


Ford p. 260, Prob. 7.10


17


UNIT B: PARTICLE MOTION IN TWO DIMENSIONS



General:
Material included: RH Chapter 4 and Section 6.3; Ford Section 7.8.


This unit involves the same kind of objec
ts and the same forces as Unit A, but combined
in such a way that the object moves in a plane instead of on a line. The suggestions
below concentrate upon projectile motion and uniform circular motion, but you are
welcome to base your experiment on someth
ing different. Typical experimental
objectives would be as follows.


Projectile motion, single trajectory
:

(positions recorded at several different times):
determine the value of “
g
” from the shape of the path; from the separate functions
and
determine the two independent components of acceleration; if air resistance
or friction cause your results to differ significantly from frictionless theory, deduce the
magnitude of the retarding force.


Projectile motion,

repeated trajectories
:

test the dependence of height, range and time of
flight on launch conditions (initial speed, direction and height). One interesting point: the
launch angle giving maximum range is not 45
o
, if the launch is not from ground level.


Uniform circular motion
: verify the dependence of the applied force needed to sustain
the motion (centripetal force) on the object's mass and orbit radius and frequency of
rotation; if positions are recorded at various times within a single cycle, determi
ne each
component of acceleration as a function of time and verify the theoretical magnitude and
direction of the acceleration.


Suggested Systems


CNV (B
-
1)
Ordinary projectile motion
. Almost any object makes a suitable projectile;
typical ones are marb
le
-
sized metal or wood spheres, golf
-

or ping
-
pong balls, tennis
balls,
etc.

It doesn't need to be small, or to be spherical, although air resistance will be
smallest for objects that are small, dense and spherical and for short
-
range trajectories,
which

may be advantageous or not depending on what you want to observe. If successive
positions in a single trajectory are to be recorded, strobe photography or video recording
are probably the only feasible methods. (The photograph can include a ruled backdr
op
for position measurements; or, so long as
some

object of known size is included in the
picture to determine the scale, measurements can be made from the photograph itself.)
The photographic method is limited to fairly small paths, but for measurements
of range,
height and time of flight there is no such limitation and larger
-
scale studies could be
carried out in an empty lecture room (if available). The projectile could be launched by
throwing, kicking or striking, or from a spring gun (compressed spri
ng in a tube).
However, to control and determine launch speed and direction such methods as launching
from a pendulum or rolling up or down a ramp (see references) are helpful. Any rotation
imparted in the launching process is harmless, since it stays co
nstant throughout the flight
and does not affect the motion of the object's center point.


18


References:

(L
-
6) “Motion in Two Dimensions”


Wall and Levine, Expt. 5 (On reserve in Science Library)


RH Sec. 4
-
2, 4
-
3


ST
R (B
-
2)
Projectile motion with reduced gravity (pinball machine).

If a small
projectile is launched in the plane of a slanted drawing board, a large part of its weight is
cancelled out and it moves over the board under the influence of a reduced componen
t of
gravity, equal to its true weight multiplied by the sine of the board's angle of inclination.
(The object's mass, of course, is unaffected.) All the general objectives listed for
projectile motion can be studied with this setup; in addition, since t
he effective value of

g
” can be controlled, the effect of the pull of gravity on projectile paths can be
investigated. For example, motions on the moon or various planets can be duplicated. A
small metal sphere makes the best projectile: its rolling mot
ion minimizes the effect of
friction, and if the sphere is small the rolling motion will not affect the projectile motion
much. Further, the sphere makes a well
-
defined

contact with the board, and if it is heavy
that contact will be easier to record. If
the board is covered with graph paper and the ball
is inked, a faint but visible trace of the path will be made on the graph paper. Video
taping is also possible.


References:

(L
-
6) “Motion in Two Dimensions”


RH Sec. 4
-
2, 4
-
3



Wall and Levine Expt. 5 (On reserve in Science Library)


Bernard and Epp, Expt. 8 (On reserve in Science Library)


STR (B
-
3)
Water jet.

Individual drops of water emerging from a nozzle are projectiles
and move accordingl
y. Therefore the steady stream displays the trajectory as a whole,
and is easily measured and subjected to various initial conditions. The nozzle may
be a
small tapered

glass tube connected through rubber tubing to
an elevated reservoir jar
.
Varying the

reservoir elevation will change the initial water speed, which can be
calculated from the flow rate (volumes passing per second) if the jet aperture is known.
The flow rate can be measured either by collecting water at the end of the trajectory or
measur
ing the rate at which the reservoir level is dropping.


References:

(L
-
6) “Motion in Two Dimensions”


RH Sec. 4
-
2, 4
-
3


Wall and Levine, Expt. 5 (On reserve in Science Library)


CNV (B
-
4)
Uniform circular motion (
UCM) maintained by stretched spring}.

In this
system the mass is held in a horizontal plane by low
-
friction guides and held in circular
motion by an attached spring
. The spring length in motion is measured by an indicator
and the whole rotated about the
vertical axis by a variable speed motor. The radius of the
motion is fixed in this device, but spring tension and rate of rotation are independently
variable.


References:

RH p. 115 Prob. 37


(L
-
7) “Uniform Circular Motion and Centri
petal Acceleration”


19



Wall and Levine, Expt. 6 (On reserve in Science Library)


Bernard and Epp, Expt. 9 (On reserve in Science Library)


STR (B
-
5)
Uniform circular motion (UCM) maintained by friction.

If a small
object
resting on a turntable holds its position, the necessary centripetal force is provided by
friction.

Thus, since the force required increases with radius (at a given rate of rotation), the
coefficient of
static

friction can be determined by finding

the largest radius in which it
can hold the object. You might thus study the laws of
static

friction (compare A
-
8); or,
by resting the object on wedges fastened to the turntable, study the dynamics of “banked”
curves (see B
-
7). You will need to measure
the exact rate of rotation of the turntable.


References:

RH Sec. 6.2 and p. 114, Prob. 25


NOV (B
-
6)
Uniform circular motion (UCM) maintained by hanging weight
. The
moving mass is attached to a string which passes through a vertical section of smooth
glass tube to a hanging weight. Thus the centripetal force is provided by the tension in
the string, which is equal to the weight of the hanging mass when the system is in
equilibrium (hanging weight not moving up or down). The glass tube is hand
-
held an
d
moved in small horizontal circles to establish the UCM. It's a simple system but sounds
tricky to operate. Ingenuity may be needed to devise ways to measure radius and rate of
rotation when equilibrium is established. Using nylon fishline should help
in reducing
friction, and the glass can also be lubricated.


References:

RH p. 114, Prob. 23 and p. 270 Ex. 5


NOV (B
-
7)
Uniform circular motion maintained by banked track
. The moving object
is a ball
bearing rolling inside a vertical glass funnel
. It
is launched in a horizontal
direction tangent to the funnel surface, and depending on its initial speed
will spiral
upward or downward until it
reaches the radius appropriate to its initial speed, where it
maintains UCM until friction destroys the equilibr
ium. In equilibrium, the vertical
component of the funnel's force on the pellet equals its weight, while the horizontal
component of the same force provides the centripetal force. In this device, therefore, the
centripetal force is known from the funnel'
s angle and the pellet's weight alone (so that,
incidentally, the centripetal force is the same for all horizontal equilibrium orbits).
Measuring the rate of rotation, with a stopwatch, should be easy; measuring the
equilibrium radius may be harder. To p
rovide a known launch speed the pellet can be
rolled down a glass tube of known elevation. Its end must be curved to meet the
condition of a horizontal, tangential contact with the funnel.


References: RH p. 105, Ex. 5


Ford p. 234,
Ex. 2


20


UNIT C: MOTION IN SIMPLE SYSTEMS



General:
Material included: RH Chapters 7, 8, 9, 10; Ford Chapters 8, 10 and Section
12.6.


The systems of this unit consist of a very small number (usually two) of “particles” like
those of preceding units, which

interact with each other; they may, or may not, also
interact with the environment. Experimental objectives may focus on the concepts of
energy and momentum such as:


verification of conservation of momentum and/or of energy, if applicable; and
otherwise

accounting for the gain or loss;

use of energy and momentum principles to measure something otherwise difficult
to determine.


Or, the objective may focus on the motions of the separate particles


symmetries and
correlations between them; or the behavio
r of the system's center of mass.


Suggested Systems


STR (C
-
1)
Forces in a train
. Three or four identical carts are connected by springs and
accelerated over the tabletop by a cord passing over a pulley to a hanging weight. The
cord represents the loco
motive coupling, and the spring extensions measure car
-
to
-
car
forces so the frictional forces on the individual cars can be deduced. The system can be
studied under conditions of acceleration, deceleration and constant speed; with wheels
locked and rollin
g; and with different masses in individual cars. The primary problem is
devising a method to measure the lengths of the springs during the motion.


References:

RH p. 93, Prob. 15


Ford p. 309, E8.25



CNV (C
-
2)
One
-
dimensional colli
sions.

Two air track gliders are the colliding objects
so the system is isolated and both bodies move at CONSTANT velocity, both coming and
going. Energy and momentum changes should be examined, for various mass
combinations,

colliding both elastically
and inelastically.


References:

(L
-
8), (L
-
8/9A) “Elastic and Inelastic Collisions”


(L
-
9) “Conservation of Linear Momentum”


RH Chapter 10


Ford p. 442, P10.25, P10.26; p. 535, E12.56



Wall and Levine, Expt. 3 (On reserve in Science Library)




21


CNV (C
-
3)
Ballistic pendulum
. This is a technique for measuring the speed of a fast
-
moving object, by letting it collide with the mass of a simple pendulum. Because of
momentum conserva
tion in the collision and energy conservation in the pendulum's
subsequent motion, the object's speed is determinable from the height of the pendulum's
motion.


References:

(L
-
10), (L10A), “The Ballistic Pendulum”


RH p. 194, Ex. 3



Ford p. 314, P8.11


Wall and Levine, Expt. 5 (On reserve in Science Library)



NOV (C
-
4)
Two
-
dimensional collisions.

This refers to the collision between two
frictionless objects which recoil along a line different from
the line of approach. The
objectives of C
-
2 are applicable, but more ingenuity is required to measure the speeds
and of course directions must also be recorded. Further, the elimination of friction is
harder for two
-
dimensional motion. You might use puc
ks on an air table (if available


consult instructor in advance), or

pieces of dry ice sliding on smooth metal.


References:

RH Sec. 10
-
6


Ford Sec. 12.6


NOV (C
-
5)
Impact forces
. For a single object colliding with a flat surfac
e, the object's
momentum change gives the product of the average interaction force and the contact
time; for a stream of such objects, the rate of change of momentum in the stream equals
the average force. If the flat surface is a horizontal pan suspended

from a spring the force
on it can be measured, and of course the object's momentum changes can be measured
independently. Therefore it is possible to verify the assertions just made; or,
alternatively, to use them to deduce the duration of the collision.

(In this connection it is
a fair approximation


GIVEN


that the average force during a collision is one half the
maximum force.) L
-
21 shows a simple method of
measuring

impact time for a dropped
rod. Other suitable objects are:


a ball dropped from a

known height and rebounding to a known height, measuring
the maximum force from the maximum spring extension during the impact;


a flexible chain hung vertically above the pan with its lower end just touching,
and dropped (how does the force on the pan va
ry with time?); water from a hose
fixed at some height above the pan, running onto the pan and then off (how does
the force on the pan depend on the height from which the water falls and its rate
of flow)?


In any of these cases it is appropriate to invest
igate whether energy is conserved, and how
any energy loss relates to the physical variables.



22


References:

(L
-
21) “Longitudinal Impact of Rods”


RH p. 183, Prob. 22, p. 208, Probs. 1 and 4;


p. 210, Probs. 25 an
d 26



NOV (C
-
6)
Variable mass system
. An arrangement can be devised to dribble fine shot
into a pan attached to an air
-
track glider, which then behaves as an isolated particle of
increasing mass. Objective: predict and verify how the object's decelerat
ion from a given
initial speed is related to the rate of inflow of the shot.


References:

RH p. 178, Ex. 10; p. 185, Prob. 40


Ford p. 313, P8.6



NOV (C
-
7)
Baton
. Masses (equal or unequal) are attached to the ends of a lightweight
r
od and the combination tossed upward so that it twirls in a vertical plane while rising
and falling. You could concentrate on the relation between how high it rises and the
number of revolutions it makes while in the air, or you could undertake a complete

study
of the translational motion of the center of mass and the rotational motion about the
center of mass.


References:

Ford, Sec. 8.3; p. 313, P8.8 and P8.9


23


UNIT D: ROTATION OF A RIGID BODY



General:
Material included: RH Chapters 11, 12, and 13; Fo
rd Chapter 9 and Section
10.9.


A rigid body rotating about a fixed axis has only one variable coordinate of position
(angle), and therefore obeys equations very similar in form to those governing linear
motion of a particle. The correspondences are tabul
ated in RH Tables 11
-
1 and 12
-
2.
The general objectives for Unit D are similar to those for Units A and C:


Verify the assertions of the Second Law of Motion (relationships among applied torque,
angular acceleration and moment of inertia).


Measure the mo
ments of inertia of various bodies, comparing them with values
calculated on the basis of size and shape.


Verify the conservation of angular momentum in isolated systems.


Study the transformation of energy, verifying its conservation or determining its
l
osses, as appropriate.


Suggested Systems


STR (D
-
1)
Flywheel.

A standard piece of apparatus consists of a steel disk attached to a
horizontal shaft rotating in low
-
friction bearings. Torque is applied by a cord wrapped
around the axle and attached to a

hanging weight, and various disks are available so both
torque and moment of inertia are known and variable. Angular acceleration can be
determined from the vertical acceleration of the falling weight. Variation: measurable
friction can be introduced b
y a cord making a half
-
turn around the axle and fastened to a
fixed spring at each end. (The difference between the tensions in the two parts of this
cord is the frictional force it is exerting.) It is possible to measure independently the
power being de
livered by gravity, and power being removed by the “brake,” and the rate
of increase of the flywheel's kinetic energy. Can you find a way to measure the frictional
torque exerted by the bearings? (This variation becomes NOV.)


References:

(L
-
11) “Rotatio
nal Kinematics and Dynamics”


RH Sec. 11
-
3, Sec. 12
-
6; Examples 12
-
3, 4, 5, 6.


Ford p. 413, Ex. 1 and Sec. 9.6


Wall and Levine, Expt. 10 (On reserve in Science Library)


STR (D
-
2
) Yo
-
yo.

This

consists of two disks on opposite ends of a cylindrical axle of
smaller radius. A cord is wrapped around the axle, its free end fastened to a support and
the yo
-
yo allowed to fall from rest. (Two strings are advisable, wrapped at opposite ends
of the ax
le, to prevent twisting about a vertical axis.) This is very similar to D
-
1 and the
same studies can be made. Some points of special interest: where do the object's kinetic

24


energy and angular momentum come from, and where do they (eventually) go to?
Var
iations: various objects can be substituted for the end disks to study moment of inertia
effects; springs can be inserted in the supporting strings to measure tension.


References:

RH p. 258, Prob. 39, p. 259, Prob. 43


Ford p. 440, P
10.19


STR (D
-
3)
Objects rolling down inclined plane
. The acceleration of such an object
down the plane is a fraction of what it would be if the object did not roll (A
-
7). This
fraction depends on the object's moment of inertia, so the measured accelera
tion can
either be used to verify the relationship or to measure the moment of inertia.
Alternatively the experiment could focus on energy transformations. Different objects of
the same mass and outer radius can be used; irregular objects can be fastened

inside a
cylindrical shell (empty can) for testing; and a can of water is especially interesting. Note
that if the can of water is stopped at the bottom of the plane by a brief touch it will start
moving again. How was the residual energy stored?


Refer
ences:

RH Sec. 12
-
7


Ford p. 414, Ex. 2; p. 436, E10.59;


p. 374, P9.22


NOV (D
-
4)
Rolling spool
. This is built like a yo
-
yo, but instead of descending from
strings fastened overhead, the object rests on a h
orizontal surface and is moved by
externally applied tension in the string. (A convenient way to control and measure the
string tension is to run it over a pulley to hanging weights.) Depending on the direction
and magnitude of string tension, the spool
may remain stationary, roll left or right, roll
without slipping, slip without rolling, or do both. The obvious objective is to find (and
interpret) the conditions of applied force (direction and magnitude) under which these
different states of motion occ
ur. Variation: to observe the direction in which the force of
friction acts (which may be hard to figure out), the horizontal surface may be a glass
sheet resting on rollers or a sheet of wood suspended by cords at the corners.


References:

RH p. 255, Qu
estion 21


Ford p. 440, P10.18


STR (D
-
5)
Moment of inertia (I): flat objects
. A simple apparatus to measure moment
of inertia can be derived from D
-
1. A
vertical

shaft rotating in bearings receives known
torque from a cord wrapped a
round it which passes over a pulley to a falling weight. The
top end of the shaft is threaded to receive a screw by which various figures (cut from
pegboard) can be clamped to it and held horizontally while they rotate, the angular
acceleration being dete
rmined from the downward acceleration of the weight. This
should be slow enough for stopwatch timing, and since the acceleration is constant it is
only necessary to time the weight over a known distance of fall to find its acceleration.
Various shapes ca
n be cut and measured, their moments of inertia being compared with
calculated values. Variation: study the dependence of I for a given object on location of
the axis of rotation, and in particular check the “parallel axis theorem.” Experimental

25


complica
tions: the system has some moment of inertia even without a cutout in place,
which must be measured and included; the

torque introduced by friction in the bearings may be of significant magnitude, in which
case it also must be measured and included.


Refe
rences:

RH Sec. 12
-
5; p. 257, Prob. 28


Ford p. 372, P9.14, P9.15


NOV (D
-
6)
Moment of inertia: your body
. This is a crude but informative experience
with angular momentum. It involves a large, low turntable holding a stool on whic
h you
can sit while it rotates. Two procedures are suggested:


1.

Sit on the turntable with arms outstretched and holding a mass of about a
kilogram in each hand, while your partner gives you a sizeable angular speed;
then quickly pull the weights into your
chest. Your partner is to measure your
angular speeds before and after this maneuver.


2. While you sit on the turntable at rest, your partner spins a bicycle wheel (with
handles on the axle) and hands it to you with its axle vertical. You then invert
i
t and your partner measures your final speed. For each procedure,
conservation of angular momentum enables you to write an equation
involving the measured angular speeds, in which your own moment of inertia
is the only unknown quantity. Thus each procedu
re gives you a value, and the
two can be compared. Complications: the turntable's moment of inertia is
combined with your own, and the turntable has sizeable frictional torque. To
deal with the first, you can calculate the table's moment of inertia prett
y
closely from its size and shape and the estimated density of wood (since we
can't dismount the turntable to measure its mass). To deal with the second
problem, the angular speeds should be measured by timing only the
last

revolution before the change an
d the
first

revolution after it, so the frictional
torque has minimal time to act. Perhaps you can devise a way to automate the
measurements for better accuracy.


References:

RH p. 270, Ex. 6 and Ex. 7; pp. 278
-
279, Probs. 23 and 26



Ford p. 362, Question 9.31; p. 369, E9.45, E9.46


CNV (D
-
7)
Rotational collision
. One horizontal disk rests on another which rests on an
air table, which is constructed similarly to the air track and is capable of providing a
frictionless air cushion be
tween the lower disk and the table, and/or between the two
disks. The two cushions are separately controlled by air valves, and the two disks are at
all times confined to rotation about the same vertical axis. With both cushions present,
the two disks ar
e given the desired initial angular speeds, then the cushion between the
disks is turned off and the final angular velocity of the pair is measured. The objective is
to study angular momentum transfer and conservation in each of the three cases: one disk
initially stationary, both initially rotating in the same direction, and initially rotating in
opposite directions. The recommended procedure for measuring angular speeds utilizes

26


reflecting strips attached to the disk rims at 45
-
degree intervals. Light
reflected from
these is registered by a photoelectric cell connected to an oscilloscope. The time interval
between light pulses is determined from the sweep speed of the scope. If you are not
familiar with use of the oscilloscope, another method can be d
evised.


References:

(L
-
12), (L12A) “Conservation of Angular Momentum”


RH p. 279, Prob. 25


Ford p. 369, E9.44


NOV (D
-
8)
Gyroscopes and tops
. Both of these are rapidly spinning objects supported
in such a way

that the axis is not fixed, and in consequence they respond to an applied
torque by “precession”: the axis of the fast rotation
itself

rotates about a different axis,
which is neither the spin axis nor the axis about which the applied torque is tending to

turn the object. The torque maybe applied by gravity, by supporting the device (in a loop
of string or on a tabletop) at a point on its axis of rotation but in an unbalanced condition;
or by a cord and hanging weight, if the spinning device is independen
tly supported.
Equipment available includes a toy top (which rests on a point that is part of the spinning
object itself), a toy gyroscope (spinning on pivots within a stationary frame which can be
supported at a point which is on the axis but motionless)
, a bicycle wheel with handles on
the axle which can be

used in the same way, and a commercial gyro suspended in “gimbal rings.” (These are a
set of three concentric rings, each pivoted for free rotation within the next. The gyro is
pivoted within the
innermost, with the result that it is supported yet free to rotate about
any direction is space.) Quantitative experiment: investigate the dependence of the speed
of precession and the axis of precession on the gyro's spin and the applied torque vectors.

Qualitative studies: gyroscopic stability and its applicability as compass, stabilizer, and
“artificial horizon” for aircraft; the behavior of a pendulum whose bob is a gyroscope,
and its relation to which way the gyro is spinning; differences in behavior

between simple
gyroscope standing on a table, and a top; conditions determining whether a top processes
progressively lower until it falls, or progressively higher until it “sleeps” in the upright
position. (Check on the kinds of objects available in the

lab before writing a proposal. If
you have suggestions for readily obtainable “toys,” we will try to supply them.)


References:

RH Sec. 13
-
2; p. 274, Question 4


Ford Sec. 9.7; p. 374, P9.24


(L
-
22) “The Gyroscope
: Elementary Discussion of a Child's Toy”


27


LABORATORY MEASUREMENTS



Measuring Length


Most of the time, this is a straightforward problem. A straight ruler or meter stick is
aligned with the length segment to be measured and only care in matching the
segment's
boundary points to ruled marks (estimating between marks) is necessary. Greater
precision, if the distances are not more than a few centimeters, is afforded by a
caliper

(see Fig. 1), which can be set to
match

the distance in question. The dist
ance is in effect
memorized by the instrument, which can then be removed to an area where it can be read
with ease. The caliper method has a very large application in length measurements. The
vernier caliper in Fig. 1 can be used to measure both inside a
nd outside diameters, and
even depths, by using the different sets of jaws or the probe at the end.







Verniers and Micrometers


A vernier scale (see Fig. 1b) provides a better method of reading between the ruled lines
of a main scale than simply esti
mating by eye. To read a vernier scale, first note how
many divisions on the vernier are equal to an integral number of main scale divisions. In
the ordinary metric vernier, ten vernier divisions are equal to nine main scale divisions.
Hence the
least co
unt
, or smallest value that can be read
directly

from a vernier scale of
this type, is one
-
tenth of a division. The caliper divisions are millimeters, so that the least
count would be one
-
tenth of a millimeter. The sketch in Fig. 1b shows the reading of
a
vernier.



28







Micrometer calipers (Fig. 2) are used to measure diameters quite precisely. Turning a
handle moves a rod forward, by a screw thread, until the object to be measured is
clamped
very gently
. In the usual metric instrument, one turn of t
he handle advances the
rod a half
-
millimeter. A circular scale around the handle reads fiftieths of this half
-
millimeter, or thousandths of a centimeter. One more figure is estimated between scale
marks. Exactly the same kind of screw, rod, and scale ar
e often incorporated into other
measuring instruments.


Both vernier and micrometer calipers should always be tested for “zero readings” when
fully closed (so of course should any other instrument).


Measuring Mass


Several types of balances for weighing o
bjects exist. With any kind, a “zero reading” (no
load and no weights) should be taken first. Care should be taken not to overload any
balance nor to spill any corrosive materials on them. The
trip balance
, or
pan balance

is
used for heavy objects, and
for general weighing. The unknown goes in the left pan, the
standard masses in the right pan.
Beam balances

are available in comparable size; there
are also more sensitive ones for weighing small objects. In using a beam balance, the
unknown is placed i
n the pan and a weight is arranged to slide along a calibrated beam.




29


Ohaus Triple Beam Balance


The range of this general laboratory mass balance is increased over the simple single
beam form by the use of several beams (see Fig. 3). One beam is for th
e largest mass
increment
--

the movable mass on this beam can be placed, in the balances used here, in
five positions besides zero. Each corresponds to an increment of 100 gm. Another beam
contains a weight that can be placed in any of ten non
-
zero posit
ions corresponding to
increments of 10 gm. In using these two beams, each movable weight must be placed
definitely in the appropriate notch. The front beam contains a weight that can slide
continuously along a marked scale to a maximum corresponding to 10

gm. The
maximum mass that could be measured without additional features would be 500 + 100 +
10 gm = 610 gm. The least count of the scale is 0.1 gm, and the weight can be estimated
to fractions of this smallest division.


Before measuring, a zero adjus
tment check is made with no mass on the pan, and all
sliding beam masses at their left
-
most positions. A thumbscrew under the pan at the left
is turned in either direction until the pointer indicates zero. Recheck the zero for each use
of the balance.


A
measurement is made by placing the mass to be measured on the pan, then moving the
largest scale
-
mass (or ‘poise’) to the highest position that does
not

cause the pointer to
change position. Then the second poise is similarly adjusted so the scale reads w
ithin ten
grams less than the unknown. Finally the front poise is moved until the scale returns to
the zero, or balanced, position. The unknown mass will have been determined to within
the smallest subdivision of the front scale, a tenth of a gram.









30


Other Balances


The
chemical balance

for weighing very small objects has two pans suspended below a
light “see
-
saw” truss. Again the unknown goes on the left (except in the special
technique of “double
-
weighing”). The standard masses for such balanc
es should be
handled only with forceps, lest perspiration etch them away. The surrounding glass case
is closed while judging a result, so that drafts cannot cause errors. Correct balance is
determined by watching the swing of a pointer,
not

by waiting fo
r it to come to rest
(friction may hold it off center).


There are also electrically operated balances which give readings of mass directly upon
dials, after prescribed settings have been made.


For most applications in our lab, the triple beam balance is
adequate.



Measurement of Elapsed Time:




Electronic Time
-
Measuring Units


Note: The following discussion is based on the Pasco Scientific modular
electronic system. An alternate system (Thornton Associates) can sometimes serve in
specific limited app
lications.


The manual mode of operation of an electronic timer is identical to that of a
digital stopwatch. The Pasco Model 8025 Timer (Fig. 4a) accumulates time in
microseconds

in the “timer” mode, while the Model 8015 timer
(Fig
. 4b) accumulates time in milliseconds
. In addition, the Model
8025, when used in the “split timer” mode, becomes two separate timers which
independently record the time elapsed in milliseconds (more on this later). Both timers
co
ntain provisions for the use of external oscillators instead of their internal circuitry for
special applications.



31





(a)




(b)

Figure 4


Parenthetically, the laboratory timers are not generally built as independent units;
you will see that they do no
t, for example, have a power plug to tap the AC (alternating
current) power strips, or even an on/off switch. Instead, they connect to a separate power
unit, the Model 8000 module, which transforms the AC power into the correct DC (direct
current) power f
or matching modules. By “matching” we mean that, with more than one
supplier's equipment in use in laboratories, one must be certain to use units such as the
timing unit and

photobridges (soon to be described) from the same manufacturer


in this case Pa
sco
Scientific. Please note that two different Pasco subsystems exist in the laboratory: most
are the newer system, based on the Model 8025 Timer, but there are a few of the older
Model 8015 timers available. Each has its own type of photogates and conne
ctors, and a
separate memory module, Model 8022, can be used with the Model 8015 to store up to
four time interval measurements generated by the 8015. Different systems and
subsystems may be similar in function but are not compatible electrically; thus, a
ttempts
to mix components from different systems will damage the units.


Manual Timing


When the timer is turned on via the connected power unit ON
-
OFF switch, a
random digital number may (or may not) appear on the display. A
RESET

button on the
timer cl
ears the display to zero. For both Models 8015 and 8025 one may choose from
the FREQUENCY, COUNTER, and TIMER modes. The first two choices are used in
special applications involving an external oscillator and will not be further discussed at

32


this time.
After the TIMER mode has been selected, one must choose either the PULSE
or GATE mode.


If the GATE mode is used, depression of the START/STOP button will start the
timer and release of this button will stop it.


If the PULSE mode is used, the first de
pression of the button will start the timer
and the second depression will stop it.


Used in this “stopwatch mode”, a manually operated timer has another feature
common to stopwatches


if it is operated after one timing operation without clearing the
ti
mer by RESET, the display simply accumulates more counts


which may or may not
be what you want to do.


Manual timing clearly has many applications in the physics laboratory. One of its
obvious limitations is the human reaction time (characteristically,
hundreds of
milliseconds) involved in recognizing the start and stop instants in a process of interest.
TO AVOID THIS LIMITATION, WE CONSIDER A
SENSING

DEVICE THAT CAN
MAKE THE PHYSICAL MOTION ITSELF SWITCH THE ELECTRONIC TIMER ON
AND OFF AUTOMATICALLY.





Photo Transistors


The electrical properties of certain materials change drastically when they are
exposed to light, compared to their behavior in its absence. Their ability to pass an
electrical current is one such property. A
phototransistor

(use
d in the application of
interest here) constructed of such material can be thought of as a light
-
actuated switch. It
is automatically turned
on

when bright light strikes its sensitive surface and turned off