NAME
__________________
LAB INSTRUCTOR________________
PARTNERS
__________________
__________________
Rotational Motion
I.
Introduction
In this we will measure the moment of inertia for a rectangular metal bar. We will apply
a constant torque tha
t produces a constant angular acceleration. From a plot of torque as a
function of angular acceleration we can find the moment of inertia for the specimens.
II.
Theory
The kinematics of
rotational
motion is completely analogous to the kinematics of
tran
slational
motion. It is a convenience of mechanics that these two types of motion can be
described independently. Every equation describing the translational motion of a body has an
equivalent counterpart describing the motion of a rotating body.
In r
otational kinematics the formula for the angular displacement and angular velocity of
an accelerating, rotating body is
(
t
)
0
0
t
1
2
t
2
x
(
t
)
x
0
v
0
t
1
2
at
2
(1a, b)
(
t
)
0
t
v
(
t
)
v
0
at
(2a, b)
The linear quantities are related to the angular quantities by the ra
dius of rotation
r
(the
distance between the center of rotation and the point whose motion is being examined).
x
(
t
)
r
(
t
)
v
r
(
t
)
r
(
t
)
a
r
(
t
0
r
(
t
)
When a rigid body is set into rotation by applying a t
orque, the body resists the change in
its current state of rotation. This rotational inertia is called the
moment of inertia
of the
body. It is equal to the ratio of the applied torque
瑯⁴桥獵汴楮i湧畬慲cce汥la瑩潮
.
I
m
F
a
(
6)
While the mass of a body depends only on the quantity of matter that the body contains,
the moment of inertia depends on both the quantity of matter and the distribution of the
matter about the center of rotation.
Rotational Motion


2
III. Apparatus
Personal Compute
r
Science Workshop 750 Interface
Rotational apparatus
Smart Pulley, String, and Weights
Rectangular metal strip (sample)
Meter stick and scale
IV. Method
In this experiment you will determine the moments of inertia of a rectangular
metal bar.
To do this, the apparatus shown below will be used. It consists of a spindle upon which the
specimens are placed. A driving torque is applied to the apparatus and specimen by a string,
one end of which is wound around the drum just above the
specimen. The other end of the
string is draped over a “smart pulley” and attached to a mass
m
. When the mass is released,
it falls with an acceleration
a
. Consequently, the specimen undergoes angular acceleration
㴠愯a
. A Macintosh is connected through a Mac 65 interface to a “smart pulley”. Every
time
Figure 1
one of the spokes of the pulley passes between the emitter and receiver of the photodiode
detector, the beam is broken. The interface reads an int
ernal clock each time a spoke blocks
the beam and thus measures the position of the pulley or the string over the pulley. The
computer will plot linear distance or linear velocity. You can use the relation between linear
motion and angular motion to find
the angular displacement and angular velocity.
The other value that you need to determine to find the moment of inertia is the torque
⸠.
F牯洠瑨攠晲ee

扯摹摩慧牡洠潦瑨攠晡汬楮i浡獳m獨潷渠潮瑨攠湥x琠灡geⰠy潵ca渠獥e瑨慴t瑨攠
睥楧桴h潦瑨攠晡汬楮i浡獳m(
W=mg
) is directed downwards, while the tension
T
in the string
(resulting from the turntable's resistance to rotation or its moment
of inertia) is directed
upward.
Rotational Motion


3
Figure 2
The imbalance between these two opposing forces is what causes the mass to fall with
an acceleration
a
. Therefore,
W

T = ma
T = W

ma
T = mg

ma
T = m(g

a)
In terms of
a
, a value which we k
now, the tension
T
is
T = m(g

a).
(7)
Since the tension in the string is responsible for turning the turntable, the torque is
= Tr,
(8)
where
r
is the radius of the turntable around which the string is wrapped. So,
= m(g

a)r.
(9)
There are also forces of friction between the turntable and the axle, and the pulley and
its axle. Both of these forces contribute
a frictional torque
f
, which opposes the torque due
to the weight. Therefore,

f
= I
or
= I
+
f
.
(10)
A plot of
vs.
y楥汤猠i瑲楧桴楮攠i桯獥汯灥猠
I
and whose y

intercept is
f
.
You will also determine the moments of inertia of
the sample mathematically. For a
rectangular strip of metal rotated about an axis which runs through its center of mass and is
perpendicular to the plane of the strip,
I
bar
1
12
a
2
b
2
(12)
In this formula,
a
is the length of the strip,
b
is t
he width of the strip, and
m
is the mass.
Rotational Motion


4
In the last section of the lab we will measure the moment of inertia for an object of
arbitrary shape. We will also find the moment of inertia of the object by using the principles
of integral calculus.
V. Proc
edures
Moment of Inertia of a Rectangular Shaped Object
A.
First, let’s set up the Science Workshop 750 interface and the rotational apparatus with
the “smart pulley”. Turn on the interface and then open Data Studio. Connect the smart
pulley by “click

a
nd

dragging” the digital sensor icon to the Digital Channel 1 and select
Smart Pulley from the sensor dialog box. Connect a graph to the smart pulley by “click

and

dragging” the graph icon to the Smart Pulley icon. Select velocity from the dialog
box.
B.
Mount the specimen on
top of the disc apparatus
. Measure the diameter of the middle
drum on which the string is
wound and compute the radius.
Place a mass of about 50
grams on the end of the string and wrap the string around the middle drum. There shoul
d
be enough string for the turntable to
undergo at least four complete
revolutions. Start
data recording
and
release weight. After the turntable has
undergone four complete revolutions
stop recording. From the graph
calculate the mean acceleration,
a
mea
n
and the standard deviation for the
accelerations,
a
sd
. Enter the value in
the table
on the next page
.
Print the
graph and show your work on the
graph. Turn in the graph with your lab.
Measure the three different diameters on the rotational apparat
us and enter them below.
small disk diameter _____________
small disk radius ______________
medium disks diameter _____________
medium disks radius ______________
large disks diameter _____________
large disk radius ______________
C.
Calculate
the angular acceleration and enter
the value
in the table
on the next page
.
D.
Calculate the torque,
a湤ne湴敲瑨攠癡汵攠
楮i瑨攠瑡扬t潮瑨攠湥x琠灡ge
⸠S桯眠y潵爠
ca汣畬l瑩潮献†
Rotational Motion


5
E.
Repeat this procedure five more times, adding 20 grams to the
weight each time.
run
mass
linear acceleration
angular acceleration
torque
1
__________
____________
____________
____________
2
__________
____________
____________
____________
3
__________
____________
____________
____________
4
__________
___________
_
____________
____________
5
__________
____________
____________
____________
6
__________
____________
____________
____________
F.
Using these five values, plot
癳⸠
mean
.
Use the standard deviation as the uncertainty
and add error bar to the graph. From the graph calculate and record the moment of inertia
of the specimen,
I
bar
.
From the fit find the uncertainty and enter it below. Print the
graph and retu
rn it with your lab report.
I
bar, exp
______________ uncertainty _______________
G
.
Now calculate the moments of inertia of the rectangular metal bar. You will first have to
weigh and measure the specimen. Record these values, along with
their
reading
uncertainty
.
mass of bar __________
uncertainty ______________
length of bar __________
uncertainty ______________
width of bar __________
uncertainty ______________
Calculation of moment of inertia
Rotational Motion


6
Calculation of uncertain
ty
I
bar, theory
______________ uncertainty _____________
Compare the
calculated
moment of inertia with the values you obtained experimentally in
procedure D, and compute the percent difference
.
percent difference ______________
Do the two
values agree to within their uncertainty? Explain.
H
.
From the graph find the frictional torque,
f
, for the tu
rntable and sample. Record the
torque
below. Can you think of another way to measure the frictional torque? Use this to
remeasure the frictional torque and compare the two values.
frictional torque from graph ___________________ unce
rtainty _____________
Explain a second way to measure the frictional torque.
frictional torque from graph ___________________ uncertainty _____________
Moment of Inertia of an Arbitrary Shaped Object
I
.
Replace the rectangular bar with t
he arbitrary shaped object and repeat procedures A
through E as before.
E
nter the data below.
run
mass
linear acceleration
angular acceleration
torque
1
__________
____________
____________
____________
2
__________
____________
____________
___________
_
3
__________
____________
____________
____________
4
__________
____________
____________
____________
Rotational Motion


7
5
__________
____________
____________
____________
6
__________
____________
____________
____________
Use these five values and plot
癳⸠
mean
as in procedure F. From the graph calculate
and record the moment of inertia of the specimen,
I
bar
.
I
bar, exp
______________ uncertainty _______________
J.
Next we will calculate the moment of inertia by using the principles of integral c
alculus.
We will do this by replacing the integral by a sum.
A
r
m
r
dm
r
2
0
A
limit
2
0
m
limit
2
We will break the object up into a large number of small masses (or areas) and then
multiply the mass of the small area by the distance from the axis of rotation square
d.
Finally we will add these all together to find the total moment of inertia.
Get a sheet of rectangular grid paper from the instructor and trace the outline of the object
on the paper. Also mark the axis of rotation
on the grid paper.
First find the
area of one
rectangle and then use this to find the total area of the object.
area of a rectangular grid __________
uncertainty ______________
area of object __________
uncertainty ______________
Weigh the objec
t and calculate the mass of
the re
ctangular grid.
mass of object __________
uncertainty ______________
mass of rectangular grid __________
uncertainty ______________
To calculate the moment of inertia we first find the distance r from the axis to the
m or
A.
Square this value
and multiple by the mass for
Calculation of moment of inertia
A and this is the contribution of this
A to the
moment of inertia. Do the same for all the other
A’s.
Calcul
ation of
the moment of inertia.
Rotational Motion


8
Calculation of uncertainty
in
the
mom
ent of inertia.
Enter the value here.
I
object,
calculation
______________
uncertainty _____________
percent difference ______________
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