7 Rotational Motion

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14 Νοε 2013 (πριν από 3 χρόνια και 11 μήνες)

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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,


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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 ______________