The Kinematics and Dynamics of Circular and Rotational Motion

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14 Νοε 2013 (πριν από 4 χρόνια)

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The Kinematics and Dynamics of

Circular and Rotational Motion

Goals:




Compare the graphs of circular motion for a rotating turntable.



Determine the coefficient of static friction between the turntable surface and a penny.



Predict the magnitude of the angul
ar velocity that causes a penny at a given radius to slip.



Compare the linear and circular motion of different points on a rotating tire.


Equipment List:


Rotating Platform with attached Rotary Motion Sensors

Stickers located at two different radii

Pull
ey

String

Hanging mass and hanger

Penny

Ruler


Computer & Equipment Set Up:


1.

Start by making certain that the string used to turn the turntable is not attached to any hanging mass.


2.

Set up Data Studio™ to read the data collected from the Rotary Motion Sens
or located at the base of
the turntable. You will need to change the default settings of the sensor; double
-
click on the
rotary
motion sensor

icon in the
experiment setup

window


this opens up the
sensor properties

window. In
the
sensor properties

windo
w select the
measurement

tab and un
-
check the box marked angular
position (deg) and check the box that reads angular position (rad), angular velocity (rad), and angular
acceleration (rad).


3.

Create a graphing window to display
Angular Position (

) vs. ime
.



4.

Check the calibration of the sensor
: Press Record; rotate the turntable exactly once; Press Stop; look at
the Angular Displacement values recorded on your graph. (They are measured in “radians”.) Decide
whether or not the graph verifies that your turn
table is correctly calibrated. (If it is not, see your TA
immediately.) Import your graph to the Word™ template and clearly explain your decision.



Lab Activity 1: Kinematics of Circular Motion


The purpose of Activity 1 is to compare the graphs of rot
ational motion for an object, the turntable,
which starts from rest and rotates with constant angular acceleration.


1.

To add to the

Angular Position (

) vs. im攠
graphing window, click and drag
Graph
1

in the
Displays
window and drop it on
Angular Velocity
a
nd
Angular Acceleration

icons in the
Data
window. You
should now have all three graphs in one window.

2.

Carefully wind the string around the base of the turntable. Place the string over the pulley and attach
the hanging mass (use 150 grams) to the other end

of it as shown in the picture above.


3.

Press Record and, releasing the turntable from rest, gather data describing the rotational motion of the
turntable as the mass falls. The angular acceleration of the turntable should be relatively constant.


4.

Using th
e “Statistics” capabilities of Data Studio

, calculate the Angular Acceleration,

, of the
turntable using
two different methods
. Explain each of your methods and state your results.


5.

Copy the graphing window (including the statistics information that you

calculated) into the Word™
template by using “Paste Special.” Paste each as if it were a “picture.”


6.

Compare your three graphs and explain, mathematically, how…



…the
Angular Position vs. Time

graph & the
Angular Velocity vs. Time

graph are related to
eac
h other.



…the
Angular Velocity vs. Time

graph & the
Angular Acceleration vs. Time

graph are related
to each other.




Lab Activity 2: The Dynamics of Circular Motion



There are two purposes for Activity 2. The first is to use the results of a penny slipp
ing off of the
turntable at a particular radius to determine the coefficient of static friction between the surface of the
turntable and a penny. The second is to use this value of the coefficient to predict the angular velocity at
which the penny will sl
ip when placed at a different radius.


Part I: Determine the Coefficient of Static Friction on the Turntable


1.

Measure the radius of the circle created by the
outer

blue sticker, R
o
. Record this value in the table
below.


2.

Carefully rewind the string aroun
d the base of the turntable and place the string over the pulley with
the hanging mass attached. (Note: The size of the hanging mass must be large enough to cause a
penny to slip at both the blue and yellow positions, but small enough to keep it from slip
ping too
quickly. Recommended: 150 grams)


3.

Place a penny at a distance, R
o
, from the center of the turntable. [Note: Do not place the penny directly
on top of the blue sticker.]


4.

Press Record and, releasing the turntable from rest, gather data describin
g the rotational motion of the
turntable as the mass falls.
Using your hand, stop the turntable at the very moment the penny
slips from the surface.

Then, Press Stop to end the collection of data.


5.

From reading your graph of Angular Velocity vs. Time, de
termine the magnitude of the Angular
Velocity of the turntable at the moment just prior to when the penny slipped. Record this value in the
table below.


6.

Record the value of the Angular Acceleration,

, of the turntable (determined in Activity 1) in the t
able
below. (Note: This value should agree with your current data.)


7.

For the moment just prior to when the penny slipped, calculate the linear (tangential) velocity, the
tangential acceleration, and the centripetal acceleration of the penny. (Note: The
tangential
acceleration will likely be much smaller in magnitude than the centripetal acceleration.) Record your
results and explain your calculations in the table below.


8.

Determine the
coefficient of static friction

between the turntable surfa
ce and the penny. Record your
results and explain your calculations in the table below.
Clearly and completely explain your method
of calculating

s
.












Quantity

Result

Explanation of how Result was obtained…

R
o

= Radius (meters)



This radius w
as measured using a ruler.



= Angular Velocity (rad/s)





= Angular Acceleration (rad/s
2
)



V = Linear Velocity (m/s)



a
t

= Tangential Accel. (m/s
2
)



a
c

= Centripetal Accel. (m/s
2
)



a
total

= Total Acceleration (m/s
2
)




s

= Coefficient of Stat
ic Friction





Part II: Predict the Angular Velocity at which the Penny will Slip at a Different Radius


9.

Measure the radius of the circle created by the
inner

yellow sticker, R
i
. Record this value in the table
below.


10.

Record the value of the coefficie
nt of static friction (calculated in Part I) in the table below. Predict the
Angular Velocity of the turntable that will cause the penny to slip when placed at a distance, R
i,

from
the center of the turntable.
Clearly and completely explain your method of

calculating

.


Quantity

Result

Explanation of how Result was obtained…

R
i

= Radius (meters)



This radius was measured using a ruler.


s

= Coefficient of Static Friction



See Table in Part I.



= r敤楣瑥i Angu污l 噥汯捩cy
(r慤⽳)





11.

Test your prediction
: Place a penny at a distance R
i

from the center of the turntable. [Note: Do not
place the penny directly on top of the yellow sticker.] Press Record and, releasing the turntable from
rest, gather data describing the rotational
motion of the turntable as the mass falls.
Using your hand,
stop the turntable at the very moment the penny slips from its position.

Then, Press Stop to end
the collection of data.


12.

From reading your graph of Angular Velocity vs. Time, determine the magn
itude of the Actual
Angular Velocity of the turntable at the moment just prior to when the penny slipped. Record this
value in the table below.


= Actual Angular Velocity

(rad/s)




13.

By what % does your Predicted value differ from the Actual value? Show
your calculation in addition
to your final answer. Does the % difference seem reasonable? Can you account for this difference in
terms of the inaccuracy of your measurements? Explain.