Physics 211R: Lab The Kinematics and Dynamics of Circular Motion

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© 2004 Penn State University

Physics 211R: Lab


The
Kinematics
and

Dynamics of

Circular
and
Rotational Motion

Physics 211R: Lab

The Kinematics and

Dynamics of

Circular Motion



Reading Assignment:


Chapter 6


Sections 2, 3, and 5

Chapter 10


Sections 2
-
5


Introduction:


There

are several different types and that each type is dependent upon a particular
frame o
f reference. Linear motion is the most basic form, involving the translation of a
particle from one point to another in one dimension. Projectile motion, although two
-
dimensional, is easily analyzed by separating it into its linear components. Circular
motion is a special case of two
-
dimensional motion in which an object translates in a
circular path. Rotational motion is observed when an object itself rotates about some
internal axis.
It is c
ircular motion

that

connects the concepts of linear and rota
tional
motion. For any object that is rotating, a particular point on that object is moving in a
circle. One of the goals of this lab activity is to explore and understand this connection.



The translational motion of any point particle can be described
in terms of
standard Cartesian coordinates. In other words, Cartesian coordinates can describe both
linear and circular motion. However, in the case of circular motion, the particle moves an
arc length, s, around a circle with a (constant) radius, r. The
refore, in this case it is
simpler to use Polar coordinates where the position of the particle can be specified by r
and the angular position,

, rather than x and y.



Notice that the right triangle defines the relatio
nship between the location of the
particle in Cartesian and Polar coordinates. The following equations describe how to
transform between Polar and Cartesian coordinates.


cos
r
x






sin
r
y



2
2
y
x
r





r
s
y
x











1
tan


S

(x, y) = (r cos

, r sin

)

X

Y

r



© 2004 Penn State University

Physics 211R: Lab


The
Kinematics
and

Dynamics of

Circular
and
Rotational Motion

The radius of a particle undergoing circular motion is always a constant. The
angula
r position, however, will change with time depending on the motion of the particle.
Since only the angular position changes with time its behavior is exactly analogous to the
behavior of the position in one
-
dimensional motion that was studied previously. T
hus, the
angular equivalent of the kinematic quantities for one
-
dimensional motion can be defined
as follows:

Arc distance traveled




r
s
s






Angular position


Linear (tangential) velocity


r
v
dt
d
dt
ds
v








Angular
velocity


Linear (tangential) acceleration

r
a
dt
d
dt
dv
a
t
t








Angular acceleration


The relation
ships between the angular position, velocity, and acceleration are
exactly the same as the relationships previously determined for one
-
dimensional motion.
For example, for motion with constant angular velocity,

:


o
t





.

For motion with constant angular acceleration,

:


o
t





.

o
o
t
t







2
2
1
.


The Net Force that causes an object of mass, m, to move in a circular path is
called the

centripetal force
, F
c
. At a particula
r constant linear speed, v, the following
equation (Newton’s Second Law) describes the dynamics of the object’s circular motion:

c
c
a
m
F





where the magnitude of the centripetal acceleration is given by:

r
v
a
c
2



For circular

motion that is not constant, the total acceleration (magnitude and
direction) of the object, at every moment, is determined by adding the centripetal (radial
component) and linear (tangential component) accelerations together. Note:
a
c

and
a
t
are
vectors

that form a right triangle when added together because they are perpendicular to
each other.


t
c
total
a
a
a







Remember that tangential acceleration is a consequence of any change in the
linear (and therefore, angular) speed of the object. Centr
ipetal acceleration is a
consequence of the rate at which the direction of the object changes at every moment.


© 2004 Penn State University

Physics 211R: Lab


The
Kinematics
and

Dynamics of

Circular
and
Rotational Motion

The centripetal force, the component of the net force directed towards the center
of the circle,
is caused by different types of forces
. Here a
re some examples for the case
of a single force causing circular motion:




Tension

in a string (such as when a ball is whirled in a horizontal circle at the end of
a rope)



The
normal force

(such as on the Rotor ride found at many amusement parks)



Gravity

(such as the orbits of planets around the sun)



Static friction

(such as a car traveling around a level curve)


Here are some examples for the case of two forces in combination causing circular
motion:




Tension

and
gravity

(such as when a ball is whirled i
n a vertical circle at the end of a
rope)



The
normal force

and
gravity

(such as when a person rides a vertical loop on a roller
coaster)



The
normal force

and
static

friction

(such as a car traveling around a banked curve)


In this lab, you will explore the

case of a penny moving in a circle on a level
surface due to the force of static friction. Recall that static friction is a variable force,
able to provide resistance up to a particular maximum value. At this point, the object is
said to be on the verge

of slipping. The following equations describe the relationship
between static friction and the dynamics of the
non
-
constant

circular motion at this
moment.


static
net
f
F





normal
s
total
F
a
m






mg
a
a
m
s
t
c







where, on a level surf
ace, the normal force is equal in magnitude to the force of gravity.
Remember that the magnitude of a total is determined by taking a vector sum.



© 2004 Penn State University

Physics 211R: Lab


The
Kinematics
and

Dynamics of

Circular
and
Rotational Motion

The Kinematics and Dynamics of

Circular Motion

Goals:




Compare the graphs of circular motion for a rotatin
g turntable.



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



Predict the magnitude of the angular 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

Pulley

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 Sensor
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
experiment setup

window select the
measurement

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

Set sample rate to 10 Hz. In the
rotary motion sensor

tab,
set the resolution to “high”.


3.

Create a graphing window to display
Angular Position (

⤠癳⸠呩me
.


4.

Check the calibration of the s
ensor
: 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
turntable 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
graph the

rotational motion for the turntable, which starts

from rest and rotates with constant angular acceleration.


© 2004 Penn State University

Physics 211R: Lab


The
Kinematics
and

Dynamics of

Circular
and
Rotational Motion

1.

C
lick and drag
Graph1

in the
Displays
window and drop it on
Angular Velocity
and
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

2
00 grams or 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 the “Statistics” capabilities of Data Studio™, calculate the Angular
Acceleration,

, of the turntable usi
ng
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
your

template by using “Paste Special.” Paste each as if it were a “picture.”


6.

Compar
e your three graphs and explain, mathematically, how…



…the
Angular Position vs. Time

graph & the
Angular Velocity vs. Time

graph
are related to each other.



…the
Angular Velocity vs. Time

graph & the
Angular Acceleration vs. Time

graph are related to each o
ther.



Lab Activity 2: The Dynamics of Circular Motion

The purpose
of this activity is

to determine the coefficient of static friction between the
surface of the turntable and a penny

using rotational motion equations
.
With this value of
the coefficient
you can then

predict the angular velocity at which the penny will slip
when placed at a different radius.


Part

I: Determine the Coefficient of Static Friction on the Turntable

Conceptual
:

Draw a force body diagram for the penny just before it starts slid
ing. Include arrows
showing the direction of
a
c
,
a
t
, and
a
total
.


Data Collection:

1.

Measure the radius of the circle created by the
outer


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


2.

Carefully rewind the string around the base of the turntable and

place the string
over the pulley with the hanging mass attached. (
Use same mass as in Activity 1.
)


3.

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

sticker.
]


4.

Press Record and, rele
asing the turntable from rest, gather data describing the
rotational motion of the turntable as the mass falls.
Using your hand, stop the
© 2004 Penn State University

Physics 211R: Lab


The
Kinematics
and

Dynamics of

Circular
and
Rotational Motion

turntable at the very moment the penny slips from the surface

(listen for the
click when the penny falls)
.

Then, Pre
ss Stop to end the collection of data.


5.

In the table below, record the values necessary to determine the coefficient of
static friction. Use values for angular velocity just prior to when the penny
slipped. You will need to 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.)



Quantity

Result

Explanation of how Result was obtain
ed…

R
o

= Radius (meters)



This radius was measured using a ruler.



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2
)


From activity 1 (does it agree with current data?)

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 Static Friction




Results:

Determine the
coefficient of static friction

between the turntable surface and the penny.
Clearly and completely explain your method of calculating

s
.

© 2004 Penn State University

Physics 211R: Lab


The
Kinematics
and

Dynamics of

Circular
and
Rotational Motion

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

Theory (Prediction):

1.

Measure the radius of the circle created by the
inner

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


2.

Record the value
s

of the coefficient of static friction
and


(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

.


Quan
tity

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.




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a搯猩




Data Collection (Test Prediction):

3.

Using the same technique as in Part I,
gather data describing the rotational motion
of the turntable as the mass falls

while the penny is located at a distance R
i

from
the center
.
[
Note: Do not place the penny

directly on top of the sticker.
]

Results:

4.

D
etermine the magnitude 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)




5.

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? Explai
n.