Lab-15-(Centripetal Force).doc
Rev. 2/18/2006
Name: ____________________________________________ Period: ______ Due Date: _____________
Lab Partners: ____________________________________________________________________________
C
ENTRIPETAL FORCE - WebAssign
Purpose: To investigate relationships among centripetal force, Fc, angular velocity, ω, and tangential
linear velocity, v, of an object (a rubber stopper on a string in this case) moving in a horizontal circle with constant
angular velocity, and a tangential linear velocity of constant magnitude. This is the extreme limit of the circular
pendulum problem.
Basic Equations:
In the diagram at the left, T is the tension
in the string attached to the rotating object. T equals the weight of
the hanging weight; T = Mg = W. TV and TH are the vertical
and horizontal components of the tension. The mass of the rotating
object is m and its weight is mg. The angle θ is relative to a
horizontal reference line, as shown. TH is the centripetal force, FC,
in this system. The angle varies with the tension, which affects the
rate at which the stopper rotates, and the radius of the circle, r,
around which the stopper travels. The length of the string is
constant, at L = 1.00 meters, in all trials. We can use simple
trigonometry to solve for all the unknowns. The known values are
L, the length of the string, and M and W, the mass and weight of
the hanging weight. To these we will add an additional known by
measuring the period of rotation, P, at each W. With these known values in hand, our goal is to calculate ω, r, m and
θ. After measuring the period of rotation at each W, we can calculate ω as follows:
Angular Velocity:
ω = 2π/ P
(where P is the measured period of rotation)
Our derivation of the remaining equations proceeds as follows: We note, first, that the radius of the circle is,
Radius of the circle
r = L cos θ
Then we exploit the centripetal force equation, as follows:
Centripetal force:
FC = TH = T cos θ = W cos θ = m r ω² = m (L cos θ) ω2
2
The underlined terms simplify to: W = m L ω
or
m = W/ (L ω2)
Equation I
Finally, we solve for θ in each trial as follows (substitute from Equation I to get the last two expressions):
By definition:
sin θ = TV/T = mg/W = mg/Mg = mg/ (m L ω2) = g/ (L ω2) = m / M
Therefore:
θ = sin-1 (m / M) and/or θ = sin-1 [g/ (L ω2)]
Equation II
In our experiments, we set string length L = 1.00 m. Substitute L = 1.00 in all these equations to evaluate your results.
Page 1 printed on 02/18/2006
Lab-15-(Centripetal Force).doc
Rev. 2/18/2006
Group Data Table
You used a data table similar to this one while making measurements outside. Your instructor, who collated the data
from all groups into grand period averages, collected the original group data sheet. You must be listed as a member of
the group on that copy of the Group Data Table.
Group Members:
_________________________________________
______
_________________________________________
______
_________________________________________
______
_________________________________________
______
Trial Data: Grand Average of All Groups
Length of string = __________ cm
Each group will take one copy of this
table outside. That copy must be
turned in to the instructor at the end of
the period.
The instructor will analyze data from
all groups over night and provide all
periods with their period averages for
all ten trials.
You should not write those period
averages in this table. Those go in the
Data Table on page 4.
This spot can be used later, as
explained in the box on the right, to jot
down your individual group results
after the labs have been graded. This
is an option, not an obligation. Your
group results will be graded based on
the data handed in at the end of the
experiment, not on what is written
here.
Leave these blank for now.
Length of string – 100 cm = __________ cm
20×P .
Time
Number
of
Washers
for 20
Revolutions
(sec) .
10
________
15
________
20
________
25
________
30
________
35
________
40
________
45
________
50
________
55
________
Page 2 printed on 02/18/2006
You will have an opportunity
to enter your group’s data on
this page after the labs are
graded and returned to you.
Until then, you will be
working with aggregated data
from all the groups in your
period. This is intended to
decrease the error in the
results by averaging many
groups together.
If you want to see or save
your group’s individual data,
it will be returned to you on
request after all the labs have
been graded.
You can keep it for reference
or to compare with the
collective average or for your
own curiosity. You do not
need the individual group data
to complete this lab write-up.
Lab-15-(Centripetal Force).doc
Rev. 2/18/2006
Procedure: This lab examines the centripetal force required to maintain an object in a horizontal circular path at
various angular velocities. The object’s mass, m, and the length of the string, L, are constant. Adding weights to the
hanging end of the string changes the centripetal force by changing the tension in the string. The angular velocity, ω, can
be measured by first finding the period, P, of the rotation at each tension. Since the mass of the revolving object is
constant (the same stopper is used in all runs), you must swing it more quickly around the circle as more washers are
added to the string on successive trials. Your ability to establish the balance between the weight of the hanging mass and
the rate of rotation will allow you to maintain the required constant radius from the handle to the revolving object.
1.
Attach a hook (paperclip) to the string. Add weights (metal washers) in groups of five to the string and lock
them on with the paperclip. Find the mass of 1000 washers and divide by 200 to find the mass of 5 washers.
Mass of 1000 washers = _________ g; Mass of 5 washers = _________ g
2.
Find the mass of a box-full of paper clip and divide to find the mass of one paperclip.
Mass of _______ paperclips = _________ g; Mass of one paperclip = _________ g
3.
The string has a mass of about
gram per meter. We are mainly interested in the mass of the string that
hangs below our apparatus and therefore contributes to the tension in the rotating string. Find the length of the
string below the mark in order to determine the mass of the string below the mark.
Length of hanging string = ____________ cm = ____________ m
Mass of hanging string = ___________m × __________g/m =_________g = ____________kg
Length of string
Linear density of string
4.
Total the masses to determine the combined mass for each trial. The combined weight of one paperclip, the
hanging string, and washers in each trial creates the tension in the string. Enter the masses in the Data Table.
5.
Pass the string through the plastic tube and attach a paperclip to the string. Add ten washers to the string for the
first trial. Add five additional washers on each consecutive trial.
6.
Swing the rubber stopper on the string. There is a mark on the string that should line up with the bottom of the
tube when one meter of string is between the stopper and the top of the tube. Adjust the angular velocity until
this mark hangs steadily at the bottom of the tube. This ensures that the string length is constant for all trials.
7.
Use a stopwatch to measure the time needed for 20 complete rotations (Count Carefully! NOT 19! NOT 21!).
Record the total time in the Group Data Table.
8.
Repeat this procedure for ten different weights (from 10 to 55 washers). Turn in the Group Data Table at the
end of the period so your instructor can average the data for all groups in your period. Enter the average values
for 20×P in the Data Table below. Divide by 20 to get P for each trial. Transfer M and P for the trials with 10
to 55 washers to the Results Table.
9.
Complete the Results Table using the averaged data, determine the angular speed for each trial. This will be:
ω = 2π radians / P seconds.
10. Read the entire Analysis section and then complete the remaining columns in the Data & Results Table.
11. As crude as it may be, this apparatus should give very good results if you do your part by attending to the details
and performing the required operations and calculations carefully and thoughtfully. There will be a penalty of ½
point, out of 10, for groups with more than 10% error in the predicted mass of the stopper. There will be a
penalty of 1 full point, out of 10, for groups with more than 20% error. The error will be determined by the
instructor’s calculations using your original measurements. Calculation errors on your part will incur additional
point deductions.
Page 3 printed on 02/18/2006
Lab-15-(Centripetal Force).doc
Rev. 2/18/2006
Data Table (Use g = 9.795 m/s2 when calculating the weights)
Mass of hanging string:
_______________ kg
Mass of one paperclip:
_______________ kg
Mass of 1000 washers:
_______________ kg
Mass of Five washers:
_______________ kg
Weight of string and paperclip:
_______________ N
Weight of Five washers:
_______________ N
Experimental Data: Enter the period averaged data for
20×P provided by your instructor.
Compute the correct hanging masses here for each trial using the data above. When finished, transfer
the information in the M and P columns to the Results Table on the last page of the handout. Then,
complete all the columns in that table following the instructions given in the Analysis section.
M
Mass
Number
of
Washers
20×P
Time
of Hanging
for
String, Washers,
and Paperclip
20
Revolutions
(kg – 6SD)
(sec – 4SD)
P
.
Time
for
1
Revolution
(sec – 4SD)
.
10
______________ ______________ _______________
15
______________ ______________ _______________
20
______________ ______________ _______________
25
______________ ______________ _______________
30
______________ ______________ _______________
35
______________ ______________ _______________
40
______________ ______________ _______________
45
______________ ______________ _______________
50
______________ ______________ _______________
55
______________ ______________ _______________
[True Mass of the Stopper = m = __________________ g = __________________ kg]
Page 4 printed on 02/18/2006
Lab-15-(Centripetal Force).doc
Rev. 2/18/2006
Analysis: (Read this entire Analysis section before you begin calculating results for the Results Table.)
Graph I: Use Graphical Analysis to perform all your calculations. Start with data in columns for M and 20×P. Add
two columns to calculate W and P. Add another column and use it to calculate ω. Set all columns to show
four Sig Figs. You are now ready to create a graph of Hanging Weight (W) vs Angular Velocity (ω).
1) Title: GraphI: Hanging Weight vs Angular Velocity.
2) Axes: Vertical: Hanging Weight in newtons; Horizontal: Angular Velocity in rad/sec.
3) Plot your points. The result should be the right half of a parabola opening upward.
4) The form of Equation I indicates that for the best fit you should select the "Variable Power" function
y = Ax^n). Set variable exponent n equal to 2.
Experimental result:
A = m L = mI = ____________________ (L = 1, B = 2)
Note: The equation of the graph is W = m L ω² so the constant for the parabola (A) should equal
the mass of the stopper in kilograms (m) times the length of the string in meters (L), but L = 1.
5) Assume that the true mass of the stopper, m, the mass of the stopper without string, as recorded on page 4,
PLUS 1/3rd of the mass of the swinging string, 0.197 g, is our reference mass. Thus, mref = m + 0.197 g.
Reference value of the stopper mass:
mref = _______________________ kg
6) Calculate %Error between the reference and experimental values of the stopper mass.
%Error = ___________________ %
Graph II: Create a new column in Graphical Analysis and use it to calculate a column of ω². Now create a new plot
the Graph II: Hanging Weight vs Angular Velocity Squared. Fit it to a straight line through the origin
(proportional line, y = Ax). The slope of that line should also equal the mass of the stopper. Report the mass
and the %Error.
Slope = m L = mII = ____________________ kg
%Error = ___________________ %
Graph III: Before you begin, go to the File menu and select Settings for …, then Degrees, then OK. Then begin by
creating three more columns in your data table in Graphical Analysis. Use the first new column to calculate
values of sin θ ( = m/M ). (Use the value of mII you just got from Graph II.) Use the second new column to
-1
2
2
calculate θ (=sin (sin θ)). Use the third new column to calculate 1/ ω . Now plot sin θ vs 1/ω . This
should produce a straight line through the origin (y=Ax) with a slope of g/L.
Slope = g/ L = gIII = ____________________ m/s2
%Error = ____________________ %
(g in Fort Worth, TX = 9.795 m/s2)
Mass, Radius, and Force Calculations: Next, you should add several more columns to the Data Table in
Graphical Analysis. Use one column to calculate the mass of the stopper (m = W/ L ω2) for each trial, and another to
calculate the radius of the circle (rc = L cos θ = cos θ) for each trial. Compute the average mass of the stopper from
the masses just calculated and write in the space at the bottom of the column. Create two more columns, one to compute
the centripetal force (Fc = m rcω²) using the average mass, mAVE, the radius, rc, and the angular velocity, ω, in each
trial, and a final column to compute the instantaneous linear velocity, v, for each trial. Print this table from Graphical
Analysis and transfer the results to the appropriate columns of the Results Table. Print one complete copy of your
data table from Graphical Analysis and attach it with the graphs at the end of the write up.
ALL entries in the Results Table on the next page must show 5 SD and 6SD for sin θ.
Page 5 printed on 02/18/2006
W*
Weight
__________
__________
__________
__________
__________
__________
__________
__________
__________
15
20
25
30
35
40
45
50
55
__________
__________
__________
__________
__________
__________
__________
__________
__________
__________
(newtons)
__________
__________
__________
__________
__________
__________
__________
__________
__________
__________
(seconds)
Revolution
time for 1
P
Period
__________
__________
__________
__________
__________
__________
__________
__________
__________
__________
(rad/sec)
2
__________
__________
__________
__________
__________
__________
__________
__________
__________
__________
(rad/sec)
Squared
Velocity
Velocity
(of the stopper)
ω2
Angular
ω
Angular
sin(θ)
the slope of the line
(Use mII value from
__________
__________
__________
__________
__________
__________
__________
__________
__________
__________
(kg)
___________
___________
___________
___________
___________
___________
___________
___________
___________
___________
sin(θ)=(m/M)
from Equation I) in Graph II)
(of Stopper,
m*
mass
__________
__________
__________
__________
__________
__________
__________
__________
__________
__________
(degrees)
θ
(=cos θ)
__________
__________
__________
__________
__________
__________
__________
__________
__________
__________
(meters)
(=√1-sin2θ)
rC
Radius
v .
v = rC ω
__________
__________
__________
__________
__________
__________
__________
__________
__________
__________
(newtons)
from below table)
__________
__________
__________
__________
__________
__________
__________
__________
__________
__________
(m/s) .
the stopper)
(use average mass
FC = T
mAVE rC ω²
ALL entries must show 5 Sig Figs in this Table, except sin(θ) must show 6 Sig Figs.
Rev. 2/18/2006
Page 6 printed on 02/18/2006
* Remember that M and W include the masses of the washers, the hanging string, and one paperclip. About 1/3rd of the mass of one meter of string (about 0.197 g) is included
in the calculated mass, m and mAVE, of the stopper. Use g = 9.795 m/s2 = 9.795 N/kg when computing W.
mAVE (mass of stopper; this automatically includes 1/3rd the mass of 1 meter of string ) = __________ g/m (Because the string mass is included, our estimate of m should be about 0.197 g high.)
__________
(kg)
10
Washers
Number String, Washers String, Washers
(Velocity of
of
& Paperclip
& Paperclip
M*
Mass
Results Table: Read the Analysis section before completing this Table
Lab-15-(Centripetal Force).doc