Centripetal Force
I. Introduction.
In this experiment you will study the centripetal force required for a mass in uniform
circular motion. You will determine the centripetal forces required for different masses,
speeds, and radii and will see if your results agree with the theoretical predictions.
For a mass M in uniform circular motion with tangential speed v at radius R, the required
centripetal force is Fc = Mv2/R. If the period of revolution is T, the linear speed of the mass
is v = 2R/T and the centripetal force is Fc = 42MR/T2.
The apparatus used in this experiment has a rotating vertical spindle with a horizontal
crossarm at the top. From this crossarm is suspended a plumb bob (to which additional
mass can be added) which will tend to move outward as the spindle rotates unless a
centripetal force is supplied. A horizontal spring supplies this centripetal force. The
vertical spindle is driven by a small DC motor connected to a power supply. Timing
information is recorded by a photogate connected to a computer operating the Data Studio
(Science Workshop) program which you have already used in an earlier experiment on
Inertial Mass.
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The required centripetal force, as calculated from the measured mass, radius, and period,
will be compared with the measured force supplied by the spring. The spring force is
determined by measuring the equivalent static force Fs required to stretch the spring to the
same length.
II. Required Equipment.
A. A Beck rotating-spindle assembly (illustrated). Protective enclosure around the
apparatus.
B. Accessory equipment: weights, a weight holder, string, two springs, photogate.
C. DC motor and power supply.
D. Macintosh computer with Pasco Signal Interface and Data Studio program.
III. Procedure.
 Initially make sure that the power supply for the DC motor is turned off and that the
voltage controls are set to their full counterclockwise (off) position.
 Before you make any measurements, the rotating-spindle assembly must be level and
balanced. The table tops are sufficiently level that no additional leveling is normally
necessary. It is necessary, however, to balance the rotating crossarm. Marks are provided
on the base of the unit at fixed radii from 14 to 20 cm. The marked radius is to the
outside edge of each bump. Set up for the first radial measurement at 16 cm. For
uniform circular motion at a given radius, the mass suspended from the crossarm should
first hang vertically downward over the corresponding mark when the crossarm is at rest
and the spring is unhooked. (Unless the bob hangs vertically, the suspension cords will
exert an additional unmeasured horizontal force on the mass during a measurement.)
 Use the weaker, more pliable spring for your first measurement.
 Whenever the crossarm is moved to change the radius, the assembly must be balanced as
follows. Unhook the spring. Adjust the crossarm until the bob is hanging directly above
the mark at the chosen radius but does not touch it. (When the bob is in this position the
mounted spring should be horizontal so that it exerts only a horizontal force on the bob.)
When the arm is balanced, it will not tend to rotate even if the base is tipped. Balance the
assembly by adjusting the position of the counterweight and then lock it in place. If you
find you cannot move the counterweight out far enough for exact balance at larger radii,
move it out as far as possible and lock it in that position. Hook up the spring again,
checking to see that it is horizontal when the bob is pulled out to hang vertically above
the chosen mark, and be sure all the screws and locknuts are tight.
 Make sure that counterweight tightening screw passes freely through the photogate.
Each revolution, the screw will cut the light beam in the photogate, providing a
measurement of the period of the circular motion.
 Before turning on the DC power supply, be sure that the protective enclosure is in place.
A. One measurement consists of determining the period of rotation for ~ 50 revolutions when
the spindle is turning at such a rate that the bob passes directly over the mark at the chosen
radius on each revolution. The DC power supply voltage must be adjusted carefully to
achieve the proper radius. The power supply has both coarse and fine voltage adjustments.
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Turn on the supply. Use the coarse adjustment to get the spindle rotating and adjust the
control slowly in order to get close to the proper radius. Then use the fine adjustment for
“fine tuning”. Observe that the system is running stably.
B. To measure the period, T, start up the Data Studio program, by double-clicking on the Data
Studio icon on the Desktop. You should see the familiar Control Window and Menu Bar.

Click and drag the digital plug image to digital channel 1 to logically connect the
photogate to that channel. From the pop-up timing menu, scroll down and select
“photogate and picket fence”.

Create the data table and graph by dragging each little image to the same channel as
you did the plug, and selecting “Time between bands (tDelta)” to display for each. In
the table heading, click on the [0.00...] button and select 4 digits to be displayed.

Start recording data once your rotating bob has reached the desired value of radius.
and Record ~ 50 revolutions. You can watch your data in real time on the graph.
Click on the stop button when you have reached 50 entries in the graph.

While recording don’t disturb the table or the apparatus. Slight shakes can affect the
time period readings.
C. Next run down the voltage on your power supply so that rotation stops. Make sure both
coarse and fine voltage controls are full counterclockwise (the off position). Turn the supply
off.

The radius R should be checked and recorded, and the error in R estimated (i.e. how
well you can determine the position of the bob over a marker).

The mass of the plumb bob (MB) must be measured using a laboratory balance and
its error (MB) estimated which should be +/- less than of the minimum value which
the balance can measure. Weigh the bob once without additional weights and use
this value to determine the mass of the bob for all cases. If additional mass is added,
that value must be recorded so that the total mass M is known.
D. Given these data, the centripetal force Fc can be calculated from the formula given in
Section I above. The error in Fc can be calculated from the errors in period, radius, and mass
by using the rules for error propagation which can be found in section 4.2 of "Measurement
and Error". The detailed formulae are listed in Section V below.
E. The equivalent force Fs for the static case (no rotational motion) is measured by passing a
string from the plumb bob over the pulley provided and by hanging weights from it until the
bob is again over the pointer. Make sure the string and the spring are both horizontal. Be
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sure to record your results. Remember that the weights are marked with their mass, not their
weight, and make sure you include the mass of the weight holder. This total mass, on and
including the weight holder, should be referred to as “m”, and should not be confused with
the mass of the bob system which is referred to as “M”. Estimate the errors in your
measurements of Fs. This may be accomplished by adding (or removing) small amounts of
mass from the weight hanger which do not significantly alter the static equilibrium (i.e. the
bob stays directly over the radial position pointer).
F.
Any one set of data allows you to check the equality of the measured static force Fs and the
calculated centripetal force Fc. It is desirable, however, to test the dependence of the
centripetal force on mass, radius and speed separately. While agreement of Fc and Fs in a
single case might be accidental, agreement for several cases under different conditions
should tend to establish the correctness of the assumptions. Using two different springs
will provide data for two different forces and hence for two different speeds with the same
mass and radius. You can add 150 grams to the plumb bob to compare two different masses
with the same radius and spring. (Be sure the slots in the added weights face outward,
clamp them securely, and make sure the spring is horizontal when the bob is in uniform
circular motion.) Two or more different radii can be used with the same mass and spring.
G.
Data should be obtained for four separate sets of conditions:
(a) Spring 1, mass 1, radius 1;
(b) Spring 2, mass 1, radius 1;
(c) Spring 2, mass 2, radius 1;
(d) Spring 2, mass 2, radius 2.
Spring 1 should be the weaker of the two springs you are given. Mass 1 should be the bob
(MB ) with no added mass; mass 2 should be the bob (MB ) plus 150 grams. Radius 1
should be 16 cm; radius 2 should be 20 cm. Remember to determine the static force in each
case before changing anything, and don’t forget to include the mass of the weight hanger
when determining the static force.
Record your data in Table 2; at this stage, you should have all the values except T and  T
for each run.
H.
The period measurements of four Runs in Data Studio will be used to find T and  T . We
will copy these to Graphical Analysis for plotting and averaging, if you wish, or you can
use Data Studio also to do this.
IV. Analysis in the Laboratory.
A.
For each of the four measurements, it is necessary to determine the period T and its error
(T) from your measurements. To do so, double-click on the Graphical Analysis icon on
the desktop. Then go back to Data Studio, by selecting it in the Finder menu (icon at right
end of the Menu bar). Select Run 1 in the data box; then in the data table select all data by
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clicking and holding on the first data entry and dragging the pointer to the last entry. From
the Edit menu select Copy. Then go back to Graphical Analysis using the Finder menu.
Select the data table and select the first box in the first open column. Then select Paste
from the Edit menu. Create additional columns as needed in the Data menu, and copy and
paste the remaining 3 data runs (one by one) into the next columns of your data table.
Label the columns according to what they contain, e.g., twb16 (for time for weak spring,
bob only, 16 cm radius), tsb16, tsmb16, tsmb20. Set the units of each column to “sec”.

From the Menu Bar, pull down the Data menu and select Data Options. Set the
number of significant figures to 5 and click on ok. Additionally from the Data
Menu, select “rename data set”. In the dialog box, enter an identifying name, such
as your lab bench and room number. This will then identify your data table, and
will identify any printing that you will do on the laser printers.

From the Graph Menu, select New Graph. Using the mouse, click and hold on the
vertical axis label and select the first data column (e.g., twb16). This quantity is the
period of the motion for a given revolution. Click and hold on the horizontal axis
and select “index”. “Index” in the Graphical Analysis program is the equivalent of
“row” in the Science Workshop program. From the Graph Menu, make sure that
the “error bars” are not selected (no check mark in front). The data should look
similar to that obtained during data recording. A regression line will also be
displayed on the data, and with error in slope and intercept written below the plot.

From the Graph Menu, select New Histogram. If the horizontal axis is not
displaying the same label as the vertical axis above, click and hold on the axis label
to select this quantity.
B.
Do a print screen for each group member.
C.
The average (mean) value of the period (T) and also the error in the period (T) may be
obtained directly from the data table itself. Select the first column, by clicking on the
header in the data table. Next pull down the data menu, and select “statistics”. The mean
value number of measurements and standard deviation for your ~ 50 values of this column
will be listed in a dialog box, along with other quantities. Take the mean value to be your
best estimate of T for this measurement: T = mean value. Additionally, take the standard
deviation of the mean to be your best estimate of the error T for this measurement: T =
(T)/ N . Write down these quantities in Table 1 (as it is not possible to print these
quantities directly).
Check to see if these numerical values for T and T are reasonable. To do so, refer to your
histogram. The mean value which you have set equal to T should be consistent with the
average value of the time in the histogram, and T should be of the same numerical order
as the half width at half height (the procedure used in the Inertial Mass experiment). For
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reference, the meaning of “mean” and “standard deviation” are described in your handout
on Measurement and Error.
D.
Repeat this procedure separately for the other three data columns.
E.
F.
You should now be able to complete Table 2
You have now completed all the required work in the laboratory. Exit from the Graphical
Analysis program.
G.
If time permits, start up the Science Workshop program and run the apparatus at the last set
up position (strong spring, radius = 20 cm, and M = MB+ 0.15kg). Set up your graph and
watch the period as a function of row number for ~300 rows. Make a print of the graph for
each group member. Describe qualitatively what you observe. Does the period of the
motion behave in a simple way or a complicated way? Do you observe any regularities?
H.
Finally, make sure all power supplies are off and all voltage controls are in the full
counterclockwise (off) position. Then exit Science Workshop.
I.
Before leaving the laboratory, you should have four printed sheets, each containing a graph
and a histogram, based on your four data sets. You should have filled in the above table.
You should have recorded the names of lab partners for reference. You should have your
teaching assistant initial all sheets.
V.
Analysis outside the laboratory.
A.
Calculate the centripetal force Fc and its error. In each case, compare the calculated force
with the measured static force. Calculate the difference between the calculated Fc and the
measured Fs; Also calculate the error in the difference using the rules of error propagation.
Insert these values in Table 3 For each measurement, decide whether the two values are
consistent. Remember to include appropriate units on all quantities in all tables. Using
units of meters, kilograms, and seconds allows us to express forces directly in Newtons.
The quantities in Table 3 are:
:
Fc = 4 2 M R / T2
Fc = Fc * SQRT [ (M / M)2 + (R / R)2 + 4 (T / T)2 ]
Fs = mg
Fs = g * m
 (Fc - Fs) = SQRT [ (Fc)2 + (Fs) 2 ].
You should refer to Section 4.2 of Measurement and Error to see how the above
expressions are determined.
B. Consider the results of your analysis. Are the values of Fc and Fs consistent in all of the
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cases? Are your results consistent with the theoretical picture for uniform circular motion?
Discuss possible sources of error in your measurements.
C. Draw the free body diagram of the bob, for both the cases, while measuring the static force
Fs and when the bob is rotating.
D. While measuring the static force Fs, if the pulley is not completely frictionless, let’s say there
is a friction f present, then comment on the relationship between Fs, Fc and f.
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TABLE 1. Data of Periods of Oscillations.
RUN Spring
1
weak
R
M (kg)
(m)
0.16 MB
N
2
strong
0.16 MB
3
strong
0.16 MB+ 0.15
4
strong
0.20 MB+ 0.15
Average
T
std dev
 (T)
err of avg
 T =  (T)
TABLE 2. Data of ‘Centripetal Force’ Experiment.
mass of the bob MB =
R
R
MB
 MB
T
T
,
m
m
 MB 
FC
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 FC
FS
 FS
FC-FS
 (FC-FS)