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NORTHERN ILLINOIS UNIVERSITY
PHYSICS DEPARTMENT
Physics 253 – Fundamental Physics Mechanics
Lab #6 Centripetal Force
Meet in FR 233.
Sections A, B, C, G: Tuesday Oct. 4; Sections D, E, T: Thursday Oct. 6.
Read Giancoli: Chapter 5
Lab Write-up due Tue. Oct. 11 (Sections A, B, C, G); Thur. Oct. 13 (Sections D, T)
Apparatus
This experiment uses a vertical shaft that can freely rotate to spin a massive bob of
mass m. The bob hangs by two strings from a horizontal bar with a counterweight on the
other side. The counterweight helps the shaft rotate evenly. A spring can connect the bob
to the shaft and provides a force to pull the hanging mass toward the shaft.
A vertical indicator is used to line up the bob. When the spring is not connected
to the bob, the bob should hang directly over the indicator. A string can be connected
from the bob over a pulley at the end of the apparatus to a hanging weight. The weight is
used to measure the spring force when the shaft is not rotating.
When the weight is disconnected from the bob, the shaft can be spun by hand. As
the shaft spins, the mass will swing away from the shaft held back by the spring. With
enough rotational speed, the mass will line up directly over a vertical post. A clock is
used to measure the number of revolutions n in a time T for the spinning bob.
Theory
Force, acceleration and velocity are all vectors. This means that they have both
magnitude and direction. Acceleration occurs when velocity changes over time. Because
velocity is a vector, this can change can be in direction rather than speed. For instance, a
car rounding a curve accelerates even when its speed is not changing.
Uniform circular motion occurs when an object moves in a circular path at a
constant speed v. The speed of the object moving in a circle can be determined by the
angular frequency  times the radius r of the circle. The angular frequency is the
number of times n an object spins around times radians divided by the time it takes to
make the n revolutions. As an equation this is
2nr
(1)
v  r 
T
For uniform circular motion the acceleration always points towards the center of
the circle. This acceleration is called centripetal acceleration. The magnitude of the
centripetal acceleration ac isac  v 2 / r or using Eq. (1) for the speed v:

2
v 2 4 2 n 2 r

(2)
r
T2
According to Newton’s Second Law a net force on an object creates an
acceleration of that object. The acceleration is in the same direction as the force and has
a magnitude equal to the net force divided by the mass of the object. Since an object in

uniform circular motion has an inward pointing acceleration, it must also have an inward
pointing force (called a centripetal force). In this laboratory, the inward pointing force is
applied by the stretched spring:
4 2 mn 2 r
Fs  mac 
(3)
T2
ac 
Here, the force exerted by the spring is balanced by an equal and opposite gravitational
force from the mass M hanging from the pulley. The inward-directed force of the spring

is then
Fs  Fg  Mg
(4)
Data Collection

(1) Measure the mass m of the bob and the diameter d of the vertical shaft and write
down their uncertainties.
(2) Slide the indicator bar to the closest position, measure the distance from the shaft to
the indicator, and add half the diameter, d/2. This total is the radius, r, of the circular
path. Record an estimated uncertainty for r.
(3) With the spring disconnected from the weight, adjust the top screw on the horizontal
bar so that the bob hangs directly over the indicator. Re-attach the spring and tighten
all screws.
(4) Attach the string to the pulley and hanging mass.
(5) Add weight to the hanging mass until the bob is directly over the indicator. Record
the total hanging mass M and its uncertainty.
(6) Practice rotating the vertical shaft pulling so that the bob moves to consistently pass
over the indicator. Hold a white sheet of paper behind the indicator to see the bob
and indicator tips more easily.
(7) When you feel ready, measure the time T it takes for the bob to make 50 revolutions
(n=50) using the photogate detector. Notice that the photogate detector double
counts. That is, it counts when the bob enters the photogate detector and a very short
time later when the bob leaves the detector. Thus, to get the proper number of
revolutions, divide the number of measurements the computer makes by two to get n.
Repeat this four more times and record the results. Find the average value of the 5
values for T and the uncertainty by calculating the standard deviation of the mean.
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(8) Slide the indicator bar to a position near the middle. Measure the radius r as in step
2. Repeat steps 3 through 7 for the new position finding M and T.
(9) Slide the indicator bar near the end to ensure you have three different r values.
Measure the radius r as in step 2. Repeat steps 3 through 7 for the new position
finding M and T.
The calculations in the Analysis section are to be finished before lab ends. You must
show your final results to your TA before you leave the lab.
Analysis
(1) Use Equation 3 to find the spring force (which is equal to the centripetal force) Fs in
Newtons. Determine the uncertainty by using the method of propagation of errors.
Your TA will give you the equation that relates the error in Fs to your measured errors
in r and T.
(2) Use Equation 4 and M from step 5 to find the spring force Fs in Newtons and its
uncertainty using the method of propagation of errors.
(3) Compare your two measured spring forces by computing the percent difference:
%difference =100 
measurement 2  measurement 1
1/2measurement 2  measurement
1

We use the “percent difference” relationship (rather than the “percent error”
described in Lab #5) because neither measurement is a commonly accepted value of
 spring force.
the
(4) Repeat these comparisons for the other values of r. Does the centripetal force agree
with the spring force to within your uncertainties? Where do the discrepancies come
from? That is, what are the sources of uncertainty?
(5) In the analyses above, you assumed a value of g=9.8 m/sec2 for the acceleration due
to gravity. Now, you will obtain an experimental value for g. In your lab notebook,
plot by hand the radius r versus MT2 for your data. Using Eqns. (3) and (4), express r
in terms of MT2. The slope is equal to g times a constant coefficient, determine the
value of g from the slope. Show this result to your lab TA before you leave.
Lab Write-up
Your write-up to be turned in next week must include:
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1) Parts (1) to (4) of the Analysis Section
2) An Excel plot of r versus MT2 where the error bars are your uncertainties in the
measured radii.
3) The value of g as calculated with a least-squares fit to a straight line. This can be
done in Excel after you’ve made the plot of the data with errors. With the plot
selected, click on the Layout tab at the top of the spread sheet. Select the Trendline
icon and at the bottom of the pull-down menu select More Trendline Options. Be
sure the Linear tab is selected and also select Display Equation. Close the box and
you should see a best find line with the equation of a line. The slope of this line is
your value of g according to a least-squares fit.
4) A discussion of your results.
a) Does your value of g agree with the commonly accepted value of 9.8 m/sec2?
b) What might be causing your value of g to be too large or too small?