Physics 181 - Summer 2013 - Experiment #7
1
Experiment #7, Centripetal Force
1 Purpose
To experimentally demonstrate the validity of Newton's Second Law, F = ma. In particular, you will
demonstrate the linear relationship between the centripetal force Fc and the centripetal acceleration ac
for an object of total mass mT rotating in uniform motion: Fc = mT ac. You will be using Hooke’s Law
and the spring constant that you measured in the Hooke’s Law experiment for the same tube.
Warning. This experiment is done with the room lights off. The flashing strobe light, around 400
flashes a minute, and the clicking sound which it makes, may cause one to feel ill. If you have a
history of being adversely affected by flashing lights, you may exit the lab while measurements are
being made. Please inform your TA.
2 Introduction
The acceleration of an object of mass m with uniform circular motion at speed v, radius R, angular
velocity ω = v/R and frequency f is always directed radially inward toward the center of the circle.
This type of acceleration is called centripetal acceleration ac. Its magnitude is given by:
2
v
2
(1)
ac = = ω R
R
By Newton's Second Law of Motion, a radially inward net force Fc (called centripetal force) must be
applied to an object of mass m to provide this acceleration. Using Newton's Second Law, one gets:
2
Fc = mac = mv = m2R = m(2f )2R
R
(2)
where f is the frequency of rotation = 1/(period), ω = 2πf = the angular velocity in radian/s. (2π
radian = 360 of arc, or 1 radian = 57.3 of arc.)
In this experiment, an object attached to a spring is rotated in uniform circular motion. You will
measure the centripetal force as a function of angular velocity ω and radius R in order to
experimentally verify Eq. (2). To do so, you will determine Fc, f and R independently. You will
calculate ac using Eq. (1), where ac = (2f )2R, and plot Fc vs. ac. You will demonstrate that this graph
is a linear graph with zero intercept and with slope equal to the value of the total rotating mass.
(Actual intercept may vary from zero.)
The angular velocity is determined using a stroboscopic light, and the centripetal force is determined
by measuring the extension of the spring. The relationship between the force applied to the spring and
the extension of the spring is known as Hooke’s law and is given by:





F = - kR (Negative, as a restoring force)
(3)
where k is the spring constant and R is the extension of the spring.
5/26/2013
3 Experimental Apparatus and Procedure
The experimental apparatus, shown in Fig. 1, consists of two sets of brass cylinders and springs
encased in a Lucite tube. Half of the tube is painted black so that you can observe only one set of the
cylinder-spring system. A shaft is mounted at the midpoint of the Lucite tube. One end of a spring is
attached to a brass cylinder and the other end is connected to the shaft. The radial distance R between
the axis of rotation and the center of the brass cylinder can be determined from the scale (in centimeters) on the Lucite cylinder. (If only one set of the cylinder-spring system is used, then an unbalance
occurs when the assembly is rotated. This unbalance, if allowed to occur, will result in the destruction
of the apparatus.)
Figure 1: Circular motion apparatus
The shaft and assembly are driven in uniform circular motion by a motor whose rotation rate is
proportional to the voltage applied to the motor. As the rotation rate increases, the spring stretches
and the radial position of the brass cylinder increases until a radius R is reached for a given angular
velocity ω. The stretched spring supplies the necessary force to keep the brass cylinder in circular
motion at angular velocity ω.
As shown in Fig. 2, the restoring force F = kR = mg = W (at equilibrium) was calculated in the
Hooke’s Law experiment and separate measurements of the elongation R were recorded. Hooke’s
Law, Eq. (3) is a linear relation between force and the displacement. The spring constant (slope) k
and intercept R1 were found for the tube (mass and spring) used.
Figure 2: Force vs. elongation graph for real spring.
In this experiment, centripetal force Fc will be calculated from spring elongation R. For a given
elongation, Fc will equal W in Fig. 2, or, Fc = k (R - R1). This expression holds if the same tube is
used in both experiments, and the spring has retained the same value of k, the spring constant.
Physics 181 - Summer 2013 - Experiment #7
3
3.1 Apparatus
1.
Circular motion apparatus consisting of a motor and a shaft, and the hollow Lucite
tube with spring and brass cylinder shown in Fig. 1.
2.
Variable voltage power supply to drive motor
3.
Stroboscope which flashes around 400 times a minute.
The stroboscope is a high-intensity flash lamp that makes moving objects appear stationary. The
duration of the flash is very short, typically 1/2000 of a second. During this interval the object's
motion is so small that it appears stationary to the eye. Since the object can be seen only when the
lamp is flashed, rotational motion is made to appear stationary if you adjust the frequency of flashing
so that the rotating object reaches the same point at each flash, i.e., if you synchronize the flash rate
with the rotation rate. The rotation speed may be determined by measuring the flashing frequency.
The frequency adjustment of our stroboscopes is calibrated directly in revolutions per minute (RPM).
False readings are obtained if the frequency is exactly ½ (or some other integer fraction of) the actual
rotational speed. Multiple stationary images occur for frequency settings exactly twice (or some
integer multiple of) the actual rotational frequency.
3.2 Procedure
Measurements with the circular motion apparatus
Caution: Be careful around this setup. Keep long hair and jewelry away from the apparatus.
Remember that although the apparatus may look stationary it is moving. If the flashing of the
stroboscope is making you feel ill, please feel free to step out into the hallway.
1.
Mount the Lucite tube assembly securely on the shaft of the motor as demonstrated by
the lab instructor. Tighten both thumb screws.
2.
3.
4.
5.
Before plugging in the power supply, check that the voltage control knob is turned full
counterclockwise (to zero voltage). Plug in the power supply and turn on the
stroboscope.
Set the stroboscope to the maximum RPM (approximately 400 RPM) setting given for
your specific spring assembly. This is the RPM for an elongation of 15.00 cm.
Slowly increase the voltage of the power supply until the rotating assembly appears to
be nearly stationary with the 15.00 cm range. Make final synchronization adjustments
using the stroboscope's frequency control. (It is easier to make the final adjustment on
the stroboscope because the voltage control is rather a coarse adjustment.)
If your elongation is not exactly 15.00 cm, that is fine. If you get something like 14.90
cm, then use it. Record your elongation as 14.90 cm and the RPM you got for this
elongation.
At this voltage the motor speed is given by the RPM (Revolutions Per Minute)
reading of the stroboscope. The voltage meter on the power supply will typically read
between 20 and 25 volts. (Be careful to avoid the false RPM readings discussed in sec.
3.1 above.)
5/26/2013
6.
Record the radial position R of the center line on the brass cylinder and the
stroboscope RPM. Each division on the strobe dial translates to 5 RPMs. Have your
first reading confirmed by your TA.
7.
Lower the voltage setting by an amount such that the radial position of the brass
cylinder is decreased by about 0.50 cm.
8.
Lower the stroboscope setting until the motion is almost stationary. Make fine
adjustments with the RPM control knob to obtain exact synchronization.
9.
Record the position of the center line on the brass cylinder R and the stroboscope
RMP.
10.
Repeat steps 7-9 to obtain readings over the range between 15.00 cm and 11.00 cm in
steps of 0.50 cm. Again, you do not have to use exact settings on elongations.
11.
Turn off the stroboscope and set the power supply voltage to zero.
12.
Measure the mass of the spring.
4 Calculations and Analysis of the Data
1.
Note the columns in the table for rotational data RPM and R. For each pair of values
of RPM and R (in meters) in the table, calculate the centripetal acceleration ac in m/s2
using Equation (5) and record it in the column marked ac. Note that the angular
velocity ω in radian/s is related to the RPM by ω = 2π(RPM)/60 so that the square of
the angular velocity and the angular acceleration are given, respectively, by:
2  RPM 2 4 π 2 (RPM) 2
) 
ω =(
60
3600
2
ac = 2R = (
2.
2  RPM 2
4 π 2 (RPM) 2
) R
R
60
3600
(4)
(5)
In the last column of your table put values of centripetal force Fc (in Newtons) for
each value of R (in centimeters) in your rotational data table.
Fc = k (R - R1)
(6)
Determine this value by using your Hooke's Law values of k and R1 from that
experiment. Below are two methods to determine Fc. You will be using the first
method, but you should understand that the second method will work too.
a.
You must use Eq. (6), with your experimental values of k and R1, and insert
the values of R from the rotational data table and calculate Fc.
Fc = k (R - R1) for any R in the spring's linear Hooke’s Law region.
b.
3.
You may use the graph directly from the Hooke’s Law experiment to read off
the force Fc corresponding to that elongation value R of the spring. This too
demonstrates that W = k (R - R1) = Fc in the linear region.
Plot Fc as a function of ac. In the lab, you do this in Excel. For your lab report, you
graph this on the 18 x 25 cm graph paper. According to Newton's Second Law, you
Physics 181 - Summer 2013 - Experiment #7
5
should expect to get a straight line with a slope equal to the total rotating mass* and
an intercept equal to zero. Do not force your line of best fit to pass through an
intercept of zero. Express the slope in both kilograms (kg) and grams using 4 figures
as 8.478x10-2 kg, so you will have 84.78 grams. The intercept in will be in Newtons.
Next, enter your linear data (as determined from your graph) into your calculator’s
linear regression function, and find your experimental value for mT, and too, the
intercept and R2. Again, express the slope in both kilograms (kg) and grams (using 4
figures). Be sure to adjust Excel in the lab to give you this result.

*Caution: The value of the rotating mass obtained in step 3 above includes the effect of
the mass of the spring. The slope of Fc vs. ac equals the sum of the mass of the brass
cylinder plus 1/3 the mass of the spring.
1
mT = mB + mS
3
4.
Using your data for mB and mS, calculate the accepted value of mT, where
1
mT = mB + mS
3
Use percent error to compare this result to your experimental mT (slope in grams) from
the Fc vs. ac calculations. Do this for your Excel graph in the lab and for the results
from linear regression in your lab report. Use the above calculated value of mT as your
accepted value for percent error. To answer this question correctly, your value of the
slope of the Fc vs. ac graph should have 4 figures. An example is 8.478x10-2 kg, so
you will have 84.78 grams.
5 Questions
1.
What law of physics is this experiment designed to experimentally validate?
2.
Based on your percent error in part 4 above, did you validate this law? Use 5.00% as
the upper limit to have achieved experimental validation. If you are higher than 5%,
try to explain why this happened.
3.
List the different possible sources of error in this experiment. Discuss these errors and
comment on which of them could be most significant to the experimental results. You
may include errors in the Hooke’s Law experiment.
4.
Use propagation of error to calculate ac for the ac calculated at an elongation of
15.00 cm. This is your first data for ac.
6 Conclusion
This section should have a clear statement of the results of the experiment and the extent to which the
results are in agreement with the theory being tested.
To make this comparison meaningful, you should include the impact of the experimental error on your
results. Please include a statement of what you have learned, a critique of the experiment, and any
suggestions you have which you think could improve the experiment or the lab handout.
5/26/2013
Centripetal Force
Data Sheet
Tube Number ________ Should be the same as in the Hooke’s Law experiment.
Mass of the spring ms = ______________ grams
Measurements with the circular motion apparatus
Data
Point
#
1
2
3
4
5
6
7
8
9
RPM
(1/min)
Elongation
R
(cm)
Elongation
R
(m)
Acceleration
ac
(m/sec2)
Estimated error in reading RPM from strobe dial.
ErrorRPM = ± RPM = ________ RPM
Estimated error in reading the scale on the spinning tube.
ErrorR = ± R =________ cm
Estimated error in reading the mass on the spring.
Errorm = ± m =________ grams
Force
Fc
(N)