Periodic Motion Lab – The Conical Pendulum
Purpose
The goals of this lab are to verify that centripetal acceleration is
given by a = v2/r and to show that the period of a conical pendulum
is given by the theoretical equation:
Procedure
A small mass is suspended by a cord and set into motion above a
“target” circle. It then moves in a horizontal circle under the
influence of gravity and the tension in the cord. As it moves, the
mass and string trace out an imaginary cone – hence it is referred to
as a conical pendulum. The radius, r, and length, l, are measured
with a ruler or meter stick. Both r and l should be measured to the
center of the mass. The period, T, is measured using a stopwatch. It
is best to time a certain number of revolutions and divide by this
number to get the period. All other values are calculated based on
these three measurements. The “height” h will not be measured
directly.
Part I – Centripetal Acceleration
Vary the value of r and measure the value of T while keeping the
ratio of l to r constant. Use a ratio of l/r = 7. (By keeping this ratio
constant, the centripetal acceleration will theoretically remain
constant according to Newton’s Laws.) In order to achieve this, you
must measure the radius of one of the circles and then adjust the
length to seven times this amount (we could have used any ratio, but
seven is a convenient value using this equipment). Once a trial is
complete, choose another radius and readjust the length and repeat.
Part II – Period dependence on h
Vary the value of l and measure the value of T while keeping r
constant. Choose one of the “midrange” circles and use the same
one each time while changing the length of the string from trial to
trial. Record the value of this radius. Then choose six lengths that
will cause a significant change and cover as wide a range of periods
as possible.
Analyses
1. Complete the tables by calculating speed, speed squared, and
acceleration for Part I, and by calculating the height h and the
constant k for Part II.
2. Produce a graph for Part I showing speed squared vs. radius.
Determine the line of best fit and corresponding equation.
3. Produce a graph for Part II showing period vs. height. This
should be a curve of the form:
. Use the mean value
for k and write this equation in finalized form on the graph.
Then plot this equation and connect with a smooth curve for
comparison with the data points.
Questions
1. (a) Make a free body diagram of the pendulum mass as it moved
in its circular path.
(b) What force or what component of force is the centripetal
force for the mass?
2. (a) Use F = ma to solve for the centripetal acceleration of the
mass in Part I. Show your work. Hint: the mass will not affect
the result, but the result is affected by the amount of tilt in the
string (which relates to the ratio of l/r).
(b) Explain why the acceleration should be constant as long as
the ratio of length to radius does not change.
3. Taking the theoretical value of acceleration found in the
previous question as the accepted value: (a) Calculate the
percent error in the mean value of the calculated acceleration
found in the table. (b) Calculate the percent error in the slope of
the graph.
4. (a) Use the mean value of k to determine the value of g. (Hint:
the theoretical equation should equate with the curve fit
equation.) (b) Calculate the percent error in this value for g.
5. The theoretical equation for period in this experiment is very
similar to simple harmonic motion equations for period. If the
mass in this experiment had been swung like a normal pendulum
(moving back and forth in one plane) would it have had the
same period as the conical pendulum (moving in a horizontal
circle)? Discuss and explain.
6. Discuss Error. As always, this means to evaluate imperfections
in the results. Make specific references to evidence of error and
the most probable sources of error responsible for these
unexpected results.
A complete report (50 pts): (5 or 6 pages in this order)

Completed data/observations tables. (10)

Speed Squared vs. Radius graph w/ line of best fit and equation.
(14)

Period vs. Height graph w/ curve of best fit and equation. (14)

On separate paper, answers to the questions using complete
sentences. (12)
Data/Observations
Part I – Centripetal Acceleration
Measured Values
Radius
Length
Period
(m)
(m)
(s)
Calculated Values
Speed
Speed2 Acceleration
(m/s)
(m2/s2)
(m/s2)
Mean
Acc:
Part II – Period of Conical Pendulum
Radius =
(Same for all trials)
Measured Values
Length (m)
Calculated Values
Period (s)
Height (m)
Mean k:
(s m-1/2)
Target Circle
Suspend the mass over this pattern and twirl it until it “hovers” over
one of the circles with its center tracing the path of the
circumference.