Experiment No. 2: The Conical Pendulum
INTRODUCTION
Have you ever seen a swing ride at the carnival, where
people sit on seats attached to a long chain, and they are swung
around in circles? This is similar to a conical pendulum. A conical
pendulum is a mass attached to a nearly massless string that is
held at the opposite end and swung in horizontal circles (Figure 1).
Why are pendulums important to engineers? Pendulums are
used in many engineered objects, such as clocks, amusement park
rides, and even earthquake seismometers. Further, engineers
understand that their physics knowledge on the motion of
pendulums is an important step towards understanding motion,
gravity, inertia and centripetal force. Other than these, engineers
also apply their understanding of these physics concepts to
Figure 1. Conical
determine the force needed to propel an object into outer space,
pendulum
the braking power required to stop a vehicle at high speeds, and
the optimal curve of a highway ramp. Teams of engineers work on
a wide range of projects and solve problems that are important to society.
OBJECTIVES
After performing this experiment, you should be able to
• describe how a conical pendulum moves;
• apply Newton's laws of motion to the motion of the pendulum;
• determine the relationship between the pendulum’s orbital period and the angle made by
the pendulum with the vertical;
• calculate the angular frequency and linear speed of the pendulum bob, the centripetal
force acting on the bob as well as the tension on the string;
• observe the motion of the pendulum when the radius of the circular path of the bob is
varied;
• explain several uses of pendulums in modern, everyday engineering applications.
EQUIPMENT & MATERIALS NEEDED
•
•
pg. 1
computer with internet connection
transparent ruler
THEORY
The conical pendulum, shown in Figure 3, consists of a mass 𝑚 (bob) at the end of a
string of length 𝐿 that moves at a constant speed around a horizontal circle of radius 𝑅. A
component of the tension in the string provides the necessary force to provide the circular motion
of the bob. Newton’s laws, along with the concept of centripetal acceleration, can be used to
predict an expression for the tension necessary to move the bob in a horizontal circle at any given
radius 𝑅.
Two external forces act on the bob, the tension in the string 𝑭 and the force of gravity or
weight mg.
𝐹cos 𝜃
𝑭
cos 𝜃 =
h

ℎ
𝐿
therefore
𝐹 sin 𝜃
ℎ = 𝐿 cos 𝜃
𝑅
𝑚𝑔
Figure 3. Conical pendulum
Figure 2. FBD of bob
The tension 𝑭 in the string can be resolved into x and y components. The y – component balances
the weight mg of the bob
𝐹 cos 𝜃 = 𝑚𝑔
(1)
Remember that centripetal force is given by
𝐹𝑐
=
𝑚𝑣 2
𝑅
(2)
From Figure 2, the centripetal force is equal to the x − component of the tension 𝑭 in the string.
𝐹 sin 𝜃 =
𝑚𝑣2
𝑅
(3)
Dividing Equation (3) by Equation (1):
𝑚𝑣 2
𝐹 sin 𝜃
= 𝑅
𝐹 cos 𝜃
𝑚𝑔
where 𝐹 and 𝑚 cancel. The speed of the rotating bob becomes
𝑣 = √𝑅𝑔 tan 𝜃
pg. 2
(4)
From rotational motion, the linear velocity 𝑣 = 𝑅𝜔 and the angular velocity 𝜔 =
the period or the time to complete one horizontal circle.
2𝜋
𝑇
, where 𝑇 is
Substituting the expression of 𝑣 in Equation (4),
𝑅𝜔 = √𝑅𝑔 tan 𝜃
Squaring both sides of the equation
𝑅 2 𝜔2 = 𝑅𝑔 tan 𝜃
From Figure 3, tan 𝜃 =
𝑅
ℎ
𝑅 2 𝜔2 = 𝑅𝑔
𝑅
ℎ
Then R cancels out and we get
𝜔2
Substituting the expression of 𝜔 =
2𝜋
𝑇
𝑔
=
ℎ
(5)
in Equation (5),
(
2𝜋 2 𝑔
) =
𝑇
ℎ
𝑇 2 ℎ
( ) =
2𝜋
𝑔
𝑇
ℎ
=√
2𝜋
𝑔
ℎ
𝑇 = 2𝜋√
𝑔
ℎ
𝑇 = 2𝜋√
𝑔
𝑇 = 2𝜋√
𝐿 cos 𝜃
𝑔
(6)
Equation (6) provides the expression for the period T of the conical pendulum, which depends
on the length 𝐿 of the pendulum and the angle 𝜃.
pg. 3
PROCEDURE
1. Open the simulated experiment The Conical Pendulum at http://ophysics.com/f4.html.
2. Accustom yourself with the simulation before you start with your measurements. Play around
with the various slider settings of the radius R and the viewing angle 𝛼.
NOTE: Do not confuse 𝛼 with the angle 𝜃 in Equation (6). The viewing angle 𝛼 is simply the
orientation of the pendulum when viewed by the observer.
3. Press RUN, PAUSE, and RESET at various settings to familiarize with the simulation. You
can also click on the show grid & axes, show free body diagrams, and show components for
added information.
4. Mark an apparent starting point within the horizontal green circle, the path of the bob. Note
this point with a small piece of paper taped to your computer monitor.
5. Adjust the radius R to 3.0 m. You can set the animation speed to a certain value that will be
convenient for you in counting the rotations or cycles. Reset the timer to zero.
6. Record the angle 𝜃 between the string and the vertical. The angle 𝜃 emerges when the viewing
angle 𝛼 is set to 0 and the RESET button is pressed.
7. Press RUN and measure the time for 5 cycles. One cycle is when the pendulum bob
completes a circle. You can start counting the number of cycles from the starting point you
marked on your computer monitor. Press RUN and start to count.
8. Press PAUSE after 5 cycles are completed. Record the time registered in the 𝑡 slider.
9. Calculate the period 𝑇 by dividing the total time for 5 cycles by 5.
𝑇=
𝑡𝑖𝑚𝑒 𝑓𝑜𝑟 5 𝑐𝑦𝑐𝑙𝑒𝑠
5
10. Record the value of 𝑇 in Table 1 and compute the percentage error of 𝑇.
11. Repeat steps 5 − 10 using different values of R from 3.3 m to 5.5 m in Table 1.
12. For each trial, calculate 𝐿 cos 𝜃. We assume the length of the string to be 6.4 m. Since R is
given, then θ can be computed from
𝑅
sin 𝜃 =
𝐿
13. Using an ordinary graphing paper, take a graph with 𝑇 2 along the ordinate (y − axis) and
𝐿 cos 𝜃 along the abscissa (x − axis). Plot the data points and draw the line of best fit from
the origin using a transparent ruler.
14. In Table 2, calculate the angular frequency 𝜔, the speed 𝑣 of the bob, the centripetal force 𝐹𝑐 ,
and the tension 𝐹 on the string.
pg. 4
Name___________________________
Subject__________________________
Date Performed____________________
Schedule__________________________
Experiment 2: The Conical Pendulum
Table 1. Period of the Pendulum
Assumed mass of bob: m = 0.0640 kg
Assumed length of pendulum 𝐿: = 6.4 m
Trial
Radius
R
𝜃
𝐿 cos 𝜃
(m)
()
(m)
1
2
3
4
5
6
6
7
8
Expected
period
using Eq. (6)
(s)
Total
time for
5 cycles
(s)
Expt’l
period
T
(s)
Square of
expt’l period
T2
(s2)
% error
of T
(%)
3.0
3.3
3.5
3.8
4.0
4.2
4.5
5.0
5.5
Table 2. Centripetal Force and Tension in String
Height
Trial
h
(m)
1
2
3
4
5
6
6
7
8
pg. 5
Angular
frequency
2𝜋
𝜔=
𝑇
(/s)
Speed of
bob
𝑣 = 𝑅𝜔
(m/s)
Centripetal
force
𝑚𝑣 2
𝐹𝑐 =
𝑅
(N)
Tension in
string
𝑚𝑣2
𝐹 sin 𝜃 =
𝑅
(N)
Name___________________________
Subject__________________________
Date Performed____________________
Schedule__________________________
Graph of ____ vs. _____
pg. 6
Calculations and Questions:
1. What relationship exists between the square of the period T of the conical pendulum and
the angle 𝜃?
2. Calculate the slope of the line of best fit. Write your complete solution here. What is the
physical significance of the slope?
3. From the slope, calculate the acceleration due to gravity. Compare this value with the
accepted value g = 9.80 m/s2 by calculating the percentage error.
4. Does period vary with the mass of the bob? Justify your answer.
5. A conical pendulum consisting of a 0.101-kg bob attached to a 0.500-m string is moving
in a uniform circular motion in a horizontal plane of radius 0.300 m.
(NOTE: Use g = 9.80 m/s2.) Calculate (to 3 significant figures each) the:
(a) angle made by the string with the vertical.
(b) tension on the string.
(c) period.
(d) speed of the bob.
(e) centripetal force.
(f) centripetal acceleration.
pg. 7