CONICAL PENDULUM If a small mass is suspended from a light string and the string slowly moved in a horizontal circular motion (of small radius), the mass will rotate in a larger circle at the end of the string. The string will describe a cone with the base of the cone being the imaginary flat surface about which the mass rotates. This phenomenon is called a conicalpendulum because of its conical appearance and because of the periodical motion of the mass as it traverses its circular path. The time required for one complete revolution of the mass is called the “period” of the pendulum. The velocity of the mass is called its “tangential velocity”. The period of the pendulum is thus the circumference of the circle (base of the cone) divided by the tangential velocity (a time, distance and velocity relationship, T = C / Vt , where “C” is the circumference, namely 2 times “pi” times the radius “r”). Newton’s first law of motion states that a body in motion tends to remain in motion. Not only does it remain in motion, but in the same state of motion, that is, with the same direction and velocity until acted upon by some exterior force. Any mass undergoing a change in direction is subjected to a centrifugal force, Fc. An example is the sidewise force on our bodies as we round a curve in a car. Another example is the force pulling on our hand as we swing a mass about our heads while it is attached to a string (recall David using his sling against Goliath). See the drawings on the sheet attached to this material. The first drawing depicts a string having length “ L ” being swung such that the angle theta is generated as shown. The “ r ” depicts the radius of the circle at the base of the cone. The height of the cone is depicted by the letter “ h ”. Note that the tangent of angle theta equals r / h. This relationship is used later in the derivation of an expression for the period T of the conical pendulum. The cosine of theta is h /L, which can be solved for the letter “ h ” as shown. h = L (Cosine theta). This equation is used to solve for the height “ h ” from our data later in the experiment. The second drawing depicts a condition of equilibrium as three forces act upon the mass while it is being swung. They are its weight (W), the tension (t) of the string, and the centrifugal force (Fc ) due to the circular motion of the mass. The centrifugal force acts outward on the mass, thereby holding the string at the angle theta from the vertical. The tension of the string can be expressed as two components ( tx and ty ) as shown in the diagram. The vertical component ty supports the weight of the mass and the horizontal component tx balances the centrifugal force. The net effect of these forces is to provide a state of equilibrium as the mass maintains its elevation and angle theta while in motion. The centrifugal force and the weight “w” are related by the tangent of the angle theta. (Tan theta = Fc /“w” ...

1. CONICAL PENDULUM
If a small mass is suspended from a light string and the string
slowly moved in a horizontal circular motion (of small radius),
the mass will rotate in a larger circle at the end of the string.
The string will describe a cone with the base of the cone being
the imaginary flat surface about which the mass rotates. This
phenomenon is called a conicalpendulum because of its conical
appearance and because of the periodical motion of the mass as
it traverses its circular path.
The time required for one complete revolution of the
mass is called the “period” of the pendulum. The velocity of
the mass is called its “tangential velocity”. The period of the
pendulum is thus the circumference of the circle (base of the
cone) divided by the tangential velocity (a time, distance and
velocity relationship, T = C / Vt , where “C” is the
circumference, namely 2 times “pi” times the radius “r”).
Newton’s first law of motion states that a body in motion tends
to remain in motion. Not only does it remain in motion, but in
the same state of motion, that is, with the same direction and
velocity until acted upon by some exterior force. Any mass
undergoing a change in direction is subjected to a centrifugal
force, Fc. An example is the sidewise force on our bodies as we
round a curve in a car. Another example is the force pulling on
our hand as we swing a mass about our heads while it is
attached to a string (recall David using his sling against
Goliath).

2. See the drawings on the sheet attached to this material. The
first drawing depicts a string having length “ L ” being swung
such that the angle theta is generated as shown. The “ r ”
depicts the radius of the circle at the base of the cone. The
height of the cone is depicted by the letter “ h ”. Note that the
tangent of angle theta equals r / h. This relationship is used
later in the derivation of an expression for the period T of the
conical pendulum. The cosine of theta is h /L, which can be
solved for the letter “ h ” as shown. h = L (Cosine theta).
This equation is used to solve for the height “ h ” from our data
later in the experiment.
The second drawing depicts a condition of equilibrium as three
forces act upon the mass while it is being swung. They are its
weight (W), the tension (t) of the string, and the centrifugal
force (Fc ) due to the circular motion of the mass. The
centrifugal force acts outward on the mass, thereby holding the
string at the angle theta from the vertical. The tension of the
string can be expressed as two components ( tx and ty ) as
shown in the diagram. The vertical component ty supports the
weight of the mass and the horizontal component tx balances the
centrifugal force. The net effect of these forces is to provide a
state of equilibrium as the mass maintains its elevation and
angle theta while in motion. The centrifugal force and the
weight “w” are related by the tangent of the angle theta. (Tan
theta = Fc /“w” where “w” = mg). The centrifugal force can
then be expressed: Fc = M g (Tan theta).
The general equation for centrifugal force is shown. The
centrifugal force Fc equals the product of the mass times the
tangential velocity squared divided by the radius “r”. The two
expressions for Fc are shown equated to one another.
Simplification of this equation produces an expression for the
square of the tangential velocity. This expression will be used
shortly.

3. In the second paragraph above I defined the period “T” and
provided the equation for it. If this expression is squared and
combined with the square of the tangential velocity equation
above, the result can be approximated by the expression: T
squared = 4 h . See this derivation on the accompanying
sheet. Or, if the square root is taken of both sides of the
equation, we find that T = the square root of the value “h”.
Note the rough sketch of T squared versus “h”. After taking our
data and calculating T and “h”, we will graph our results and
see how close we come to the theoretical slope of “four”. In my
experiment I obtained a slope of 3.90 which is quite close to the
theoretical value.
PROCEDURE
Tie a small “nut” on the end of a light string of about 1.1 meters
length. Sit at a table and lay your meterstick on the table,
running left to right. Place two pencils on your side of the stick
at 90 degrees to the stick, separated by a distance of 0.2 meters.
All of the data in the experiment will be taken at this separation
distance.
Measure a length of 0.2 meters from the nut on the string and
grasp it at that point between your thumb and forefinger. Don’t
cut or break the string. Toss the excess back over your shoulder
to get it out of your way. Put your elbow on the table near the
pencils and rotate the string to cause the mass (nut) to revolve
in a circle. The pencils serve as markers for the ends of the
diameter of the circle and give you a target to stay within as you
maneuver the mass in its circle. You will find it reasonably
easy to make a decent circle. Once you get fairly good at it,
take the stopwatch in your other hand and start timing as the
mass crosses one of the pencils. Stop timing as the mass
completes 20 revolutions. Take this data twice for each length
of string. Calculate the average time and divide by 20 to find
the period T and enter onto the data sheet. Repeat this process

4. for each of the lengths of string.
As longer lengths of string are used, you will find it easier to
stand by the desk as you perform the experiment. Later still,
you may find it easier to place the meterstick (with pencils) on
the floor. You may choose to stand as you do these longer
measurements. Suit yourself.
The radius for each of the circles is 0.1 meters. Recall that the
sine of theta is the radius divided by the length L. You may
thus use “inverse sine” to find the angle. With this angle on the
calculator, select “cosine” and multiply the result by the length
“ L ”. This will provide the value “ h ” (height). Repeat this
process for all of your data points. The angle itself is not
important. We just want to get a good determination of the
value of height “h”.
On your data sheet, divide the average values for the period “ T
” by 20 to get the experimental value of the period, T.
Calculate these values for each data point. Before clearing the
value of “ T ” from your calculator, square the number and log
this value of “T squared” for each data point. Log the values of
T and “T squared” to three digit accuracy: .842, 1.04, etc.
Having completed all calculations on your data sheet, graph “T
squared” on the vertical axis versus the value of “h” on the
horizontal axis. Choose good scales so as to get a good full-
page depiction of the graph. Then draw your “best line”
through the data and calculate the slope of the line. It should be
approximately 4.0.
CONCLUSIONS
You should have obtained a straight line from the graph of your
data and its slope should have been 4.0. Recall the equation
shown earlier: T squared = 4 h. If your data graphed as I’ve

5. described, it is a credit to you in several ways: (1) you
followed directions well (2) you manipulated the string well
to make good circles with the revolving mass (3) your
calculations were correct, and finally (4) you made a good
graph and calculated your slope correctly.
What this means in the bigger picture is that our assumptions
for the effect of the centrifugal forces and their calculations
were correct. This proves that the centrifugal forces were
exactly adequate to cause the mass to be displaced by the
amounts of radius and angle as described in the drawings.