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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 conical
pendulum 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” 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.
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 =
4h.
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
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 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.