Background Theory
When viewed from above, the path taken by a conical pendulum's bob is circular. Freebody diagrams can help us understand the forces acting on the bob.
Vertically, the pendulum bob is in dynamic equilibrium, T cos(θ) = mg. However, the horizontal component of the tension, T sin(θ), supplies an unbalanced force towards the center of the circle. This is the source of the centripetal force that allows the bob to follow its circular trajectory.
T sin(θ) = mv^{2}/r
Solving these equations simultaneously by dividing T sin(θ) by T cos(θ) yields,
y: T sin(θ) = mv^{2}/r x: T cos(θ) = mg
tan(θ) = v^{2}/rg
This result is true for all horizontal conical pendulums for which the angle, θ, is measured from the pendulum's position of vertical equilibrium.
Materials needed:
 2 meters of string
 meter stick
 felt tip marker
 stopper
 case
 20 washers
 timer to record 40 seconds
Secure the stopper on one end of the string after passing the string down and back up through the stopper. After tying a good solid knot mark off four distances: 50 cm, 75 cm, 100 cm, and 125 cm. Measure each distance from the middle of the stopper, not from the top or bottom of the stopper. Make sure that your 4 marks are DARK and can be easily seen. Then thread the string through an empty pen case (orienting the smooth edge of the case towards the stopper). Finally tie 10 washers to the other end of the string.
Data Collection
Holding the apparatus only by the case, go outside to a balcony and practice spinning the stopper so that one of your dark marks hovers at the top of the pen case. If you twirl the stopper too rapidly, the string will feed out the top of the pen case. If you twirl too slowly, the string will slide back into the case. The perfect speed will allow the mark to hover at the top of the case while the stopper traces out a consistent level cone of maximum amplitude. REMEMBER, you may never allow the washers to touch the ground!
Once you achieve a good cone, begin counting the number of revolutions the stopper makes in 40second intervals. You need to repeat the experiment for each mark two times.
Table 1

mass of stopper 
_____ 
kg 

mass of 10 washers

_____ 
kg 

Table 2

mass of stopper 
_____ 
kg 

mass of 20 washers

_____ 
kg 

radius 
total number of revolutions in 40 seconds

average number of revolutions

0.50 m

_____
_____

_____

0.75 m

_____
_____

_____

1.00 m

_____
_____

_____

1.25 m

_____
_____

_____


radius 
total number of revolutions in 40 seconds

average number of revolutions

0.50 m

_____
_____

_____

0.75 m

_____
_____

_____

1.00 m

_____
_____

_____

1.25 m

_____
_____

_____


Data Analysis
Formulas:
f = total revolutions / total time v = 2πr/T = 2πrf since f = 1/T
Table 3 10 washer data

20 washer data

You may use the string's length as the pendulum's radius since we are unable to calculate θ at this junction in the lab.

average number of revs

string length (m)

frequency (hz)

velocity (m/s)

v^{2} (m/s)^{2} 

0.50





0.75





1.00





1.25





average number of revs

string length (m)

frequency (hz)

velocity (m/s)

v^{2} (m/s)^{2} 

0.50





0.75





1.00





1.25





Graphical Analysis
Use the EXCEL file entitled conical_2013.xls to plot a combined graph of
Velocity^{2} vs Length
for both data charts. Line #1 for the 10 washer data; line #2 for the 20 washer data. The spreadsheet will calculate the slope and yaxis intercept for each data set's regression line. Print your graph.
