Lab
Conical Pendulums
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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 40-second 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 = 2
π
r/T = 2
π
rf 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 y-axis intercept for each data set's regression line. Print your graph.
What was the filename of your graph?
What is the mass of your stopper in kg?
What is the mass of your 10 washers in kg?
What is the mass of your 20 washers in kg?
Conclusions
1. On your graph's printout, color the data points for the 10-washer line. Using the same color, write in the equation of the 10-washers line along the graph and also state its equation here.
On your graph's printout, use a second color to mark the data points for the 20-washer line and write its equation along the graph. Also state its equation here.
2. In the margin on your graph include a dimensional argument to determine the units you should use when discussing the slope of each line. Based on your analysis, what is the correct unit for each line's slope?
3. Why is the stopper considered to be accelerating?
Answer the next four questions by analyzing the patterns in your data tables."
4 (a). What happened to the frequency of the stopper when the radius increased but the centripetal force (supplied by the washers) remained unchanged.
increased
remained the same
decreased
4 (b). What happened to the frequency of the stopper when the centripetal force (supplied by the washers) increased but the radius remained unchanged.
increased
remained the same
decreased
5 (a). What happened to the tangential velocity of the stopper when the radius increased but the centripetal force (supplied by the washers) remained unchanged.
increased
remained the same
decreased
5 (b). What happened to the tangential velocity of the stopper when the centripetal force (supplied by the washers) increased but the radius remained unchanged.
increased
remained the same
decreased
Now we will examine why we did not need try to measure the experimental angle made by the string to your case.
Notice in the
left freebody diagram
that the component T sin(θ) is the unbalanced force acting towards the center, or the centripetal force. Notice in the
right length diagram
, that the radius of the conical pendulum is equal to L sin(θ) where L is the length of the string.
6. Based on your data from the trial using a 0.75-meter string length with 20 washers, what was the experimental tension in the string attached to the stopper? Show your calculations on your graph's printout.
7. Sketch a freebody diagram for the 20 washers. [Although the washers were also behaving as a conical pendulum, their circular radius should have been minimal. You may use a freebody diagram for a stationary mass hanging from the end of a string to answer this question.] After establishing the equation which relates the tension in the string and the weight of the washers, calculate the experimental mass of the 20-washers (in kg) using the experimental tension found in question #7. Show your calculations on your graph's printout.
8. Calculate a percent error for the mass of the 20 washers found in question #8 with the actual value measured at the beginning of the experiment. Show your calculations on your graph's printout.
9(a). Describe a procedural source of error in your experiment.
9(b). Provide a suggestion on how to reduce this error's negative impact on the experiment.
After you submit your results online, your final lab report will include a copy of Tables 1, 2, and 3 followed by a well documented (see conclusions #1, 2, 6, 7, 8) printout of your EXCEL graph of Velocity
^{2}
vs Length.
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