PhysicsLAB Lab
Conical Pendulums

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Background Theory
 
When viewed from above, the path taken by a conical pendulum's bob is a horizontal circle.   
 
 
Freebody diagrams can help us understand the forces acting on the bob.
 
         conicalpendulumforces.gif (2456 bytes)
 
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) = mv2/r
 
Solving these equations simultaneously by dividing T sin(θ) by T cos(θ) yields,
 
y: T sin(θ) = mv2/r
x: T cos(θ) = mg
 
tan(θ) = v2/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
  • 2-hole stopper
  • pen case
  • 10 washer counterweight
  • 20 washer counterweight
  • 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 pen 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.
 
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 designated length three times. Alternate class mates so that everyone gets an opportunity to experience twirling the stopper. Record your class data in the tables below.
 
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 or measure θ at this junction in the lab.
average
number
of revs
string
length
(m)
frequency
(hz)
velocity
(m/s)
v2
(m/s)2
  0.50      
  0.75      
  1.00      
  1.25      
average
number
of revs
string
length
(m)
frequency
(hz)
velocity
(m/s)
v2
(m/s)2
  0.50      
  0.75      
  1.00      
  1.25      
 
 
Analysis
 
What was the mass of the stopper in kg? 

What was the mass of the 10-washers in kg? 

What was the mass of the 20-washers in kg? 

 
Conclusions
 
1a. Why is the stopper considered to be accelerating?
 

1b. What force acting on the stopper demands that the string attached to the stopper be angled downward? That is, why could you never swing the stopper in a perfect "flat" horizontal circle?
 

1c. What force acting on the string prevented the stopper from being swung either two slowly or too fast?
 

2. What happened to the frequency of the stopper when the centripetal force (supplied by the washers) increased but the radius remained unchanged.
 
3. What happened to the tangential velocity of the stopper when the centripetal force (supplied by the washers) increased but the radius remained unchanged.
 
4a. Graph your your two sets of data for v2 vs string length plotting separate lines for the 10-washer data and the 20-washer data. Record your slopes in the spaces provided.

4b. What are the units on the slope values provided in your previous answer? 

4c. Based on the units in Question 4b, what does each slope represent?
 
In the experiment, the centripetal force was provided by the horizontal component of the tension, T sin(θ). The actual radius of the stopper's circle was not the length of the string, but L sin(θ). So it appears as though the angle should be critically important to the experimental value of the string's tension but we did not make any attempt to measure its value. Let's now show you why the value of the angle was theoretially not important to our error calculations.

Using net F = ma, we can write T sin(θ) = mstopper ac = mstopper [4 π2 r f2 ]

In this equation T represents the tension in the string attached to the stopper and f represents frequency (rev/sec) at which the stopper moved along its circular path. But remember that the radius of the circular path, r, is equal to L sin(θ). Substituing in this value for the radius allows us to cancel the value of sin(θ) from both sides of the equation.

Our final equation then becomes Tension = mstopper [4 π2 L f2 ] which is the slope of your lines from the graphs of v2 vs string length
 
5a. Calculate the experiemental tension in the string attached to the stopper when 10-washers were used as the counterweight. 

5a. Calculate the experiemental tension in the string attached to the stopper when 20-washers were used as the counterweight. 

6. Sketch a freebody diagram for the suspended washers and select which of the following relationships between the tension in the string and the weight of the washers is correct. We are assuming that the washers remained relatively motionless even though they had small circular oscillations.
 
7a. Calculate the percent error for the tension in the string when 10-washers were used as the counterweight. 

7a. Calculate the percent error for the tension in the string when 20-washers were used as the counterweight. 

8a. Which set of data (10-washer vs 20-washer) had the smaller percent error?
 
8b. Why do you think one set of data had a smaller error? Do not "blame" a classmate. Think along the lines of the magnitudes of the forces involved in the experiment.
 




 
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