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Tension and Angles: Analyzing Force Relationships

By: Pooya Gozlo, Taryn Murray, Nate DeLamarter

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Experimental Objective

To find and test the validity of a derived equation for the tension on a string, caused by some weight, while knowing only the angle values and the weight on the string.

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Experimental Setup

- We are trying to derive an equation similar to the one found in the textbook, question 63: . However, this equation only works when the weight is halfway between two ends,making it symmetrical. We are trying to find the equation that applies to situations where the weight is not halfway. In other words the general equation for tension when a weight is on a string.

Equipment Used:

- 2 force sensors

- 3 table clamps

- 4 rods

- 5 rod clamps

- 1 KG Weight with Hook

- ~45 cm String

- Angle Measure

- Pasco Capstone Software

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Equipment

Angle Measure

1 kg weight

Force Sensor

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Equipment (Continued)

String

Support Rod

Table Clamp

Rod Clamp

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Experiment Execution

Set up equipment according to the diagram.
Connect force sensors to rods and computer.
Tie string to the force sensors via double knot.
Extend two supports outward to get rid of all the string slack.
Zero the force Sensor.
Place weight on string, ~3cm in from the left force sensor.
Wait 125 seconds, record force readings from both sensors
Record angles from both sides.
Using a marker or ticks on the string, place weight ≈ 2.5 centimeters in towards the center.
Repeat steps 5-8 for 10 trials.

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Data we Collected and How it Was Used

Angle measurements for calculating the left and right tensions.
Measured tensions for comparing our calculated tension.

Measured Tension Left

Measured Tension Right

Collected:

Left Angle

Right Angle

Used:

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Vector Math From Professor Sean

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Equations Used (Our Hypothetical Equation)

Fg

TR

TL

θL

θR

TxR

TxL

TyR

TyL

To find some equation for the tensions across the string, given Fg, θR, and θL:

1.) Find an expression for the y forces:

2.) Find an expression for x forces: (Because the weight does not move. And there are no other forces)

Free Body Diagram

(+)

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Equations Used (Continued)

3.) Substitute the relationship found, into the equation for y:

With symmetry we have:

General Equations for tension on a string

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Data Tables

This is only to check the accuracy and precision of our experiment.

Measured Experiment Results

1

Totals:

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Graphs

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Final Results

-Standard deviations were decent in their own right

-Percentage error was very large

Average Total Calculated Tension: 48.1 N

Experimental Standard Deviation (Total Calculated Tension): 2.31 N

Experimental Standard Deviation of the Mean (Total Calculated Tension): 0.730 N

Average Total Measured Tension: 20.1 N

Experimental Standard Deviation (Total Measured Tension): 2.17 N

Experimental Standard Deviation of the Mean (Total Measured Tension): 0.732 N

Percent Error (Total Calculated Tension Compared to Total Measured Tension): 139%

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Summary

-Data does not support the hypothetical equation

-Hypothetical equation is most likely still accurate

-Poor experimentation led to minimal support

-Minimal support came in the form of the measured tension not being large enough.

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What Went Wrong

All possible sources of error:

-Potential inaccuracies with the force sensors

-Differences in tension readings after zeroing the force sensors

-Inconsistent angle measurements

-Tared the force sensor and didn’t measure the inherent tension of the string (overall poor force measurement)

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Eliminating Certain Errors

Inaccuracies with the force sensor

—> Disproven:

Differences in tension readings after zeroing the force sensors —> Disproven:

After Zeroing

At Max Tension

Tension Difference: 22.45 N

Tension Difference: 22.51N

Tension Reading: 9.78 N

After we saw how off our experiment was we went back into the lab to run some additional experiments to test some of our hypothesis for why we came to such an inaccurate conclusion.

The first possible error was disproved by simply putting the weight on the force sensor and seeing that it was almost exactly 9.8 N, which is accurate.

The second possible error was disproved by a small experiment in which we zeroed one force sensor and left another unchanged. We then pulled the string tighter and tighter

to see if there would be any discrepancy between the readings. As you can see from the before and after, our original difference in Newtons was 22.45 and our final difference was 22.51.

Showing that there is no difference in the amount the tension will increase whether or not it was zeroed.

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Why our Angle Measures are Inconsistent

-Imprecise angle measurer.

-Small scale.

-Measured angles after taking the force readings.

Our angle measures are very inconsistent because of a few reasons. One of which being the fact that the measuring device was not very precise, especially for the very small angles we were working with.

As you can see from the image the string of the angle measurer was almost the width of the distance between two angles. This made it extremely hard to tell the angle, and we were probably only accurate to +- 1 degree.

This coupled with the fact that we were only increasing by about 1 degree per measurement, caused very inconsistent angle measures. On top of this we did not originally think we had to measure the angle

We thought it could have been calculated from the tension readings. However, since those were significantly off we had to go back after all of the tension readings had taken place, and approximate where exactly

The weight was on the string. Although we managed to find the same spots on the string to roughly +- 3mm, that is still a significant difference at this small scale.

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How the Angle Errors Affected the Data

Difference of ~3N

1 degree change

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Mystery Tension Decrease

- As time went on, the tension force on the left and the right went down significantly.

- Staring at roughly 12.5 N on the left and 11.5 N on the right, we decreased to around 6.75 N on the left and 5.75 N on the right.

-We took our measurements after 2 minutes, this could have contributed to the very low tension measurements.

≈ 5.75

≈ 6.75

≈ 12.5

≈ 11.5

Note: There is a disruption at 600 seconds but we can see that the overall graph still looks fairly consistent

Left and Right Tensions Over Time

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We shouldn’t have tared the sensor…

- Strings have inherent tension when taut.

- We got rid of that inherent tension after taring the force sensors.

-From memory we found that the tension on the taut string about 30 N.

The difference between 39.8 N and 7.55 N is 32.25 N.

T1

T2

Since a = 0

T1 = T2

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We shouldn’t have tared the sensor… (continued)

We see the difference between every single measured and calculated force reading at their given angles, and their averages:

Avg = 31.6

Avg = 31.8

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Conclusion

We think our hypothesis is actually correct:

-If we had not tared the sensor.

-If we had not waited so long to take our measurements.

-The result most likely would have supported our hypothesis.

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What We Would do Differently

-Heavier weight to increase the angle.

-Wait only about 10 seconds to take the force measurement to decrease affect from the mystery tension decrease.

-No taring the force sensors before putting the weight on the string.

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THANK YOU

Any Questions?