Technical Report Brian Hartman 3.0

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Apr 3, 2024

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Experiment 7 Centripetal Force Brian Hartman Bh018420@ohio.edu Tuesday 2:30 – 4:20 Zach Bernens 10/3/2023 Centripetal Force Abstract: In this experiment we used the laws of tension to calculate the tension of a pendulum with the PASCO 750 Interface, then calculate the tension yourself to see how well they match up and calculate precent error. In the second part of the experiment, we measure the tension of the centripetal force on a mass, then calculate the tension to see how well the numbers compare and calculate the precent error. If the numbers recorded are correct, they should match up to the tensions that the PASCO 750 Interface recorded. The results for the first experiment were the average tension was 0.4175N and the average velocity was 0.4325m/s. For the second experiment the average force was 0.44N and the average period was 1.496. Theory: Newton’s Second law of motion states that a body remains at rest or in uniform motion in a straight line unless acted on by a non-zero external force. (1) ࠵?࠵? = ࠵? !" !# = ࠵? $%# %&#%’$() A force with a constant magnitude applied consistently at right angles to the velocity of a body for a time it was cause the body to move in a circle making a radius. The acceleration can be found by (2) ࠵? * = " ! ’ Centripetal force is found by (3) ࠵? * = ࠵?࠵? → +" ! ’ (−࠵?) When there are two external forces on mass it’s the force of gravity and tension. (4) ࠵?࠵? = ࠵?࠵? + ࠵? , At any point if motion, tension, or variables act at a right angle it creates centripetal force upward. (5) ࠵? * = ࠵?࠵? - ࠵? = −࠵?࠵? + ࠵? , This is the equation to find how tension is on the string while it is being pulled in. Although if this is going in a circle the mass must have forces acting on it to keep it positive. (6) ࠵? , = ࠵?࠵? - ࠵? + ࠵?࠵?

For the second part of the experiment The tension is providing the centripetal force, when linear and angular velocity are constant, they can be written in terms of angular speed w and period T. (7) ࠵? = ࠵?࠵? = 2࠵?࠵? ࠵? So centripetal force is (8) ࠵? * = ࠵? , = 4࠵? -+ ࠵? - ࠵? Experimental Details: Equipment: Simple pendulum, Rotating table and mass for circular motion, PASCO 750 Interface, Computer, photogate Figure 1: Rotating Table with pendulum For part A of the experiment, the first thing to do it set up the pendulum onto the force sensor but before hooking it on measure the length of the pendulum string, this will be your radius in equation 6, make sure you tare the sensor as well. The next thing is to open PASCO on the computer and move the photo gate so that it is in the middle of the table, once you are ready to record, make sure the pendulum is not moving and record for 10 seconds to find the weight of the pendulum, record that information down. After that then lift the pendulum up about 15 ° and hit record then let it go. Let it swing back and forth through the photogate for 30 seconds. Then highlight only the bottom part of the graph where the tension is the highest, also record the velocity. Do not let the pendulum hit the photogate. Do this 4 times then use the formula 6 to calculate the tension and see how well it matches what you recorded. After finding the tension calculate the error percentage and standard deviation for tension and velocity. For part B of the experiment, you are to set up a car on the track and have the string connected to the force sensor. Measure the length from the middle of the track to the middle of the car, that is your radius. Then measure the mass of the cart. Add the various weight to the mass, first will be 50g, next is 100g, last is 150g. Move the photo gate back to the outside where when the table spins the mass goes through the photogate. Make sure the photogate is tared. Set the motor onto the side of the table and allow the table to spin. Record and let the table spin for 20 revolutions and write down the mean force, the period, and the weight of all of it. Do that for the 3 different weights then calculate the centripetal force and compare to what you recorded. Then find the error precent and the standard deviation for the force. Some of the systematic errors that could occur is the photo gate not being calibrated, the force sensor being tared, and the length of time that you test. These all can give you bad data.

When finding tension, we also want to calculate standard deviation by (9) ࠵? = 5∑ (& " /0) ! 2 We also want to calculate the percentage difference by (10) 100 ࠵? |࠵? 3 − ࠵? - | ࠵? 3 + ࠵? - 2 Then we calculate error percentage. (11) %error = 4& #"$%& /& %’() 4 & #"$%& 100% Result and Discussion: Figure 2: Weight of pendulum Pendulum String / Radius = 37cm or .37 ± .1m When taring the force sensor, it was not going to zero it was .03. Cylinder mass = 51.3g or 0.0513 ± .1kg Weight of Cylinder = 0.42g

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Table 1: Tension Tests Run Tension(N) Velocity(m/s) Calculated Tension(N) % Difference (%) 1 0.43 ± .0 0.48 ±.01 0.53 20.83 2 0.30 ± .0 0.47 ±.01 0.53 55.42 3 0.48 ± .0 0.37 ±.01 0.52 9.09 4 0.46 ± .0 0.41 ±.01 0.52 14.14 To calculate tension you need the mass, velocity, and radius. When we ran the tests, we found that the mass from weighing the pendulum, the velocity came from the force sensor, and the radius came from the length of the pendulum string. Putting all those together with gravity allows you to find tension. Figure 3: Tension and Velocity Car on a track Mass of Cart = 17.8 ±.1 g Distance/Radius = 23.2cm Table 2: Centripetal Force Run Force(N) Period Mass(g) Calculated Force(N) % Difference (%) 1 0.58 ±.01 1.31 117.8 ±.1 0.62 6.6 2 0.40 ±.01 1.14 70 ±.1 0.49 20.2 3 0.34 ±.01 2.04 167.8 ±.1 0.36 5.71 To calculate centripetal force you need mass, radius, and the period. The mass was calculated by weighing the cart and the mass added to it, the radius was found from the length of the track to the cart, and the period was the time it took to make a full loop. Putting all those together with 4 ࠵? - and you can find centripetal force.

Figure 4: Force and Period Discussion: We did precent difference instead of precent error because we needed to get the plus and minus of the error not the percentage number that we were off by. I would say the pendulum was not a great example of centripetal force because it was not going in a full circle it was just rocking back and forth so we had to do calculation to make it seem like it was going in a full circle. The sources of the errors are most likely from the photogate and the force sensor, our force sensor was not zeroing out like it should have so it might have messed with our numbers. The size of the errors can be big or small depending on what numbers the computer outputs. These errors affect our answers because we were getting numbers that won’t make sense due to gravity, we should have had bigger numbers but for some reason we didn’t. Conclusion/Result: In our first experiment we calculated tension with a force sensor and then calculated tension with the other values and compare the two. Our first test didn’t seem to go so well our numbers didn’t really match and the numbers we got we should never have gotten. Our force sensor was not taring correctly so I believe that’s why our numbers went off. Another reason why our numbers might have been off is because of the photo gate might have read the measurements wrong as well. With the second part of the experiment our numbers were a little better our error percentage was not as far off as the last experiment. In this lab we learned how to calculate the tension on a string and we also found how to find centripetal force on a mass. The results for the first experiment were the average tension was 0.4175N and the average velocity was 0.4325m/s, the precent difference was 24.87% . For the second experiment the average force was 0.44N and the average period was 1.496 and the precent difference was 10.83%.

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