Lab5-CentripetalForce.1

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Jan 9, 2024

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PHY 131/221 Centripetal Force (revised 4/11/13 J) Name ___________________ Bianca Marie delos Reyes Date: ______________ Lab Partners ___________________________________________ Alina, Gabe, Annie Introduction : As you now know, a force is required to change the velocity of an object. Velocity can change in two ways, a change of direction and/or a change in magnitude. As you have learned, if the net force on an object, and thus the acceleration, is perpendicular to the velocity at an instant in time then only the direction of the velocity will change and the speed will remain constant. If the net force, and thus the acceleration, is parallel, or anti-parallel, to the velocity then only the speed will change and not the direction. If the net force is somehow created such that it is always perpendicular to the velocity then the particle will move at constant speed and the velocity will continually change direction. The path of the object will be a circle if the force is constant in magnitude. The force, or combination of forces, that causes an object to move, or revolve, in a circle, or an arc, is called the centripetal force . The centripetal force is not a new force, but rather it is the name given to the various forces that cause the object to move in a circle. Examples are the tension in a string as you whirl a ball over your head, static friction acting on the wheels of a car as it navigates a turn on a road, and gravity that acts on the Moon to keep it in an (almost circular) orbit around Earth. This lab is meant to take you through a process of discovery; a discovery of the correct relationship between the magnitude of the centripetal force and other quantities like the mass of the revolving object, the object’s speed and the radius of the circular path. From past experience you may know an equation for this relationship, but for the purpose of this lab please pretend that you have forgotten that equation . In other words, nowhere in this lab should you use that equation to calculate anything . Only in the end, in your conclusion, can you compare what you discovered in this lab with what you thought was true from past experience. You will use the apparatus shown in the picture below. 1
The apparatus consists of a nearly frictionless vertical rod that is free to rotate. Mounted on top of this rod is another horizontal rod that serves to support the rotating mass and provide counterbalance so the whole system does not wobble. The mass that you will be forcing to travel in a circular path is the black object with the pointy bottom and the eye-bolt on top. We’ll call this mass the bob . (A bob is often the name given to a hanging, or swinging mass.) A spring is connected between the vertical rod and the bob. This spring will be providing the centripetal force. If the spring is not there then when you rotate the vertical rod, the bob will swing outward and a component of tension in the support string would be providing the centripetal force. Do not try this! The support string could break sending the bob flying off at a high speed possibly resulting in injury or death! Ok, probably not death to you, but certainly death to your lab grade. Calibration Steps: For all steps in this lab you will want the support string to be in a vertical plane as you rotate the vertical rod so that the centripetal force is provided solely by the spring. You may also need to adjust the length of the support string so that the spring connecting the bob to the vertical rod is horizontal. A vertical metal pointer is provided to help you ensure the bob’s path is indeed circular as you rotate it at a constant speed. You would first adjust the pointer so that it is a specified distance from the center of the vertical rod. Then you would disconnect the spring from the bob and adjust the horizontal rod so that the bob hangs directly over the pointer. Before you do any rotating you must be sure the horizontal rod and the counterweight are secure. It is a good idea to wear safety goggles in the event that something comes loose . Once everything is secure you may reconnect the spring. Now that you have established a radius for the bob’s circular path and reconnected the spring you will notice that spring pulls the bob in toward the vertical rod. When you rotate the vertical rod, the bob will move outward. If the angular velocity is just right the bob will pass directly over the pointer on each revolution. The spring is now stretched by a certain amount and provides the centripetal force that keeps the bob moving in the circular path. You can directly measure the force (the centripetal force) that the spring will exert on the bob when the bob is revolving in its circular path. To do this, stop the rotation and, without disconnecting the spring, loop the string attached to the bob, opposite the spring, over the pulley and add hanging weights until the bob is directly over the pointer. The spring is now stretched the same amount as when the bob was rotating. The weight of the hanging weights will be a direct measure of the force the spring is providing when the bob is revolving in its circular path. 2
Setting the radius of the circular path is easy, but how do you measure the bob’s speed? The easiest way to do this is to measure the amount of time that it takes for the bob to revolve 20 to 30 times, divide this time by the number of revolutions to get the time for one revolution, the period, T, and then divide the circle’s circumference by T to get the speed. Part I: The First Trial: Remove the bob from the supporting string and from the spring and measure its mass. Record this in the Data section. Reattach the bob. Adjust the eye-bolt, connected to the spring, on the vertical rod by loosening the wing nut so that the eye-bolt is centered on the vertical rod. Adjust the vertical pointer to a position midway between its limits and secure it in place. Adjust the horizontal rod so that the bob hangs freely above the pointer without the spring attached. Reconnect the spring and directly measure the spring force (as described above) that will exist when the bob is revolving in a circle with this radius. Record this data in the Data section for Part I. Decide on the number of revolutions that you will be timing and record this number in the Data section. One lab partner should practice rotating the vertical rod so that the bob moves at a constant speed and passes over the pointer every time. Once you are satisfied that this can be done precisely do three trials to determine the total time for the number of revolutions that you decided on earlier. Record these times in Table 1. Using these times calculate the average time per revolution and then the speed of the bob. Record these numbers in the appropriate spots below Table 1. What you have now done is established a starting point that we can use to compare how the centripetal force changes as we change variables such as the mass of the bob, the speed of the bob, and the radius of the circular path. Part II: Dependence of Centripetal Force on Speed: Do not change the position of the pointer, or the mass of the bob. We only want to change the speed of the bob as it revolves in the same circular path as in Part I. How do we change the speed? Try spinning the vertical rod slower than you had in Part I. Does the bob revolve in the same circular path? What should you change so that the bob does revolve in the same circular path but at a slower speed? Discuss this with you lab partners then with your instructor. Make the necessary change and do three trials as in Part I where you measure the time for a certain number of revolutions, find the average time per revolution, find the bob’s speed, and measure the centripetal force directly. Enter this data in Table 2 and in the area below Table 2. (Hint: The change(s) you make may involve a significant change in how “things” are connected, but you should not change the radius of the circular path and thus you should not move the horizontal rod or vertical pointer. Be creative.) Do the above procedures again, but try to give the bob a speed faster than you had in Part I. What change is necessary (without changing the bob’s mass, or the radius of the circle) to keep the bob traveling in the same circular path but with a higher speed? Enter this data in Table 3 and the space below Table 3. Since our goal is to determine how the centripetal force depends on speed we’d like to experiment with several different speeds to be sure we find the correct relationship. So let’s do one more speed. You can make a choice here, do you want to consider an even slower speed or an even faster speed? Make the necessary adjustments (again without changing the radius or mass) and enter your data in Table 4. Now you have four different speeds and the corresponding directly measured centripetal forces. Part III: Dependence of Centripetal Force on Radius of path: 3
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Move the vertical pointer out to its maximum distance and lock it in place. You will now need to loosen the horizontal rod and disconnect the spring from the bob to be sure it is hanging freely above the pointer. Secure the horizontal rod after making the appropriate adjustment. Your goal now is to make the bob travel in a larger circular path, but with the same speed as in Part I. This is the most difficult part of this lab so read carefully. You know speed is determined by dividing circumference by the period. So the speed in Part I equals , where is the radius of the circular path in Part I and is the time per revolution in Part I. This speed should be equal to the speed for this new larger radius, in other words, 2 π r 1 T 1 = 2 π r 2 T 2 , where is the new radius that you just set, and is the time per revolution that you will need to have so that the speeds from Part I and this part are equal. Solving for , we have . Calculate and enter it into the data section for Part III. As before, you will want to time more than one revolution so let’s agree that you will do 20 revolutions. What adjustments do you have to make (without changing the mass of the bob, or the radius of the path) to make the time for 20 revolutions equal to ? Make the necessary adjustment and rotate the mass for 20 revolutions to see if the adjustment you made is correct. You will need to fine tune your adjustment to get the total time for 20 revolutions to be as close to as possible. Once you are satisfied that the bob revolves around a circular path of radius in time , as calculated above, make a direct measurement of the centripetal force and record this in the data section for Part III. Now change the radius to the smallest radius allowed by the pointer. Call this radius . Calculate the time per revolution, , that you will need to ensure the speed of the bob is equal to what it was in Part I. Again, make fine adjustments so that the total time for 20 revolutions is equal to . Measure and record the centripetal force. Finally, to ensure we can find the correct relationship between centripetal force and radius, choose another radius that isn’t the same as the radius in Part I or the two extreme radii that you just tested. Call this radius r 4 and determine the time T 4 . Make the necessary adjustments so that when you rotate the bob the measured time for 20 revolutions is correct and then directly measure the corresponding centripetal force. Proceed to the Questions section. 4
Data Mass of bob with eye- bolt 451 Total number of revolutions 12 Data For Part I: Trail # Total Time (s) Time per Revolution (s) 1 17.72 s 1.4 77 s 2 17.81 s 1.4 8 4 s 3 17.47 s 1.45 6 s Table 1 Average Time per Revolution, 1.472 s Radius of circular path, 19.8cm Speed of bob 0.846m/s Mass of Hanger and Masses 0. 190 k g Direct measurement of centripetal Force 1.862 N Show calculation for the speed of the bob: (2pi*radius)/ave time per rev [2pi(0.198m)/1.472s] 0.846m/s 5
Data For Part II: Slower Speed Same Radius as Part I Trail # Total Time (s) Time per Revolution (s) 1 18.56 s 1. 547 s 2 1 9.62 s 1.635 s 3 1 9.50 s 1. 625 s Table 2 Average Time per Revolution 1 .602 s Speed of bob 0. 777 m/s Mass of Hanger and Masses 0. 159 k g Direct measurement of centripetal Force 1.558N Show calculation for the speed of the bob: (2pi*radius)/ave time per rev [2pi(0. 198 m)/1. 602 s] 0. 777 m 6
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Greater Speed Same Radius as Part I Trail # Total Time (s) Time per Revolution (s) 1 1 4.63 s 1.2 19 s 2 15.75 s 1.313 s 3 15.50 s 1.292 s Table 3 Average Time per Revolution 1.275s Speed of bob 0.9 76 m/s Mass of Hanger and Masses 0.240kg Direct measurement of centripetal Force 2.352 N Show calculation for the speed of the bob: (2pi*radius)/ave time per rev [2pi(0.198m)/1. 275 s] 0. 976 m /s 7
Even Greater or Even Slower Speed Same Radius as Part I Trail # Total Time (s) Time per Revolution (s) 1 15.06s 1.255s 2 16.12 s 1.343s 3 15.81s 1.318s Table 4 Average Time per Revolution 1.305s Speed of bob 0.953m/s Mass of Hanger and Masses 0.240kg Direct measurement of centripetal Force 2.352N Show calculation for the speed of the bob: (2pi*radius)/ave time per rev [2pi(0.198m)/1. 305 s] 0.9 53 m/s 8
Data For Part III: Larger Radius Same Speed as Part I Radius of circular path, 20. 9 cm Desired time per revolution for larger radius, 1.5 22 s Desired time for 20 revolutions, 30.44s Direct Measurement of Centripetal Force for 2.45N Smaller Radius Same Speed as Part I Radius of circular path, 15.8cm Desired time per revolution for larger radius, 1.173s Desired time for 20 revolutions, 23.46s Direct Measurement of Centripetal Force for 1.568 N Fourth Radius Same Speed as Part I Radius of circular path, r 4 18.4 cm Desired time per revolution for larger radius, T 4 1.366s Desired time for 20 revolutions, 20 T 4 27.32s Direct Measurement of Centripetal Force for r 4 2.352N Show calculations for Part III below: ( ( 2pi*r)/t ) = ((2pi*R)/T ) ((2pi*0.198)/1.47 ) = ((2pi*0.205)/T ) T=1.522 ((2pi*r)/t ) = ((2pi*R)/T ) ((2pi*0.198)/1.47 ) = ((2pi*0. 158 )/T ) T=1. 173 ((2pi*r)/t ) = ((2pi*R)/T ) 9
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((2pi*0.198)/1.47 ) = ((2pi*0.184)/T ) T=1.366 10
Questions 1. What change did you make to give the bob a slower speed than in Part I without changing the mass of the bob, or radius of the circular path? Explain how you did this. 2. 3. 4. 5. If we increase the time period then we can observe that the speed is changing without changing the mass or radius. 6. What change did you make to give the bob a greater speed than in Part I without changing the mass of the bob, or radius of the circular path? Explain how you did this. If we decrease the time period, then we can observe the speed increasing 7. Make a graph of Measured Centripetal Force as function of speed (with constant radius and mass) for the four different speeds that you in Part I and Part II. (Be sure to properly label the graph, and its axes.) What kind of function fits this data best? Is it linear, quadratic, or exponential? What does this tell you about how centripetal force depends on speed? Explain. 8. 11
The function that fits t he data the best is linear. If you increase the speed then centripetal force will increase time . If you increase speed by two, then centripetal force will increase four times 9. 10. 11. 12. 13. 12
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14. 15. Make a graph of Measured Centripetal Force as a function of radius (with constant speed and mass) for the four different radii you had in Part I and Part III. (Be sure to properly label the graph, and its axes.) What kind of function fits this data best? Is it linear, quadratic, exponential, or reciprocal relationship? Or is it something else? What does this tell you about how centripetal force depends on radius? Explain. 16. 17. Radius and force are inversely proportional. 18. What were the most significant sources of error in this lab and what did you do to minimize their affect? There could be random and systematic errors that played into significance sources of erro rs in this lab. An example of systematic error that could have happened is when observers does not read the measurement correctly, when the instrument that is given to us is flawed which causes consistent inaccurate in reading, etc. Random errors that could have also played a role is when the observers judgments leads to a random inaccuracies. There are ways to minimize their effect, an example would be double checking all the measurement for accuracy, double checking formulas, make sure that the instruments that are going to be used had the highest precision, etc. 13