Lab Members: Carol Ge, Andy Chen, Aayush Chopra, Adithya Kalyan
Research Question: How does speed of an object in uniform circular motion affect its acceleration ?
Variables & Controls of the Experiment
Variables
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Controls
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Methods
To Control the Variables: First, mark the end of the radius on the string with a marker so the person doing the spinning can keep the radius consistent -- controlling a potential confounding variable. Next, calibrate the force sensor connected to Logger Pro from zero by holding the force sensor with the hook facing upward, ensuring that measurements of force, which are used to calculate acceleration, are as precise as possible. |
Then, start a metronome with a consistent beat that the bottle cap should be spun to, with one full rotation per beat. Have one person spin the bottle cap on one end of the string with a stick with the string going through using one hand, and hold the force sensor that is hooked securely to the other end with their other hand. The second person, who is measuring the time of the total rotation for one metronome bpm, should allow them to reach a horizontal rotation that matches the metronome before they starts the timer. This will keep their speed consistent throughout the spins. The person that starts the timer will be standing slightly away from the person doing the spinning to maintain precision. A third person, along with the first two, should watch the bottle cap spinning closely to count the number of rotations it makes before the timer alarm sounds. The amount of time on the timer should be the same during every spin for consistency, but the metronome bpm should be different to control the independent variable.
At least three people are required to carry out this procedure as it will be difficult for a single person to swing the bottle cap, count the rotations, and start/end a timer at the same time while maintaining precision and accuracy.
At least three people are required to carry out this procedure as it will be difficult for a single person to swing the bottle cap, count the rotations, and start/end a timer at the same time while maintaining precision and accuracy.
To Collect the Data:
Lay out the string used to attach the bottle cap to the force sensor on a flat surface and pull it taut. Measure the radius of the circle created by the circular motion of the spinning object with a yard stick starting from the tip of the string attached to the bottle cap. As stated in the section, "To Control the Variables", have all three (or more) people carrying out the procedure count the rotations made by the bottle cap in a specific amount of time. Now, we have the information to calculate the independent variable -- the speed of the bottle cap being swung at each different bpm.
During each group of rotations made according to a certain bpm, the force sensor has recorded the force of tension, which is displayed on Logger Pro. In addition, measure the mass of the bottle cap using an electronic balance. With this information, we can calculate the dependent variable -- the acceleration of the bottle cap .
Procedure
- Measure mass of bottle cap with electronic balance (0.01 kg)
- Mark a radius (0.5 m) for the spin with a marker
- Attach force sensor to one end of string and bottle cap to the other
- Start the metronome (at 90, 100, 110, 120 bpm)
- Pull the stick so it is exactly below the dot made on the string
- Start swinging the bottle cap with the string below the stick vertical and the string above horizontal
- Start timer once the rotations are determined to be horizontal and at a constant speed
- Measure number of rotations by counting
- Stop the timer after counting at least 25 rotations
- Record the bpm, time of rotations, number of rotations, and force of tension
- Total trials conducted: 4
Raw Data - Table
Mass of Bottle Cap: 0.01 kg
Radius of Rotation: 0.5 m The total time was recorded with a stopwatch, rotations were counted by the lab members and I, and the force of tension was recorded by a force sensor and reported by the Logger Pro application. Uncertainty
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Processed Raw Data - Table
Rotations per second = rotations/ total time (s)
Speed = rotations per second * (2π * radius) Acceleration = force of tension/ mass of bottle cap *All of these calculations were made with Google Sheets by typing out the equation used to calculate them and dragging the equation down to automatically be applied to other rows. |
Processed Raw Data - Graphical Representation
The image pictured above is a graph generated by the Logger Pro application. The independent variable (x-value) is Speed (m/s) and the dependent variable (y-value) is Acceleration (m/s^2).
A nonlinear, simple quadratic model (y = Ax^2) was the best fit for this graph as it made better predictions. We did not choose a proportional fit because the graph would only align with clusters of data -- not all of them simultaneously. The A value of this graph was 3.227, and the equation itself was y = 3.227x^2.
A nonlinear, simple quadratic model (y = Ax^2) was the best fit for this graph as it made better predictions. We did not choose a proportional fit because the graph would only align with clusters of data -- not all of them simultaneously. The A value of this graph was 3.227, and the equation itself was y = 3.227x^2.
Conclusion
The purpose of this lab was to determine the relationship between the speed of an object in uniform circular motion and its acceleration, which we concluded is a simple quadratic relationship due to the simple quadratic model that best fits our data. As the speed of an object in uniform circular motion increases, its acceleration also increases. In addition, we have determined the relationship between the measurements of our raw data: the force acting upon the object, the mass of the object, the number of rotations per second the object is being swung in, and the radius of the circular motion. Because the mass of the object and the radius of the circular motion should stay constant, the force acting upon the object increases as the number of rotations per second the object is being swung in increases.
The acceleration that is occurring as an object is swung with a uniform circular motion is called centripetal acceleration -- acceleration that points towards the center of the rotation. The string that contains a force of tension on the object being swung keeps it moving in a circle, undergoing centripetal acceleration. Moreover, the object itself is not speeding up or slowing down -- it is just being pulled towards the center, which indicates a force of tension acting upon it. Subsequently, it has a non-zero acceleration.
According to Newton's First Law of Inertia, "an object will continue to move at a constant velocity unless it feels an unbalanced push or pull". Thus, an object traveling in a circle will remain traveling in a circle. However, if, while spinning the object, we were to cut the string or the string broke at any point in the circular spin, the force of tension has stopped acting upon the object. Then, it will travel in a straight path in the specific direction it was traveling right at the moment the force of tension stopped acting upon the object. The direction of the motion will be perpendicular to the radius, and the velocity of the object will be constant. Moreover, the direction of motion can also be observed as tangential to the circle at the time, which proves that tangential acceleration -- the acceleration of an object moving tangent to its original path -- can be calculated. |
In real life, one may see centripetal acceleration in action in a centrifuge or a merry go round. In a centrifuge, the objects inside are being spun rapidly at a constant speed, but the force of the container they are in keeps them from flying out and traveling in a tangential motion. In a merry-go-round, the force of friction allows a person to stay standing on the merry-go-round and not fly off of it. If a person were to jump up or let go of the handle bars, they would fly off in a direction tangential to the centripetal motion. Essentially, any object undergoing uniform circular motion will keep traveling in a circle unless the force keeping it traveling in a circle stops acting upon it. Moreover, if this speed were to change, the acceleration would also change in the direction that the speed has changed.
Evaluating Procedures
Weaknesses and Limitations
This lab had multiple weaknesses. First of all, the swinging of the object was extremely inconsistent. There was often a different radius for each rotation, and the speed of the swings was often faster or slower than the metronome beat. In addition, it was not always swung in a circle that is parallel to the ground, so the net vertical force was not 0 N. Another limitation of this lab was that not enough trials were conducted. We had only conducted a total of 4 trials due to time constraints and difficulty with setting up the lab, so with more time, more trials may be conducted and increase confidence in our results. The force sensor was improperly calibrated for the first trial, the one at 90 bpm, so the measurement of force was incorrect and had to be adjusted post-experiment. Following that instance, the force sensor may have still been improperly calibrated, so the force measurement was still not as accurate as it could be. Lastly, we had difficulty counting the amount of rotations the object made during the specific time frame due to the fast rotation speed.
Uncertainty
Improvement
To improve this lab, I would have conducted repeated trials to double check the speed of the object. Moreover, instead of using dots to mark the radius, I would use tape so the stick doesn't shift on the string, which subsequently decreases the uncertainty of the radius. I would also use a robot instead of an object to do the swinging and counting the rotations in this lab as a robot likely has a more consistent swing than a human.
In addition, I would use a more precise motion detector to determine the exact position and time of the cart to create a more exact calculation of acceleration. Lastly, I would increase my confidence in my data by collecting more data points, around 7, of 70, 80, 90, 100, 110, 120, 130 bpm. This will also help to create a greater range of data. Moreover, I would conduct repeated trials of rotations at each bpm as there are too many factors creating uncertainty.
This lab had multiple weaknesses. First of all, the swinging of the object was extremely inconsistent. There was often a different radius for each rotation, and the speed of the swings was often faster or slower than the metronome beat. In addition, it was not always swung in a circle that is parallel to the ground, so the net vertical force was not 0 N. Another limitation of this lab was that not enough trials were conducted. We had only conducted a total of 4 trials due to time constraints and difficulty with setting up the lab, so with more time, more trials may be conducted and increase confidence in our results. The force sensor was improperly calibrated for the first trial, the one at 90 bpm, so the measurement of force was incorrect and had to be adjusted post-experiment. Following that instance, the force sensor may have still been improperly calibrated, so the force measurement was still not as accurate as it could be. Lastly, we had difficulty counting the amount of rotations the object made during the specific time frame due to the fast rotation speed.
Uncertainty
- The radius of the circular motion, which was used to calculate the speed of the object, may have differed during each rotation because it was difficult to keep track of where the stick that was being held by the person doing the swinging should be on the string. The dots made to indicate the radius were too small and faint.
- The total time while the object was spinning could have been more accurate. Only one person was recording the time as the other person swung the bottle cap, so the time that the person started spinning with proper uniform circular motion and stopped spinning may be different from the starting and stopping time collected by the other person.
Improvement
To improve this lab, I would have conducted repeated trials to double check the speed of the object. Moreover, instead of using dots to mark the radius, I would use tape so the stick doesn't shift on the string, which subsequently decreases the uncertainty of the radius. I would also use a robot instead of an object to do the swinging and counting the rotations in this lab as a robot likely has a more consistent swing than a human.
In addition, I would use a more precise motion detector to determine the exact position and time of the cart to create a more exact calculation of acceleration. Lastly, I would increase my confidence in my data by collecting more data points, around 7, of 70, 80, 90, 100, 110, 120, 130 bpm. This will also help to create a greater range of data. Moreover, I would conduct repeated trials of rotations at each bpm as there are too many factors creating uncertainty.