Whirly Durly Lab
Researchers: Emmy Xu, Grace Chung, Jasmine Zhao
Date Conducted: Friday, December 4, 2021
Experimental Design
Research Question: What is the relationship between the tangential speed of an object traveling in circular motion and its acceleration?
Independent Variable: tangential speed (m/s)
Dependent Variable: acceleration (m/s^2)
Controls: radius of string (m), mass of the bob (kg)
In this set up, a bob was attached to a string and spun around in a circular motion parallel to the ground. To stabilize the motion, the string was strung through a plastic tube, with the length of the radius measuring from the top end of the tube to the bob. To ensure that the radius remained constant throughout the trials and within each trial, red marks were drawn on the string at the location of the top of the tube and the bottom of the tube. The amount of time it took for the bob to pass 20 rotations was measured in each trial. This time was then later on used to calculate the angular speed, and thus the tangential speed. The trials gradually increased in tempo, and a metronome was used to keep track of these speeds. In order to calculate acceleration later on, the tension force needed to be measured for each trial as well. Thus, the bottom of the string was attached to a force sensor, with the string being perpendicular to the circular path of the bob. Because the radius of the string influences tangential speed and because the mass of the bob influences acceleration, it was essential that both factors remained constant throughout the experiment.
Method - Procedure
Method:
The method of data collection for this experiment involved the use of a timer to measure the amount of time it took for the bob to make 20 rotations and the use of a force sensor to measure the tension force for the bob's motion throughout the various tempos. The force sensor was attached to the Logger Pro software, which was used to determine the average force for each trial. Five different tempos were used for this experiment: 75 BPM, 90 BPM, 100 BPM, 120 BPM, 130 BPM. For each trial, the radius of the string remained .6 m and the mass of the bob remained .01 kg.
Procedure:
1. Calibrate the force sensor on Logger Pro, ensuring that it is close to 0.
2. Attach the end of the string to the force sensor.
3. Have one group member stand with the force sensor in one hand and the other hand holding the plastic tube.
4. Start the metronome, setting it to a tempo of 75 BPM.
5. Have the same group member start spinning the bob above their heads.
6. Have another team member set up with a timer and act as the counter.
7. Once the timer starts, have this group member count 20 rotations, setting a spot in the room as a marker for rotations.
8. Record the time as well as the average tension force as detected by Logger Pro.
9. Repeat steps 1-8 with the four different tempos: 90, 100, 120, 130 BPM.
Raw Data
The BPM data was recorded based on the metronome tempos selected. The time for each trial was recorded on an iPhone stopwatch, and the values reported account for the duration of 20 rotations. The force reported is based off of data recorded by the force sensor and Logger Pro software. The mass of the bob was measured using a scale and the radius using a meter stick.
Processed Data
Angular Speed: To calculate angular speed, the formula (omega/angular speed) = (change in theta) / (change in time) was used. The change in theta was calculated by multiplying the number of rotations (20) by the number of radians traveled within each rotation (2 pi). The values used for change in time are the time values recorded during the experiment.
Ex. For the trial with BPM 75, the following calculations were made: (20 rotations * 2pi) / (14.435 s) = 8.705 radians/second.
Tangential Speed: To calculate tangential speed, the formula (velocity/tangential speed) = (omega/angular speed) * (radius) was used. Thus, for each trial, we multiplied the afore calculated angular speed by the radius constant of .60 m.
Ex. For the trial with BPM 75, the following calculatins were made: (8.705 radians/second) * (.60 m) = 5.223 meters/second.
Acceleration: To calculate acceleration, the formula (Net Force) = (mass) * (acceleration) was used. For each trial, we divided the force values determined by the force sensor and Logger Pro by the bob mass constant of .01 kg.
Ex. For the trial with BPM 75, the following calculations were made: (0.5 Newtons) / (.01 kg) = 50 m/s/s. *** N/kg = m/s/s
Graphical Analysis
A "baby" quadratic graph was chosen to represent this relationship. In a quadratic equation with the form y = Ax^2 + Bx + C, a baby quadratic only has the A value: y = Ax^2.
Acceleration = 2.036 (tangential speed)^2
Upon initial observation of the graph, it was hypothesized that the relationship could be linear or proportional. This was largely due to the limited range and amount of data available.
It was determined that a linear relationship could not be the right fit given that the x and y-intercepts are not (0,0). In the context of this problem, having a nonzero x-intercept value would indicate that the acceleration could be 0 while the bob was in motion, traveling in the circular path. This is impossible because if the speed is nonzero, that means the bob must be moving, which means it is spinning and constantly changing directions. Because change in direction is a form of acceleration, it is impossible for the acceleration to be 0 when the speed is not. Having a nonzero y-intercept in the context of this problem is also not possible. If the bob is not in motion, if it is not spinning and therefore not changing direction nor speed, it cannot be accelerating.
It was determined that a proportional relationship also could not be the right fit, given that it goes into the section of the graph where speed is negative and acceleration is also negative. In this experiment, a "positive" acceleration was assigned to the direction towards the center of the circular motion, and a "negative" direction the direction away from the center. In the context of this problem, this would mean that when the bob is traveling in a counterclockwise direction, it is accelerating away from the center. This is impossible because the tension force is constantly pulling the bob towards the center, therefore even if the bob were moving counterclockwise, the acceleration should still be positive.
Based on the quadratic model chosen: as the tangential speed of an object traveling in circular motion increases, its acceleration increases exponentially, it is changing direction faster. The y-intercept is (0,0), indicating that the bob is not changing direction when it is not moving. The acceleration is never negative, indicating that when the bob is in motion, it is always accelerating towards the center, in a circular motion.
Conclusion Pt. 1: Centripetal Acceleration
The purpose of this experiment was to identify the relationship between the tangential speed of an object traveling in a circular path and its acceleration. In this experiment, the specific type of acceleration observed was centripetal acceleration. Centripetal means “center seeking”; in this example, the acceleration is center seeking in that it points towards the center of rotation. If one were to draw a force diagram to depict the motion of the bob throughout the trials, there would be a force of gravity and a force of tension acting on the bob. In separating the horizontal and vertical components of this tension force, we are able to determine that the vertical component cancels out with gravity, leaving the net force as the horizontal component, which is consistently pulling the bob inwards, towards the center of its circular path.
Using vectors to visualize this, we see here to the left that the direction of the velocity vectors is consistently outwards, outside of the path of motion. The bob, therefore, has a tendency to leave its circular path. However, the direction of the acceleration vectors are consistently to the center. Thus, the acceleration the bob experiences, which has now been determined to be the horizontal component of the tension force, pulls the bob directly towards the center of its circular path, keeping the bob in continual orbit.
Conclusion Pt.2: Deriving the Equation
Based on the quadratic equation found during this experiment, the formula relating centripetal acceleration with velocity and radius can be derived. The equation found in this experiment followed the format y = Ax^2, where acceleration (y) is measured in m/s/s and speed (x) is measured in m/s; given that the equation has “x^2”, the units used would then be m^2/s^2. Thus in order for the units on either side of the equation to cancel out, the units for coefficient A must be (1/m). The formula for centripetal acceleration would indicate that this A value is the radius, the distance between the object in motion and the center of its circular path. The formula for centripetal acceleration is a(c) = v^2 / r, where v indicates velocity and r indicates radius. In comparing the two equations, then, we can observe that x^2 is velocity squared, A is radius, and y is centripetal acceleration.
Evaluating Procedures - Improving the Investigation
The two greatest sources of uncertainty in this lab were both in regard to data collection: firstly, the limited range of data collected, and secondly, the limited amount of data collected. In terms of the speed at which the bob was spun at, we were limited by the fact that if the tempo was too slow, the bob would no longer be on a circular path parallel to the ground. At the same time, there was a moderate upper bound on the tempo as well because physically, it would’ve been difficult to attain a consistent tempo of spinning (with a string radius of 60cm) at such a fast pace. Additionally, attempting to do so would likely be in violation of safety protocol. Thus, the range of our data was limited to 75 beats per minute at the lower bound and 135 beats per minute at the upper bound.
In addition to a limited range, the experiment was also hindered by the limited amount of data collected. In total, there were only five trials conducted, largely due to time and safety constraints. In evaluating the set up of this experiment, five trials seems far too little, and it was. When attempting to determine the proper fit for the data graph, we ran into problems. Upon first look, the graph appeared either linear or proportional. In the graphing analysis section of this lab, it is explained why neither of these models are the right fit. The most effective way to improve this investigation would be to conduct a larger number of trials within a larger range of data. This would result in a more comprehensive view of the pattern of the data points. With only five data points, it was easy to mistake a quadratic relationship for a linear one. Nevertheless, with the proper logic applied, it was still possible to determine the proper quadratic fit for the data.