Rotational Dynamics

Rotational Dynamics Abstract Rotational dynamics is the study of the many angular equivalents that exist for vector dynamics, and how they relate to one another. Rotational dynamics lets us view and consider a completely new set of physical applications including those that involve rotational motion. The purpose of this experiment is to investigate the rotational concepts of vector dynamics, and study the relationship between the two quantities by using an Atwood machine, that contains two different masses attached. We used the height (0. Mom) of the Atwood machine, and the average time (2. 5 s) the heavier eight took to hit the bottom, to calculate the acceleration (0. 36 m/SAA) of the Atwood machine. Once the acceleration was obtained, we used it to find the angular acceleration or alpha (2. 12 radar/SAA) and moment of force(torque) of the Atwood machine, in which then we were finally able to calculate the moment of inertia for the Atwood machine. In comparing rotational dynamics a nd linear dynamics to vector dynamics, it varied in the fact that linear dynamics happens only in one direction, while rotational dynamics happens in many different directions, while they are both examples of vector dynamics.Laboratory Partners Divine Kraal James Mulligan Robert Goalless Victoria Parr Introduction The experiment deals with the Rotational Dynamics of an object or the circular motion (rotation) of an object around its axis. Vector dynamics, includes both Rotational and Linear dynamics, which studies how the forces and torques of an object, affect the motion of it. Dynamics is related to Newton's second law of motion, which states that the acceleration of an object produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.This is where the famous law of F=ma, force equals mass times acceleration, which directly deals with Newton's second law of motion. The important part of Newton's second law and how it relates to rotational dynamics and circular motion, is that Newton's second law of rotation is applied directly towards the Atwood machine, which is Just a different form of Newton's second law. This equation for circular motion is: torque=FRR=l(alpha), which is important for helping us understand what forces are acting upon the Atwood machine. It is important to test the formulas because it either refutes or provesNewton's second law of rotation and more importantly helps us discover the moment of inertia and what it really means. Although both rotational and linear dynamics fall under the category of vector dynamics, there is a big difference between the two quantities. Linear dynamics pertains to an object moving in a straight line and contains quantities such as force, mass, displacement, velocity, acceleration and momentum. Rotational dynamics deals with objects that are rotating or moving in a curved path and involves the q uantities such as torque, moment of inertia, angular velocity, angular acceleration, and angular momentum.In this lab we will be incorporating both of these ideas, but mainly focusing on the rotational dynamics in the Atwood Machine. Every value that we discover in the experiment is important for finding the moment of inertia for the Atwood machine, which describes the mass property of an object that describes the torque needed for a specific angular acceleration about an axis of rotation. This value will be discovered by getting the two masses used on the Atwood machine and calculating the weight, then getting the average time it takes for the smaller weight to hit the ground, the height of theAtwood machine, the radius, the circumference, and the mass of the wheel. From these values, you can calculate the velocity, acceleration, angular acceleration, angular velocity, and torque. Lastly, the law of conservation of energy equation is used to find the formulas used to finally obtain the moment of inertia. Once these values are obtained, it is important to understand the rotational dynamics and how it relates to vector dynamics. It is not only important to understand how and why they relate to each other, but to prove or disprove Newton's second law of motion and understand what it means.Purpose The purpose of this experiment is to study the rotational concepts of vector dynamics, and to understand the relationship between them. We will assume the relationships between the two quantities hold to be true, by using an Atwood machine with two different masses attached to discover the moment of inertia for the circular motion. Equipment The equipment used in this experiment is as follows: 1 Atwood machine 1 0. 20 kilogram weight 1 0. 25 kilogram weight 1 scale 1 piece of string 1 stopwatch with 0. 01 accuracy Procedure 1 . Gather all of the equipment for the experiment. 2.Measure the weight of the two masses by using the scale, making sure to measure as accurately as possible. 3. Measure the length of the radius of the wheel on the Atwood machine. Then after obtaining this number, double it to obtain the circumference. 4. After measuring what is need, proceed to set up the Atwood machine properly. Ask the TA for assistance if needed. 5. First start by tying the end of the string to both weights, double knotting to make sure that it is tight. 6. Set the string with the weights attached to the groove of the Atwood machine wheel, making sure that it is properly in place. 7.Then set the lighter mass on the appropriate end of the machine, and hold in place, so that the starting point is at O degrees. 8. Make sure that the stopwatch is ready to start recording time. 9. When both the timer and the weight dropper are ready to start, release the weight and start the time in sync with one another. 10. At the exact time the mass makes contact with the floor, stop the time as accurately and precise as possible. 1 1 . Repeat this process three times, so t hat an average can be obtained of the three run times, making the data a much more accurate representation of the time it takes he weight to hit the ground. 2. Now that the radius, masses, and time are recorded, it is time to perform the calculations of the data. 13. Calculate the velocity, acceleration, angular acceleration, moment of force or torque, and finally moment of inertia. 14. Finally, compare the relationships of the rotational concepts inquired and draw conclusions. Notes and Observations The Atwood machine contained four outer cylinders that stuck out of the wheel, which cause air resistance in rotation, and contribute to the moment of inertia. The timer, was hard to stop at the exact right time when the weight made contact with he floor.Lastly, there was friction of the string on the wheel, when the weight was released and it rubbed on the wheel. Data Mass of the first weight: 250 g=O. Keg Mass of the second weight: egg=O. Keg Weight 1=MGM= 2. 45 N Weight 2=MGM= 1. 96 N Time 1: 2. 20 seconds Time 2: 2. 19 seconds Time 3: 2. 06 seconds Height: 82. 4 CM= 0. 824 m Radius: 17 CM= 0. 17 m Circumference (distance)= 0. 34 m Mass of the wheel= 221. G x 4= egg= 0. Keg 2 x (change in a= (change in 0. 36 urn,'92 a=r x (alpha) alpha= alarm = 2. 12 radar/92 Velocity'=d/t -?0. 58 m/s E(final) E(final) + Work of friction (l)g(change in height)= h + m(2)g(change in height) + h + h law v/r Moment of Inertia= 0. 026 keg x m/SAA summation of . 876 Error Analysis There was error to account for in this lab, which first started with the four cylinders that stuck out of the Atwood machine in a circular pattern. This caused air resistance in which we could not account for. We only measured the weight of the four cylinders for the total weight of the Atwood machine, because the wheel itself was massages in comparison.Even though it accounted for very little error in our experiment, it effected the other numbers that we calculated in our data, making them a little less ac curate. When finding the amount of time it took the heavier weight to make contact with the rubber pad, there was human error in the reaction time of the timer in which we accounted for, making our data more accurate and precise. This is why we averaged all of the values in order to make the times more precise. Lastly, there was error for the friction of the string making contact with the wheel, which we did not account for, because there was no way of accounting for it.The reason why the force f the tension and the weight were not equal to each other was because of this friction force that existed, which we were not able to find. Conclusion Throughout this experiment we examined the circular dynamics of a pendulum when outside act upon it, making the pendulum move in a circular motion. We measured many values, including the period, in order to determine the theoretical and experimental forces acting on the pendulum. From this we were able to draw conclusions about how the experimen tal and theoretical forces relate to each other.We also were able to test Newton's second law of motion determining whether or not t holds to be true. The values that we obtained to get our experimental and theoretical forces started with setting up the cross bar set-up, and attaching the string with the pendulum to the force gauge and obtaining the tension in the string which was 3 Newton's, by reading the off of the gauge, while the pendulum was swinging in a circle. We then measured the mass of the pendulum with a balance scale to be 0. 267 kilograms, which were then able to find the weight to be 2. 63 Newton's.Next we were able to find the length of the string and force gauge attached to the pendulum. Instead of measuring Just the string attached to the pendulum, we also measured the force gauge, because without it our readings would be inaccurate. After placing the wall grid under the pendulum, we received the numeric value of 0. 5 meters of the radius by reading it off of the chart, by measuring from the origin, to the end of the where the pendulum hovered the graph. Then we found the period by using the stopwatch, which was 1. 71 seconds. We started the time at the beginning of the first crossbar and ended it at the same place.With these numbers that we measured we were able o calculate the angle of the string to the crossbars when it was in motion to be 35. 5 degrees. Then we found the constant velocity by using V = nor/t, in which we obtained the value of 1. 84 meters/second. From this we used the formula a = ‘ГËÅ"2/r to calculate the constant acceleration which was 6. 67 m/SAA, which we came to the understanding that the pendulum was moving very quickly, and that it took a while to slow down. From this we used Newton's famous second law, which was F=ma, to solve for the Force that was subjected on the pendulum.We knew that if this value was airily close to our experimental value that his theory would be proven correct. Me modified the equa tion to fit for the situation that was involved, in which we used F = m x ‘ГËÅ"2/r to receive the value of 1. 81 Newton's. Lastly, by using all of the data that we obtained from the experiment, we used the formula Force Experimental= Ft(sin B) to get an experimental force value of 1. 74 Newton's, which lead us to believe we solved for the correct formulas, and followed the procedure for the experiment correctly. Some of the discrepancy in our data comes from the instability of the crossbar set- up.This is because our crossbar holders were not in place correctly, which we couldn't correct, so we obtained our data as accurately as we could. Another error in our data came from the force gauge, in that it didn't stand still when we set the pendulum in motion. We couldn't read exactly what was on the force gauge and it also kept changing numbers, so we had to estimate based on what we saw. Lastly, the error in reaction time of the stopwatch changed our data. Without these erro rs existing, I believe our experimental values would be closer to our theoretical values. Even though this may be true, our values were only different by 0. Newton's, meaning we performed the experiment correctly for the most part. From the results that we obtained from the experiment, we now understand what we would have to do to improve our results in collecting data and obtaining the Experimental Force acting on the pendulum. Our error could have been improved by using a different table with more stability, improving our reaction time, and obtaining multiple values for the force gauge then averaging the results. We figured out that even though there was error in our experimentation, that our values were still pretty accurate Judging by the theoretical value.Theoretical values are based on what is discovered by physicists performing the experiment over and over again. So to use these values and get a number only fractions off, shows that the way we performed our experiment was not very far off. We proved Newton's second law to be true, because by doing the experiment and getting similar values shows that his concept holds to be true. The forces that we used to move the pendulum showed the dynamics of the pendulum, and how this can be used to understand concepts of the planets rotating around the sun in the universe, Just at a much smaller scale.