Everything included in the attached file
Simple Harmonic Oscillation of a Pendulum This experiment will be broken down in three parts. 1. Computational Exploration of Pendulum Motion • Simple Pendulum • Damped Simple Pendulum • Damped Driven Pendulum • Physical Pendulum 2. Experimental • Use Phyphox to take gyroscope data of pendulum motion of the phone. 3. Analysis • Compare the data taken by using Phyphox and compare it to the computational models explored in step 1. Purpose The purpose of this lab is to explore real world analysis to the Simple Harmonic Motion models. Excel (or programing language of your choice) will be used to model and graphically display these relationships and compare the undamped motion to the damped motion. The Euler method will be used in order to make modeling easier for the more compl icated motions. Theory Undamped Simple Pendulum A simple pendulum is described as a mass particle, m, attached to a massless string of length, L. When the mass is displaced an angular displacement from the equilibrium position and released. The gravitatio nal force applies a force causing the object to accelerate, as shown in the picture. This results in an oscillation. ∑ = − sin = = 2 2= 2 2= , where = and α is angular acceleration . Resulting in = 2 2= − sin If we assume theta is small then, 2 2= − This results in a small angle approximation provides an angular position relationship with respect to time, Figure 1 Diagram of the forces acting on a pendulum, taken from Duke Math Site ()= cos ( + ), where = √ and is the phase angle. Damped Simple Pendulum A damped oscillation is when a damping force is present. This damping force is = − = − where b is the dampening constant. If applied to the previous force equation, ∑ = − sin − = = 2 2= 2 2= = 2 2= − sin − This equation becomes more complicated to solve, but if we assume theta is small, then the small approximation yields an angular position function, 2 2= − − ()= −2cos ( + ), where = √ − ( 2)2. Damped Driven Pendulum A driven pendulum is a pendulum that has some damping, but it also has a sinusoidal driving torque. This yields an even more complicated function, = 2 2= sin − + cos () where A is the driving torque divided by the moment of inertia (basically a driven angular acceleration) and is the driving frequency. In order to solve this problem, one must use numerical analysis like the Euler Method. Physical Pendulum A physical pendulum is when you can not use the ball and string method but now you must take into consideration the moment of inertia of the object as it is a defined shape. ∑ = − sin = = 2 2 Resulting in = 2 2= − sin If we assume theta is small then, 2 2= − Figure 2 Physical Pendulum, image taken from HyperPhysics. This results in a small angle approximation provides an angular position relationship with respect to time, ()= cos ( + ), where = √ and is the phase angle. Note this is the equation for a undamped oscillation. If this pendulum is ei ther damped or driven, the necessary terms would be added to the equation. Part 1 – Computational Exploration of Pendulum Motion Exercise #1 Compare the small angle formula to the large angle formula. In order to analyze the large angle formula, you need t o use the Euler Method. The Euler method is a first -order numerical procedure to solve ordinary differential equations. Basically, you take a small step of time and multiply it by the derivate of the value this provides a small change in the value which yo u then add to the prior value to get the new value. For example, if you know the initial position, you can find the next position by adding the initial position to the initial velocity times the change in time. The velocity can be found a similar way. (+ )= ()+ ()∗ (+ )= ()+ ()∗ Using Excel graph both the small angle relationship to the large angle relationship. Are they comparable at small angles? At what angle does the small angle does not match the big angle? Show a graph showing th e difference between the small angle approximation and the Euler method Large Angle approximation. Exercise #2 Now see what happens when you add damping to your pendulum. How does the graph change? What happens if you vary the damp en ing co onstant , b? Is t here still a difference between the small angle approximation and the Euler method large angle approximation at large angles, Is it like Exercise #1? If you change the dampening constant, how does it affect the graph (just do this for one of the models i.e . Small Angle)? Show a graph showing the impact of dampening your pendulum. Exercise #3 Now add the driven torque term , A . Play around and see what happens when you vary A and the driven frequency , . How does it compared to the values learned in Exercises #1 and #2. Note you will need to use the Euler method for this, therefore compare it to the other large angle relationships (in Ex. #1,#2) . Show a graphical relationship of the affect driving a pend ulum compared to just a damped oscillation and a non -damped oscillation. Part 2 – Data Attention ! This part of the lab involves swinging your cell phone. Make sure to perform this experiment over a soft surface and be aware of the cell phones motion. Durha m Technical Community College is not responsible for broken cell phones. Materials Cellphone Phyphox app Either string or a meter stick Tape or a very tight knot . Procedure: 1. ATTENTION!! Make sure you are in a carpeted area and that you are careful with the phone. 2. Weigh your phone. Some phone companies provide this in the specs but don’t forget if you have a case. 3. Either use a rod like a meter stick and attach your cellphone by tape or use string and attach it to your phone, make sure the ph one will not fall off. 4. Open Phyphox, in the menu select Gyroscope , as shown on the left. 5. Press the play button in the gyroscope sensor. Start oscillating the phone, try keeping the angle of oscillation small. You will start seeing data that will look simil ar to the picture on the right. Notice how the Gyroscope z data gives the clearest oscillation graph. This is the graph you will be using. 6. Click on the Gyroscope z data, it will open just the z axis data. Once you have isolated the Gyroscope z data, click on the three vertical dots in the menu bar (next to the trash can). A list will appear. Click on Export Data, in order to send the data in spreadsheet form to your email. Feel free to select what data format you want to use. 7. Now use the data to compare to the models you used in part 1. NOTE: M ake sure you adjust your parameters. The mass is the mass of your cell phone. You may have to shift your data along the time axis as you weren’t able to precisely achieve data the moment it started isolating. Part 3 – Analysis In this part you will be comparing computational analysis to real data. After making adjustments to the data (shifting the axis so your max point is at t=0, this will help with analysis), overlay the theoretical models from Part1 on the data graph from Part2. Play around with the variables and models to see what fits best with your data. Note you, can’t change the mass of your phon e, but you can play around with the idea that it is a physical pendulum instead of a simple pendulum. Is it a damped oscillation? Is it a driven pendulum motion (sometimes holding the string, we tend to influence the motion by adding a driving force.)? Ta sks for the write -up: There will be one writ e up. Include the names of all partners if you had any (one submission per group if you worked in a group.) a. Introduction: Explain what models you are looking at. What variables you kept constant and what variables you didn’t. Then discuss the experiment, the details about the experiment and the parameters: mass and length of string, w hat model you think (hypothesis) will best fit your data. b. Data: Discuss the data that was found, include a graph of the data, note this can be included in the Analysis section. Do you know notice anything interesting in the data? Does it look rather const ant the oscillation or is the amplitude decreasing overtime? Or is the amplitude increasing over time. What did you find when changing models and variables. Answer the questions posed in the lab manual. Include comparison graphs to show how it compares to the data. c. Analysis: Compare the data to the models. Figure out what model and parameters best fit the data and why. (Include a graph of the comparison .) Are there any errors, where could that have come from (no human errors)? d. Conclusion: Overal l understanding of the graphs and the impact the dampening constant and driven force play on the oscillatory motion. Briefly re -iterate what you discovered and why you think that it matches the best with the data. Explain any errors or ways for improvement in conducting the experiment.