PHYS 2142 Lab #7 - Simple Harmonic Motion and Sine Function [Virtual] There are many objects and systems which exhibit S

[answerhappygod]

Phys 2142 Lab 7 Simple Harmonic Motion And Sine Function Virtual There Are Many Objects And Systems Which Exhibit S 1 (67.83 KiB) Viewed 54 times

PHYS 2142 Lab #7 - Simple Harmonic Motion and Sine Function [Virtual] There are many objects and systems which exhibit Simple Harmonic Motion (SHM). SEM is a periodic repetition of a pattern such as: the pendulum of a clock, a mass oscillating on a spring, a steel ball rolling up-and-down the wall of a curved bowl, a child swinging on a swing, and some electrical circuits (called oscillators) are just a few examples of SHM. So, why do scientists love SHM? To answer this question, we will first look at the Sine Wave function and its graph, which are shown below. y = 2 + sin(3+x+4) +1 The above Sine Wave can be expressed by the General Equation below which we will call Equation #1: y asin(bx + c) + d Note the following for Equation #1 above: Wave Amplitude [4]: The distance between a Crest (or a Trough) and the green dashed Equilibrium Line. For the above wave, the Equilibrium Line is at y-1, the Crests are at y3, and the Troughs are at y=-1. So, the Crests are 2 units above the green line while the Troughs are 2 units below the green line. That is, they both have an Amplitude of two (1.e., a - 2). Wave Frequency 1/1: If you measure the time interval between two consecutive Crests, you would have measured the Wave's Period (usually symbolized by T in seconds). The Period represents the time it takes a wave to complete ONE cycle. A wave's Frequency equals the reciprocal of the Period (i.6., £-//7) -) and can be expressed in the following units: cycles per second [cps), Hertz (Hz), or reciprocal seconds (s). The Frequency indicates how my cycles have occurred in ONE second. For this lab, you will determine a wave's Frequency by curve fitting data to Equation #1 using Vernier's Graphical Analysis program. However, Equation #1 does NOT give you directly. To calculate the wave's Experimental Frequency, we will use the following equation: 6-27 1 => Ju b (2x Equation #2
Horizontal Shift [el: This value determines the extent to which a wave is shifted along the X-axis. However, the c-value is counterintuitive. That is, a Positive c-value means the wave is shifted to the LEFT, while a Negative e-value means the wave is shifted to the RIGHT. Vertical shift [d]: This value determines the extent to which a wave is shifted along the Y-axis. A Positive d-value means the wave is shifted UP, while a Negative d-value means the wave is shifted DOWN. Returning to the question, "why do scientist love SHM? Answer: When you can express the behavior of an object or system by an equation, then you are able to PREDICT the object's/system's behavior under conditions that have NOT yet been observed. In most of your labs, you had to determine how well the experiment work. Up to now, you did this by calculating: a) Percent Error: Here, you strive for Zero Percent Error. and/or In b) R-value: You saw this when you created an Excel graph for "Lab 5- Atwood's Machine". Here, you strive for a R²-value of One. R-value is also called the Coefficient of Determination. general, it measures the strength of the relationship between the dependent (e.g., y) and independent variables (e.g., x). A R-value of 1.000 indicates a strong relationship. A R²-value of 0.9500 indicates 95% of the data can be explained by the fitted equation. Now, it is time for you to learn another method called Root-Mean-Square-Error (aka, RMSE). Below is a graph with data points and the best fitting straight- line equation through those data points. Force Force vs. Displacement 7.000 5.000 predicated40.019*x+0.0072 5.000 4000 1.000 2.000 1.000 0.000 10000 0.02:00 0.0400 40500 0.0000 03000 0.1200 0.1400 0.1500 Displacement im 2
Here is a table which gives the data points (blue dots) and the predicted values from: F40.019+x+0.0072 If we sum the error values in Column #4, we obtain the small error value of -0.001 N. So, we see it is possible to have "large" individual errors for each data point, and yet, have an overall error close to or equaled to ZERO. Displacement (m) Procedure: 0.0000 0.0250 0.0500 0.0750 0.00990 0.1270 F [N] 0.000 1.120 1.850 3.120 3.820 5.180 (N) 0.007 1.006 2.006 3.009 ://www.vernier.com/product/graphical-analysis-4/ 5.090 [N] -0.007 0.112 -0.158 0.111 -0.149 0.090 I (F-F/= RMSEX (F-F) (N²) 0.000 0.013 0.025 0.012 0.022 0.008 0.013 To overcome this problem, we square the values in Column #4 and write the results in Column #5. Now, we sun the values in Column # 5, and then, divide the sum by six (the number of data points) to obtain a Mean Square value of 0.013 M. However, this value has units of newton squared. Therefore, we take the Square Root of 0.013 to express the Error in newtons (1.e., 0.116 M). Note: Just like Percent Error, we strive for a RMSE value of Zero. 0.116 I used Logger Pro's Microphone to collect four sound waves created by a Tone Generating app on my phone. Using the FREE Vernier Graphical Analysis program, I was able to create a Sine wave pattern for each tone. In addition, I used the program's Curve Fitting tool to obtain parameters a, b, c, and d which I inserted into Equation 1 [i.e., y asin(b*x + c) + d]. 1) In this folder, you will find pictures of my Vernier Graphical Analysis program results. 2) For each picture, do the following: a) The SINE box contains the values for parameters: a, b, c, and d. Write the General Equation for the wave by substituting these parameters into Equation 1 (page 1). Do NOT round off the parameter values. b) Record ALL the digits for the RMSE value. c) Use the value of parameter b with Equation 2 (page 1) to obtain the wave's Frequency [] to Two Decimal Places. d) The wave's True Frequency is given at the top of the graph. Using your result in Part c, calculate the Percent Error for this analysis. 3
PHYS 2142 Name: b) 250 Hz Data (You only need to show your calculations for this data set) Note: You will find the parameter values on the 250 Hz graph. Note: 250 Hz is the True Frequency for this graph. a) ya+sin(b+x+ c) + d (Write your equation below using all the digits) RMSE: 600 Hz Data d) % Error - b) RMSE: 6 b) RMSE: 1,725 Hz Data b) RMSE: Lab #7 - Simple Harmonic Motion and Sine Function [Virtual] _| /__/ | 100 3,000 Hz Data (2x) c) f- c) - No.: c) Ime - +100 d) Error - d) Error - d) Error