Module Overview Summary of Module Description For full details, go to Module Descriptor. Aims The aim of this module is to: • Develop individuals for a career in business and management • Enhance and develop employability , professional and lifelong learning skills and personal development Learning Outcomes Learners will be able to critically evaluate the acquisition of a range of academic and professional skills using a number of theoretical frameworks. Assessment – Summary Category Assessment Description Duration Word Count Weight (%) Written Assignment Essay 1 Reflective Essay N/A 3000 45 For full details, go to Assessment. Additional Information Remember that a variety of Resources is available to support your learning materials.Skills and character audit This document provides an initial picture of your skills and character. It will also provide the basis of further documents that make up the first assignment on the module. It is based on the skills statements that form a fundamental part of your Masters programme which were approved by a validation panel that consisted of members of staff in the Business School, academic staff from other higher education institutions and employers. The statements in the form are there for you and you will not be judged on whether your responses are positive or negative. The responses should enable you to identify what you are good or bad at from which you can create a personal SLOT analysis (Strengths, Limitations, Opportunities, Threats). From this SLOT analysis you can then concentrate on developing certain areas that will enhance your academic and professional development. We would very much like to” get to know” you through this document and would encourage you to also complete the notes section. In this you could give us a rationale for your responses to the questions. As a guide to how you should gauge your response consider the following: Strongly agree – I have a wide range of experience in this area and have been commended by a tutor or employer for my efforts in this area Agree – I am comfortable with this aspect and have been able to demonstrate my ability Disagree – I am Ok with this but realise that I do need to improve Strongly disagree – I know I am weak in this area and need to focus on this as I could fine this weakness to be detrimental to my progression Explain why – please take the room to consider the reasons for your answer as this is the reflection that is of most value. Do not worry if your section spills onto the next page.   Intellectual (thinking) skills Strongly Agree Agree Disagree Strongly Disagree I am a creative person who can adapt my thinking to circumstances I am able to organise my thoughts, analyse, synthesise and critically appraise situations I can identify assumptions, evaluate statements in terms of evidence, detect false logic or reasoning, identify implicit values, define terms adequately and generalise appropriately Explain why: Professional/Vocational skills Strongly Agree Agree Disagree Strongly Disagree I use a wide range of techniques in approaching and solving problems. I am comfortable with a range of research techniques I am able to analyse and interpret quantitative data I am able to analyse and interpret qualitative data My leadership skills are well developed and I can adapt them to different situations I am able to manage people effectively Motivating myself and others comes easy to me I am aware of my responsibilities to myself, the organisation and other people I treat people with respect and consideration Explain why:   Key/Common skills Strongly Agree Agree Disagree Strongly Disagree I am able to use mathematical techniques to analyse data I can effectively interpret numerical data including tables and charts I am able to use a wide range of software on a PC I use a range Information Technology devices to communicate and access information I am a good listener I am able to communicate my ideas well in a face-to-face situation I can adapt my written style to suit an audiences needs I am comfortable presenting my ideas to an audience Whenever I have completed a task I always reflect on the experience with a view to seeking continuous improvement I manage my time effectively I am always prompt when asked to complete a task I am aware of the need to be sensitive to the cultural differences to which I have been exposed I am keen to learn about other people and their country and culture I enjoy working with others to complete a task I know my own character and am sensitive of this in a group situation I understand that a group is made of individuals and I am sensitive to the needs and preferences of others I will always ensure that I get my views across in a meeting I am willing to accept the viewpoint of others I always give 100% in a group task Explain why: SLOT Analysis Having responded to the statements above you should now be in a position to look forward and recognise those areas on which your development will be based. The SLOT analysis can help you to arrange this. Strengths – can be those skills and characteristics to which you have responded positively to in the previous section. It is worth noting that whilst you may be strong in these areas that does not mean you ignore their development. Indeed you may be able to utilise these strengths in the development of areas identified as weaknesses or to overcome strengths, this will enhance those skills and characteristics. Limitations – All of us can identify some sort of limitation to our skills. None of us should be afraid of doing this as this is the first stage on the improvement and development of these weaknesses. Opportunities – These arise or can be created. When thinking of this look ahead at opportunities that will arise in a professional, academic or social context within which your development can take place. Threats – Many threats from your development can come from within – your own characteristics e.g. poor time management can lead to missing deadlines. However we could equally identify a busy lifestyle as a threat to our development. Once again think widely in terms of where the threat will come from. Do not worry if you find that a strength can also be a limitation. This is often true as a characteristic you have may be strength in one situation but a limitation in another. E.g. you may be an assertive person, which is positive, but this could be negative in a group situation. Please try and elaborate this in the notes section at the foot of the table. SLOT Analysis (you may need to use two pages to set out this analysis) Strengths Limitations Opportunities Threats Analysis of the Bullet points in the SLOT table Objectives Having undertaken some analysis of your skills and characteristics the aim of this next section is to identify various aspects of your development during the course of this module, other modules on your course, and extra-curricular activities. Make sure the objectives are SMART:- S – Specific. Clearly identified from the exercises undertaken M – Measurable. The outcomes can be easily demonstrated (to yourself, and where possible others) A – Achievable. They can be done given the opportunities available to you R – Relevant. They form part of your development either on this award, in your employability prospects or in your current job role T – Timebound. They can be achieved within a given timescale Whilst there are 5 rows in the table below, please feel free to add more. However be sure that you need to do this development and that they fit within the scope of the above criteria. Area What I am going to do. How I am going to do it When I am going to do it by Force Field Analysis This technique was designed by Kurt Lewin (1947 and 1953). In the business world it is used for decision making, looking at forces that need to be considered when implementing change – it can be said to be a specialised method of weighing up the pros and cons of a decision. Having looked at your personal strengths and weaknesses we would like you to use this technique to become aware of those factors that will help/hinder, give you motivation for or may act against, your personal development. Whilst you could do this for each of your objectives we want you to think in terms of where you would like to be at the end of your Masters programme. In the central pillar, put in a statement of where you want to be at the end of the course. Then in the arrows either side look at those factors/forces that may work in your favour. Be realistic and please add as many arrows that you think may be necessary; use a separate page for the module if it makes it easier to structure your thoughts. Forces or factors working for achieving your desired outcome Where I want to be Forces or factors against working against you achieving your desired outcome

Module Overview Summary of Module Description For full details, go to Module Descriptor. Aims The aim of this module is to: • Develop individuals for a career in business and management • Enhance and develop employability , professional and lifelong learning skills and personal development Learning Outcomes Learners will be able to critically evaluate the acquisition of a range of academic and professional skills using a number of theoretical frameworks. Assessment – Summary Category Assessment Description Duration Word Count Weight (%) Written Assignment Essay 1 Reflective Essay N/A 3000 45 For full details, go to Assessment. Additional Information Remember that a variety of Resources is available to support your learning materials.Skills and character audit This document provides an initial picture of your skills and character. It will also provide the basis of further documents that make up the first assignment on the module. It is based on the skills statements that form a fundamental part of your Masters programme which were approved by a validation panel that consisted of members of staff in the Business School, academic staff from other higher education institutions and employers. The statements in the form are there for you and you will not be judged on whether your responses are positive or negative. The responses should enable you to identify what you are good or bad at from which you can create a personal SLOT analysis (Strengths, Limitations, Opportunities, Threats). From this SLOT analysis you can then concentrate on developing certain areas that will enhance your academic and professional development. We would very much like to” get to know” you through this document and would encourage you to also complete the notes section. In this you could give us a rationale for your responses to the questions. As a guide to how you should gauge your response consider the following: Strongly agree – I have a wide range of experience in this area and have been commended by a tutor or employer for my efforts in this area Agree – I am comfortable with this aspect and have been able to demonstrate my ability Disagree – I am Ok with this but realise that I do need to improve Strongly disagree – I know I am weak in this area and need to focus on this as I could fine this weakness to be detrimental to my progression Explain why – please take the room to consider the reasons for your answer as this is the reflection that is of most value. Do not worry if your section spills onto the next page.   Intellectual (thinking) skills Strongly Agree Agree Disagree Strongly Disagree I am a creative person who can adapt my thinking to circumstances I am able to organise my thoughts, analyse, synthesise and critically appraise situations I can identify assumptions, evaluate statements in terms of evidence, detect false logic or reasoning, identify implicit values, define terms adequately and generalise appropriately Explain why: Professional/Vocational skills Strongly Agree Agree Disagree Strongly Disagree I use a wide range of techniques in approaching and solving problems. I am comfortable with a range of research techniques I am able to analyse and interpret quantitative data I am able to analyse and interpret qualitative data My leadership skills are well developed and I can adapt them to different situations I am able to manage people effectively Motivating myself and others comes easy to me I am aware of my responsibilities to myself, the organisation and other people I treat people with respect and consideration Explain why:   Key/Common skills Strongly Agree Agree Disagree Strongly Disagree I am able to use mathematical techniques to analyse data I can effectively interpret numerical data including tables and charts I am able to use a wide range of software on a PC I use a range Information Technology devices to communicate and access information I am a good listener I am able to communicate my ideas well in a face-to-face situation I can adapt my written style to suit an audiences needs I am comfortable presenting my ideas to an audience Whenever I have completed a task I always reflect on the experience with a view to seeking continuous improvement I manage my time effectively I am always prompt when asked to complete a task I am aware of the need to be sensitive to the cultural differences to which I have been exposed I am keen to learn about other people and their country and culture I enjoy working with others to complete a task I know my own character and am sensitive of this in a group situation I understand that a group is made of individuals and I am sensitive to the needs and preferences of others I will always ensure that I get my views across in a meeting I am willing to accept the viewpoint of others I always give 100% in a group task Explain why: SLOT Analysis Having responded to the statements above you should now be in a position to look forward and recognise those areas on which your development will be based. The SLOT analysis can help you to arrange this. Strengths – can be those skills and characteristics to which you have responded positively to in the previous section. It is worth noting that whilst you may be strong in these areas that does not mean you ignore their development. Indeed you may be able to utilise these strengths in the development of areas identified as weaknesses or to overcome strengths, this will enhance those skills and characteristics. Limitations – All of us can identify some sort of limitation to our skills. None of us should be afraid of doing this as this is the first stage on the improvement and development of these weaknesses. Opportunities – These arise or can be created. When thinking of this look ahead at opportunities that will arise in a professional, academic or social context within which your development can take place. Threats – Many threats from your development can come from within – your own characteristics e.g. poor time management can lead to missing deadlines. However we could equally identify a busy lifestyle as a threat to our development. Once again think widely in terms of where the threat will come from. Do not worry if you find that a strength can also be a limitation. This is often true as a characteristic you have may be strength in one situation but a limitation in another. E.g. you may be an assertive person, which is positive, but this could be negative in a group situation. Please try and elaborate this in the notes section at the foot of the table. SLOT Analysis (you may need to use two pages to set out this analysis) Strengths Limitations Opportunities Threats Analysis of the Bullet points in the SLOT table Objectives Having undertaken some analysis of your skills and characteristics the aim of this next section is to identify various aspects of your development during the course of this module, other modules on your course, and extra-curricular activities. Make sure the objectives are SMART:- S – Specific. Clearly identified from the exercises undertaken M – Measurable. The outcomes can be easily demonstrated (to yourself, and where possible others) A – Achievable. They can be done given the opportunities available to you R – Relevant. They form part of your development either on this award, in your employability prospects or in your current job role T – Timebound. They can be achieved within a given timescale Whilst there are 5 rows in the table below, please feel free to add more. However be sure that you need to do this development and that they fit within the scope of the above criteria. Area What I am going to do. How I am going to do it When I am going to do it by Force Field Analysis This technique was designed by Kurt Lewin (1947 and 1953). In the business world it is used for decision making, looking at forces that need to be considered when implementing change – it can be said to be a specialised method of weighing up the pros and cons of a decision. Having looked at your personal strengths and weaknesses we would like you to use this technique to become aware of those factors that will help/hinder, give you motivation for or may act against, your personal development. Whilst you could do this for each of your objectives we want you to think in terms of where you would like to be at the end of your Masters programme. In the central pillar, put in a statement of where you want to be at the end of the course. Then in the arrows either side look at those factors/forces that may work in your favour. Be realistic and please add as many arrows that you think may be necessary; use a separate page for the module if it makes it easier to structure your thoughts. Forces or factors working for achieving your desired outcome Where I want to be Forces or factors against working against you achieving your desired outcome

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CIV ENG 280 Computer-based Engineering Analysis Assignment for Lab 5 1. The number of annual precipitation days of one-half of the 50 largest U.S. cities is listed below. Find the mean, mode, median, range, standard deviation and variance of the data. 135 128 78 116 77 111 79 44 97 116 123 88 102 26 82 156 133 107 35 112 98 45 122 125 2. Please go through the steps described in the instruction manual (from page 112 to 136). Your lab report should include the following exercises in the manual. a) Binomial distribution (Page 112) b) Poisson distribution (page 119) c) Normal distribution (page 125) d) t distribution (page 132) 3. A math exam contains 10 multiple-choice questions, each with four choices. Since you have not spent any time preparing the exam, you decided to guess at each question by flipping a coin twice (i.e., two heads for A, head and tail for B, tail and head for C, two tails for D). Let X = the number of questions answered correctly. a) Plot the probability mass function (pmf) of the random variable X. (using the chart type “column”). b) If you have to get at least 5 questions answered correctly to pass the exam, what is the probability that you will pass.

CIV ENG 280 Computer-based Engineering Analysis Assignment for Lab 5 1. The number of annual precipitation days of one-half of the 50 largest U.S. cities is listed below. Find the mean, mode, median, range, standard deviation and variance of the data. 135 128 78 116 77 111 79 44 97 116 123 88 102 26 82 156 133 107 35 112 98 45 122 125 2. Please go through the steps described in the instruction manual (from page 112 to 136). Your lab report should include the following exercises in the manual. a) Binomial distribution (Page 112) b) Poisson distribution (page 119) c) Normal distribution (page 125) d) t distribution (page 132) 3. A math exam contains 10 multiple-choice questions, each with four choices. Since you have not spent any time preparing the exam, you decided to guess at each question by flipping a coin twice (i.e., two heads for A, head and tail for B, tail and head for C, two tails for D). Let X = the number of questions answered correctly. a) Plot the probability mass function (pmf) of the random variable X. (using the chart type “column”). b) If you have to get at least 5 questions answered correctly to pass the exam, what is the probability that you will pass.

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Vermont Technical College Electronic Applications ELT-2060 Lab 05: DC characteristics, input offset voltage and input bias current Reference: Operational Amplifiers with Linear Integrated Circuits Fourth edition William D. Stanley, pages 154-155 (Problems 3-21, 3-22 and Lab exercises LE 3-1 to LE 3-4) For the following exercises, make sure to record all calculations, estimations and measured results. Components: 2 741 Op Amps, 10k Ω Potentiometer, 4-10kΩ, 1kΩ , 100kΩ , 100Ω , 560kΩ , 5.6M Ω, resistors Objectives: a. Voltage offset Null Circuit and Closed-loop Differential Circuit b. Measurement of dc Input Offset Voltage c. Measurement of dc Bias and Offset Currents a. Voltage offset Null Circuit and Closed-loop Differential Circuit In this exercise, investigate the use of a null circuit to reduce the output dc offset to its minimum possible value. Refer to the “Voltage Offset Null Circuit” describe in the 741 op amp data sheet from Appendix C of your text book. Although there are no specific closed-loop configurations shown, use a closed-loop differential Amplifier shown in Figure 1. The differential nature of this type of circuit makes it particularly sensitive, therefore well suited, to illustrate the concept dc voltage offset. 1. Connect the closed-loop difference amplifier of Figure 1 with R=10k Ω and A=1. Using a 10kΩ potentiometer connect the “Voltage Offset Null Circuit” between nodes 1 and 5 as shown in the 741 data sheet. Keep in mind that a potentiometer is a three terminal device. You will need to connect the potentiometer wiper terminal to the lowest potential in the circuit -VCC. 2. Connect the two external circuit inputs (v1 and v2) to ground, measure the dc voltage. From the data sheet the expected value of offset voltage at room temperature is 2mV typical and 6mV maximum. Voltages at these levels will be hard to measure with the laboratory multimeter. 3. Adjust the potentiometer until the dc output magnitude is either zero or it’s minimum possible value. Record the minimum value of voltage attained. 5. Do not break down you difference amplifier. Next, build the non-inverting amplifier as shown in figure 2 with Ri=1k Ω and Rf =100k Ω. Attach the output of the difference amplifier to the input of the non-inverting amplifier. This will amplify your offset by 101. 6. Adjust the potentiometer until the dc output magnitude is either zero or it’s minimum possible value. Record the minimum value of voltage attained. 7. In effect we amplified the voltage offset from the difference amplifier by 101. Please describe any possible flaws in using this approach. Compare this result to what was measured in step 2. 8. Write an equation that expresses the expected output voltage Vo in terms of the two input voltages V1 and V2. 9. Apply dc input voltage for the following six combinations, compare the results to the expected values you calculate with the equation from step 8 a. V1=10V, V2=0V b. V1=0V, V2=10V c. V1=V2=10V d. V1=10mV, V2=0 e. V1=0, V2=10mV f. V1=V2=10mV b. Measurement of dc Input Offset Voltage ( Stanley Problem 3-21 page 151) A circuit and equation to measure the input offset voltage Vio is show in figure 3. With the proper selection of resistors Ri, Rf, and Rc the effects of offset due to input bias currents can be neglected. When the input terminals are both held to ground the resulting output voltage should be a direct measurement of Vio. 1. Build the circuit in Figure 3 with Ri=100 Ω and Rf=10k Ω measure and record Vo. Compare your results with the specification of input offset voltage provided in the data sheet. 2. Increase the value of Rf to 100k Ω, and measure Vo again. Did the output increase by approximately 10x the value recorded in step 1, if so explain how that validates the assumption the input bias currents are negligible. 3. Be sure to include a comparison of the measured values in steps 1 and 2. Include a discussion on how there relationship demonstrates that neglecting input bias current was a valid assumption. c. Measurement of dc Bias and Offset Currents (Stanley Problem 3-22 page 152) Consider the three circuits of figure 4 .The resistance R is chosen large so that the contribution to the output from bias currents is considerably larger than the contribution from the input offset voltages. The accompanying equations will predict the values of Ib+, Ib- and Iio. 1. Start with setting R=560k Ω and build each circuit in figure 4 one at a time. Going from one configuration to the next configuration should be quick, all that is changing is the placement of the resistors. Measure Voa, Vob and Voc for each circuit and calculate Ib+, Ib-, and Iio, compare your measurements to the values in the data sheet. 2. Increase the value of R to 5.6M Ω. Measure Voa, Vob and Voc for each circuit and calculate Ib+, Ib-, and Iio, compare your measurements to the values in the data sheet and to the results in part 1.Did the output increase by approximately 10x the value recorded in step 1, if so explain how that validates the assumption the input offset voltage effect is negligible. 3. Be sure to include a comparison of the measured values in steps 1 and 2. Include a discussion on why neglecting input offset voltage was a valid assumption. LAB write up: This lab requires a semi-formal lab report. Record all calculations, estimations, and measured results. No MultiSim will be required for this report. Please include a written English language paragraph for all lab steps that required an explanation.

Vermont Technical College Electronic Applications ELT-2060 Lab 05: DC characteristics, input offset voltage and input bias current Reference: Operational Amplifiers with Linear Integrated Circuits Fourth edition William D. Stanley, pages 154-155 (Problems 3-21, 3-22 and Lab exercises LE 3-1 to LE 3-4) For the following exercises, make sure to record all calculations, estimations and measured results. Components: 2 741 Op Amps, 10k Ω Potentiometer, 4-10kΩ, 1kΩ , 100kΩ , 100Ω , 560kΩ , 5.6M Ω, resistors Objectives: a. Voltage offset Null Circuit and Closed-loop Differential Circuit b. Measurement of dc Input Offset Voltage c. Measurement of dc Bias and Offset Currents a. Voltage offset Null Circuit and Closed-loop Differential Circuit In this exercise, investigate the use of a null circuit to reduce the output dc offset to its minimum possible value. Refer to the “Voltage Offset Null Circuit” describe in the 741 op amp data sheet from Appendix C of your text book. Although there are no specific closed-loop configurations shown, use a closed-loop differential Amplifier shown in Figure 1. The differential nature of this type of circuit makes it particularly sensitive, therefore well suited, to illustrate the concept dc voltage offset. 1. Connect the closed-loop difference amplifier of Figure 1 with R=10k Ω and A=1. Using a 10kΩ potentiometer connect the “Voltage Offset Null Circuit” between nodes 1 and 5 as shown in the 741 data sheet. Keep in mind that a potentiometer is a three terminal device. You will need to connect the potentiometer wiper terminal to the lowest potential in the circuit -VCC. 2. Connect the two external circuit inputs (v1 and v2) to ground, measure the dc voltage. From the data sheet the expected value of offset voltage at room temperature is 2mV typical and 6mV maximum. Voltages at these levels will be hard to measure with the laboratory multimeter. 3. Adjust the potentiometer until the dc output magnitude is either zero or it’s minimum possible value. Record the minimum value of voltage attained. 5. Do not break down you difference amplifier. Next, build the non-inverting amplifier as shown in figure 2 with Ri=1k Ω and Rf =100k Ω. Attach the output of the difference amplifier to the input of the non-inverting amplifier. This will amplify your offset by 101. 6. Adjust the potentiometer until the dc output magnitude is either zero or it’s minimum possible value. Record the minimum value of voltage attained. 7. In effect we amplified the voltage offset from the difference amplifier by 101. Please describe any possible flaws in using this approach. Compare this result to what was measured in step 2. 8. Write an equation that expresses the expected output voltage Vo in terms of the two input voltages V1 and V2. 9. Apply dc input voltage for the following six combinations, compare the results to the expected values you calculate with the equation from step 8 a. V1=10V, V2=0V b. V1=0V, V2=10V c. V1=V2=10V d. V1=10mV, V2=0 e. V1=0, V2=10mV f. V1=V2=10mV b. Measurement of dc Input Offset Voltage ( Stanley Problem 3-21 page 151) A circuit and equation to measure the input offset voltage Vio is show in figure 3. With the proper selection of resistors Ri, Rf, and Rc the effects of offset due to input bias currents can be neglected. When the input terminals are both held to ground the resulting output voltage should be a direct measurement of Vio. 1. Build the circuit in Figure 3 with Ri=100 Ω and Rf=10k Ω measure and record Vo. Compare your results with the specification of input offset voltage provided in the data sheet. 2. Increase the value of Rf to 100k Ω, and measure Vo again. Did the output increase by approximately 10x the value recorded in step 1, if so explain how that validates the assumption the input bias currents are negligible. 3. Be sure to include a comparison of the measured values in steps 1 and 2. Include a discussion on how there relationship demonstrates that neglecting input bias current was a valid assumption. c. Measurement of dc Bias and Offset Currents (Stanley Problem 3-22 page 152) Consider the three circuits of figure 4 .The resistance R is chosen large so that the contribution to the output from bias currents is considerably larger than the contribution from the input offset voltages. The accompanying equations will predict the values of Ib+, Ib- and Iio. 1. Start with setting R=560k Ω and build each circuit in figure 4 one at a time. Going from one configuration to the next configuration should be quick, all that is changing is the placement of the resistors. Measure Voa, Vob and Voc for each circuit and calculate Ib+, Ib-, and Iio, compare your measurements to the values in the data sheet. 2. Increase the value of R to 5.6M Ω. Measure Voa, Vob and Voc for each circuit and calculate Ib+, Ib-, and Iio, compare your measurements to the values in the data sheet and to the results in part 1.Did the output increase by approximately 10x the value recorded in step 1, if so explain how that validates the assumption the input offset voltage effect is negligible. 3. Be sure to include a comparison of the measured values in steps 1 and 2. Include a discussion on why neglecting input offset voltage was a valid assumption. LAB write up: This lab requires a semi-formal lab report. Record all calculations, estimations, and measured results. No MultiSim will be required for this report. Please include a written English language paragraph for all lab steps that required an explanation.

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Using the Earnings_and_Height dataset, perform the following exercises. 1. Test whether the difference in mean height for men and women is statistically significant? Is it? 2. Does the distribution of height for men and women in the US follow the normal distribution? Answer by looking at detailed summary statistics and high order moments of the data. 3. If you were to select one al observation from the data at random, what is the probability that individual is than sixty seven inches tall? 4a. Run the following regressions and interpret the coefficient on the height variable. I. Regress earnings on height II. Regress log of earning on height III. Regress log earnings on log of height 4b. which model is preferred? 5a. Regress log of earnings on height and height² 5b. is there a non-linear relationship between height and log earnings? 5c. Give a-formula for the effect of a change in height on the change in log earnings. 6. Create the following variables: I. A dummy variable for being-Hispania. ii. A dummy variable for being black. iii. A dummy variable for being female. iv. A set of region dummy variables. 7a. Run the following regression separately by gender: Regress log earnings on height education age black Hispanic 7b. Is there a difference in the estimated effect of height on earnings by gender? 8. Run the following regression: Regress log earnings on height education age black Hispanic female and a set of region indicators, and perform the following tests (and interpret the results): I. Test for the equality of coefficients on the Hispanic and black variables. ii. Test the hypothesis that the coefficients on female, black and Hispanic are all zero. 9a. Run the following regression for men: Regress log earnings on height height² education age black Hispanic and a set of region indicators. Is there evidence of a non-linear relationship between height andlog earnings for men? 9b. Estimate the effect of a one inch increase in height on log earnings for a man starting an average height) 10. Discuss the following threats to internal validity regarding the model in (8): I. measurement error focusing on earnings and height) ii. Omitted variables bias

Using the Earnings_and_Height dataset, perform the following exercises. 1. Test whether the difference in mean height for men and women is statistically significant? Is it? 2. Does the distribution of height for men and women in the US follow the normal distribution? Answer by looking at detailed summary statistics and high order moments of the data. 3. If you were to select one al observation from the data at random, what is the probability that individual is than sixty seven inches tall? 4a. Run the following regressions and interpret the coefficient on the height variable. I. Regress earnings on height II. Regress log of earning on height III. Regress log earnings on log of height 4b. which model is preferred? 5a. Regress log of earnings on height and height² 5b. is there a non-linear relationship between height and log earnings? 5c. Give a-formula for the effect of a change in height on the change in log earnings. 6. Create the following variables: I. A dummy variable for being-Hispania. ii. A dummy variable for being black. iii. A dummy variable for being female. iv. A set of region dummy variables. 7a. Run the following regression separately by gender: Regress log earnings on height education age black Hispanic 7b. Is there a difference in the estimated effect of height on earnings by gender? 8. Run the following regression: Regress log earnings on height education age black Hispanic female and a set of region indicators, and perform the following tests (and interpret the results): I. Test for the equality of coefficients on the Hispanic and black variables. ii. Test the hypothesis that the coefficients on female, black and Hispanic are all zero. 9a. Run the following regression for men: Regress log earnings on height height² education age black Hispanic and a set of region indicators. Is there evidence of a non-linear relationship between height andlog earnings for men? 9b. Estimate the effect of a one inch increase in height on log earnings for a man starting an average height) 10. Discuss the following threats to internal validity regarding the model in (8): I. measurement error focusing on earnings and height) ii. Omitted variables bias

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Biomedical Signal and Image Processing (4800_420_001) Assigned on September 12th, 2017 Assignment 4 – Noise and Correlation 1. If a signal is measured as 2.5 V and the noise is 28 mV (28 × 10−3 V), what is the SNR in dB? 2. A single sinusoidal signal is found with some noise. If the RMS value of the noise is 0.5 V and the SNR is 10 dB, what is the RMS amplitude of the sinusoid? 3. The file signal_noise.mat contains a variable x that consists of a 1.0-V peak sinusoidal signal buried in noise. What is the SNR for this signal and noise? Assume that the noise RMS is much greater than the signal RMS. Note: “signal_noise.mat” and other files used in these assignments can be downloaded from the content area of Brightspace, within the “Data Files for Exercises” folder. These files can be opened in Matlab by copying into the active folder and double-clicking on the file or using the Matlab load command using the format: load(‘signal_noise.mat’). To discover the variables within the files use the Matlab who command. 4. An 8-bit ADC converter that has an input range of ±5 V is used to convert a signal that ranges between ±2 V. What is the SNR of the input if the input noise equals the quantization noise of the converter? Hint: Refer to Equation below to find the quantization noise: 5. The file filter1.mat contains the spectrum of a fourth-order lowpass filter as variable x in dB. The file also contains the corresponding frequencies of x in variable freq. Plot the spectrum of this filter both as dB versus log frequency and as linear amplitude versus linear frequency. The frequency axis should range between 10 and 400 Hz in both plots. Hint: Use Equation below to convert: Biomedical Signal and Image Processing (4800_420_001) Assigned on September 12th, 2017 6. Generate one cycle of the square wave similar to the one shown below in a 500-point MATLAB array. Determine the RMS value of this waveform. [Hint: When you take the square of the data array, be sure to use a period before the up arrow so that MATLAB does the squaring point-by-point (i.e., x.^2).]. 7. A resistor produces 10 μV noise (i.e., 10 × 10−6 V noise) when the room temperature is 310 K and the bandwidth is 1 kHz (i.e., 1000 Hz). What current noise would be produced by this resistor? 8. A 3-ma current flows through both a diode (i.e., a semiconductor) and a 20,000-Ω (i.e., 20-kΩ) resistor. What is the net current noise, in? Assume a bandwidth of 1 kHz (i.e., 1 × 103 Hz). Which of the two components is responsible for producing the most noise? 9. Determine if the two signals, x and y, in file correl1.mat are correlated by checking the angle between them. 10. Modify the approach used in Practice Problem 3 to find the angle between short signals: Do not attempt to plot these vectors as it would require a 6-dimensional plot!

Biomedical Signal and Image Processing (4800_420_001) Assigned on September 12th, 2017 Assignment 4 – Noise and Correlation 1. If a signal is measured as 2.5 V and the noise is 28 mV (28 × 10−3 V), what is the SNR in dB? 2. A single sinusoidal signal is found with some noise. If the RMS value of the noise is 0.5 V and the SNR is 10 dB, what is the RMS amplitude of the sinusoid? 3. The file signal_noise.mat contains a variable x that consists of a 1.0-V peak sinusoidal signal buried in noise. What is the SNR for this signal and noise? Assume that the noise RMS is much greater than the signal RMS. Note: “signal_noise.mat” and other files used in these assignments can be downloaded from the content area of Brightspace, within the “Data Files for Exercises” folder. These files can be opened in Matlab by copying into the active folder and double-clicking on the file or using the Matlab load command using the format: load(‘signal_noise.mat’). To discover the variables within the files use the Matlab who command. 4. An 8-bit ADC converter that has an input range of ±5 V is used to convert a signal that ranges between ±2 V. What is the SNR of the input if the input noise equals the quantization noise of the converter? Hint: Refer to Equation below to find the quantization noise: 5. The file filter1.mat contains the spectrum of a fourth-order lowpass filter as variable x in dB. The file also contains the corresponding frequencies of x in variable freq. Plot the spectrum of this filter both as dB versus log frequency and as linear amplitude versus linear frequency. The frequency axis should range between 10 and 400 Hz in both plots. Hint: Use Equation below to convert: Biomedical Signal and Image Processing (4800_420_001) Assigned on September 12th, 2017 6. Generate one cycle of the square wave similar to the one shown below in a 500-point MATLAB array. Determine the RMS value of this waveform. [Hint: When you take the square of the data array, be sure to use a period before the up arrow so that MATLAB does the squaring point-by-point (i.e., x.^2).]. 7. A resistor produces 10 μV noise (i.e., 10 × 10−6 V noise) when the room temperature is 310 K and the bandwidth is 1 kHz (i.e., 1000 Hz). What current noise would be produced by this resistor? 8. A 3-ma current flows through both a diode (i.e., a semiconductor) and a 20,000-Ω (i.e., 20-kΩ) resistor. What is the net current noise, in? Assume a bandwidth of 1 kHz (i.e., 1 × 103 Hz). Which of the two components is responsible for producing the most noise? 9. Determine if the two signals, x and y, in file correl1.mat are correlated by checking the angle between them. 10. Modify the approach used in Practice Problem 3 to find the angle between short signals: Do not attempt to plot these vectors as it would require a 6-dimensional plot!

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Morgan Extra Pages Graphing with Excel to be carried out in a computer lab, 3rd floor Calloway Hall or elsewhere The Excel spreadsheet consists of vertical columns and horizontal rows; a column and row intersect at a cell. A cell can contain data for use in calculations of all sorts. The Name Box shows the currently selected cell (Fig. 1). In the Excel 2007 and 2010 versions the drop-down menus familiar in most software screens have been replaced by tabs with horizontally-arranged command buttons of various categories (Fig. 2) ___________________________________________________________________ Open Excel, click on the Microsoft circle, upper left, and Save As your surname. xlsx on the desktop. Before leaving the lab e-mail the file to yourself and/or save to a flash drive. Also e-mail it to your instructor. Figure 1. Parts of an Excel spreadsheet. Name Box Figure 2. Tabs. Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com EXERCISE 1: BASIC OPERATIONS Click Save often as you work. 1. Type the heading “Edge Length” in Cell A1 and double click the crack between the A and B column heading for automatic widening of column A. Similarly, write headings for columns B and C and enter numbers in Cells A2 and A3 as in Fig. 3. Highlight Cells A2 and A3 by dragging the cursor (chunky plus-shape) over the two of them and letting go. 2. Note that there are three types of cursor crosses: chunky for selecting, barbed for moving entries or blocks of entries from cell to cell, and tiny (appearing only at the little square in the lower-right corner of a cell). Obtain a tiny arrow for Cell A3 and perform a plus-drag down Column A until the cells are filled up to 40 (in Cell A8). Note that the two highlighted cells set both the starting value of the fill and the intervals. 3. Click on Cell B2 and enter a formula for face area of a cube as follows: type =, click on Cell A2, type ^2, and press Enter (note the formula bar in Fig. 4). 4. Enter the formula for cube volume in Cell C2 (same procedure, but “=, click on A2, ^3, Enter”). 5. Highlight Cells B2 and C2; plus-drag down to Row 8 (Fig. 5). Do the numbers look correct? Click on some cells in the newly filled area and notice how Excel steps the row designations as it moves down the column (it can do it for horizontal plusdrags along rows also). This is the major programming development that has led to the popularity of spreadsheets. Figure 3. Entries. Figure 4. A formula. Figure 5. Plus-dragging formulas. Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com 6. Now let’s graph the Face Area versus Edge Length: select Cells A1 through B8, choose the Insert tab, and click the Scatter drop-down menu and select “Scatter with only Markers” (Fig. 6). 7. Move the graph (Excel calls it a “chart”) that appears up alongside your number table and dress it up as follows: a. Note that some Chart Layouts have appeared above. Click Layout 1 and alter each title to read Face Area for the vertical axis, Edge Length for the horizontal and Face Area vs. Edge Length for the Graph Title. b. Activate the Excel Least squares routine, called “fitting a trendline” in the program: right click any of the data markers and click Add Trendline. Choose Power and also check “Display equation on chart” and “Display R-squared value on chart.” Fig. 7 shows what the graph will look like at this point. c. The titles are explicit, so the legend is unnecessary. Click on it and press the delete button to remove it. Figure 6. Creating a scatter graph. Figure 7. A graph with a fitted curve. Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com 8. Now let’s overlay the Volume vs. Edge Length curve onto the same graph (optional for 203L/205L): Make a copy of your graph by clicking on the outer white area, clicking ctrl-c (or right click, copy), and pasting the copy somewhere else (ctrl-v). If you wish, delete the trendline as in Fig. 8. a. Right click on the outer white space, choose Select Data and click the Add button. b. You can type in the cell ranges by hand in the dialog box that comes up, but it is easier to click the red, white, and blue button on the right of each space and highlight what you want to go in. Click the red, white, and blue of the bar that has appeared, and you will bounce back to the Add dialog box. Use the Edge Length column for the x’s and Volume for the y’s. c. Right-click on any volume data point and choose Format Data Series. Clicking Secondary Axis will place its scale on the right of the graph as in Fig. 8. d. Dress up your graph with two axis titles (Layout-Labels-Axis Titles), etc. Figure 8. Adding a second curve and y-axis to the graph Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com EXERCISE 2: INTERPRETING A LINEAR GRAPH Introduction: Many experiments are repeated a number of times with one of the parameters involved varied from run to run. Often the goal is to measure the rate of change of a dependent variable, rather than a particular value. If the dependent variable can be expressed as a linear function of the independent parameter, then the slope and yintercept of an appropriate graph will give the rate of change and a particular value, respectively. An example of such an experiment in PHYS.203L/205L is the first part of Lab 20, in which weights are added to the bottom of a suspended spring (Figure 9). This experiment shows that a spring exerts a force Fs proportional to the distance stretched y = (y-yo), a relationship known as Hooke’s Law: Fs = – k(y – yo) (Eq. 1) where k is called the Hooke’s Law constant. The minus sign shows that the spring opposes any push or pull on it. In Lab 20 Fs is equal to (- Mg) and y is given by the reading on a meter stick. Masses were added to the bottom of the spring in 50-g increments giving weights in newtons of 0.49, 0.98, etc. The weight pan was used as the pointer for reading y and had a mass of 50 g, so yo could not be directly measured. For convenient graphing Equation 1 can be rewritten: -(Mg) = – ky + kyo Or (Mg) = ky – kyo (Eq. 1′) Procedure 1. On your spreadsheet note the tabs at the bottom left and double-click Sheet1. Type in “Basics,” and then click the Sheet2 tab to bring up a fresh worksheet. Change the sheet name to “Linear Fit” and fill in data as in this table. Hooke’s Law Experiment y (m) -Fs = Mg (N) 0.337 0.49 0.388 0.98 0.446 1.47 0.498 1.96 0.550 2.45 2. Highlight the cells with the numbers, and graph (Mg) versus y as in Steps 6 and 7 of the Basics section. Your Trendline this time will be Linear of course. If you are having trouble remembering what’s versus what, “y” looks like “v”, so what comes before the “v” of “versus” goes on the y (vertical) axis. Yes, this graph is confusing: the horizontal (“x”) axis is distance y, and the “y” axis is something else. 3. Click on the Equation/R2 box on the graph and highlight just the slope, that is, only the number that comes before the “x.” Copy it (control-c is a fast way to Figure 9. A spring with a weight stretching it Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com do it) and paste it (control-v) into an empty cell. Do likewise for the intercept (including the minus sign). SAVE YOUR FILE! 5. The next steps use the standard procedure for obtaining information from linear data. Write the general equation for a straight line immediately below a hand-written copy of Equation 1′ then circle matching items: (Mg) = k y + (- k yo) (Eq. 1′) y = m x + b Note the parentheses around the intercept term of Equation 1′ to emphasize that the minus sign is part of it. Equating above and below, you can create two useful new equations: slope m = k (Eq. 2) y-intercept b = -kyo (Eq. 3) 6. Solve Equation 2 for k, that is, rewrite left to right. Then substitute the value for slope m from your graph, and you have an experimental value for the Hooke’s Law constant k. Next solve Equation 3 for yo, substitute the value for intercept b from your graph and the value of k that you just found, and calculate yo. 7. Examine your linear graph for clues to finding the units of the slope and the yintercept. Use these units to find the units of k and yo. 8. Present your values of k and yo with their units neatly at the bottom of your spreadsheet. 9. R2 in Excel, like r in our lab manual and Corr. in the LoggerPro software, is a measure of how well the calculated line matches the data points. 1.00 would indicate a perfect match. State how good a match you think was made in this case? 10. Do the Homework, Further Exercises on Interpreting Linear Graphs, on the following pages. Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com Eq.1 M m f M a g               , (Eq.2) M slope m g       (Eq.3) M b f        Morgan Extra Pages Homework: Graph Interpretation Exercises EXAMPLE WITH COMPLETE SOLUTION In PHYS.203L and 205L we do Lab 9 Newton’s Second Law on Atwood’s Machine using a photogate sensor (Fig. 1). The Atwood’s apparatus can slow the rate of fall enough to be measured even with primitive timing devices. In our experiment LoggerPro software automatically collects and analyzes the data giving reliable measurements of g, the acceleration of gravity. The equation governing motion for Atwood’s Machine can be written: where a is the acceleration of the masses and string, g is the acceleration of gravity, M is the total mass at both ends of the string, m is the difference between the masses, and f is the frictional force at the hub of the pulley wheel. In this exercise you are given a graph of a vs. m obtained in this experiment with the values of M and the slope and intercept (Fig. 2). The goal is to extract values for acceleration of gravity g and frictional force f from this information. To analyze the graph we write y = mx + b, the general equation for a straight line, directly under Equation 1 and match up the various parameters: Equating above and below, you can create two new equations: and y m x b M m f M a g                Figure 1. The Atwood’s Machine setup (from the LoggerPro handout). Figure 2. Graph of acceleration versus mass difference; data from a Physics I experiment. Atwood’s Machine M = 0.400 kg a = 24.4 m – 0.018 R2 = 0.998 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 0.000 0.010 0.020 0.030 0.040 0.050 0.060  m (kg) a (m/s2) Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com 2 2 9.76 / 0.400 24.4 /( ) m s kg m kg s g Mm      To handle Equation 2 it pays to consider what the units of the slope are. A slope is “the rise over the run,“ so its units must be the units of the vertical axis divided by those of the horizontal axis. In this case: Now let’s solve Equation 2 for g and substitute the values of total mass M and of the slope m from the graph: Using 9.80 m/s2 as the Baltimore accepted value for g, we can calculate the percent error: A similar process with Equation 3 leads to a value for f, the frictional force at the hub of the pulley wheel. Note that the units of intercept b are simply whatever the vertical axis units are, m/s2 in this case. Solving Equation 3 for f: EXERCISE 1 The Picket Fence experiment makes use of LoggerPro software to calculate velocities at regular time intervals as the striped plate passes through the photogate (Fig. 3). The theoretical equation is v = vi + at (Eq. 4) where vi = 0 (the fence is dropped from rest) and a = g. a. Write Equation 4 with y = mx + b under it and circle matching factors as in the Example. b. What is the experimental value of the acceleration of gravity? What is its percent error from the accepted value for Baltimore, 9.80 m/s2? c. Does the value of the y-intercept make sense? d. How well did the straight Trendline match the data? 2 / 2 kg s m kg m s   0.4% 100 9.80 9.76 9.80 100 . . . %        Acc Exp Acc Error kg m s mN kg m s f Mb 7.2 10 / 7.2 0.400 ( 0.018 / ) 3 2 2           Figure 3. Graph of speed versus time as calculated by LoggerPro as a picket fence falls freely through a photogate. Picket Fence Drop y = 9.8224x + 0.0007 R2 = 0.9997 0 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 1.2 t (s) v (m/s) Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com EXERCISE 2 This is an electrical example from PHYS.204L/206L, potential difference, V, versus current, I (Fig. 4). The theoretical equation is V = IR (Eq. 5) and is known as “Ohm’s Law.” The unit symbols stand for volts, V, and Amperes, A. The factor R stands for resistance and is measured in units of ohms, symbol  (capital omega). The definition of the ohm is: V (Eq. 6) By coincidence the letter symbols for potential (a quantity ) and volts (its unit) are identical. Thus “voltage” has become the laboratory slang name for potential. a. Rearrange the Ohm’s Law equation to match y = mx + b.. b. What is the experimental resistance? c. Comment on the experimental intercept: is its value reasonable? EXERCISE 3 This graph (Fig. 5) also follows Ohm’s Law, but solved for current I. For this graph the experimenter held potential difference V constant at 15.0V and measured the current for resistances of 100, 50, 40, and 30  Solve Ohm’s Law for I and you will see that 1/R is the logical variable to use on the x axis. For units, someone once jokingly referred to a “reciprocal ohm” as a “mho,” and the name stuck. a. Rearrange Equation 5 solved for I to match y = mx + b. b. What is the experimental potential difference? c. Calculate the percent difference from the 15.0 V that the experimenter set on the power supply (the instrument used for such experiments). d. Comment on the experimental intercept: is its value reasonable? Figure 4. Graph of potential difference versus current; data from a Physics II experiment. The theoretical equation, V = IR, is known as “Ohm’s Law.” Ohm’s Law y = 0.628x – 0.0275 R2 = 0.9933 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 Current, I (A) Potential difference, V (V) Figure 5. Another application of Ohm’s Law: a graph of current versus the inverse of resistance, from a different electric circuit experiment. Current versus (1/Resistance) y = 14.727x – 0.2214 R2 = 0.9938 0 100 200 300 400 500 600 5 10 15 20 25 30 35 R-1 (millimhos) I (milliamperes) Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com EXERCISE 4 The Atwood’s Machine experiment (see the solved example above) can be done in another way: keep mass difference m the same and vary the total mass M (Fig. 6). a. Rewrite Equation 1 and factor out (1/M). b. Equate the coefficient of (1/M) with the experimental slope and solve for acceleration of gravity g. c. Substitute the values for slope, mass difference, and frictional force and calculate the experimental of g. d. Derive the units of the slope and show that the units of g come out as they should. e. Is the value of the experimental intercept reasonable? EXERCISE 5 In the previous two exercises the reciprocal of a variable was used to make the graph come out linear. In this one the trick will be to use the square root of a variable (Fig. 7). In PHYS.203L and 205L Lab 19 The Pendulum the theoretical equation is where the period T is the time per cycle, L is the length of the string, and g is the acceleration of gravity. a. Rewrite Equation 7 with the square root of L factored out and placed at the end. b. Equate the coefficient of √L with the experimental slope and solve for acceleration of gravity g. c. Substitute the value for slope and calculate the experimental of g. d. Derive the units of the slope and show that the units of g come out as they should. e. Is the value of the experimental intercept reasonable? 2 (Eq . 7) g T   L Figure 6. Graph of acceleration versus the reciprocal of total mass; data from a another Physics I experiment. Atwood’s Machine m = 0.020 kg f = 7.2 mN y = 0.1964x – 0.0735 R2 = 0.995 0.400 0.600 0.800 1.000 2.000 2.500 3.000 3.500 4.000 4.500 5.000 1/M (1/kg) a (m/s2) Effect of Pendulum Length on Period y = 2.0523x – 0.0331 R2 = 0.999 0.400 0.800 1.200 1.600 2.000 2.400 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 L1/2 (m1/2) T (s) Figure 7. Graph of period T versus the square root of pendulum length; data from a Physics I experiment. Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com EXERCISE 6 In Exercise 5 another approach would have been to square both sides of Equation 7 and plot T2 versus L. Lab 20 directs us to use that alternative. It involves another case of periodic or harmonic motion with a similar, but more complicated, equation for the period: where T is the period of the bobbing (Fig. 8), M is the suspended mass, ms is the mass of the spring, k is a measure of stiffness called the spring constant, and C is a dimensionless factor showing how much of the spring mass is effectively bobbing. a. Square both sides of Equation 8 and rearrange it to match y = mx + b. b. Write y = mx + b under your rearranged equation and circle matching factors as in the Example. c. Write two new equations analogous to Equations 2 and 3 in the Example. Use the first of the two for calculating k and the second for finding C from the data of Fig. 9. d. A theoretical analysis has shown that for most springs C = 1/3. Find the percent error from that value. e. Derive the units of the slope and intercept; show that the units of k come out as N/m and that C is dimensionless. 2 (Eq . 8) k T M Cm s    Figure 8. In Lab 20 mass M is suspended from a spring which is set to bobbing up and down, a good approximation to simple harmonic motion (SHM), described by Equation 8. Lab 20: SHM of a Spring Mass of the spring, ms = 25.1 g y = 3.0185x + 0.0197 R2 = 0.9965 0.0000 0.2000 0.4000 0.6000 0.8000 1.0000 0 0.05 0.1 0.15 0.2 0.25 0.3 M (kg) T 2 2 Figure 9. Graph of the square of the period T2 versus suspended mass M data from a Physics I experiment. Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com EXERCISE 7 This last exercise deals with an exponential equation, and the trick is to take the logarithm of both sides. In PHYS.204L/206L we do Lab 33 The RC Time Constant with theoretical equation: where V is the potential difference at time t across a circuit element called a capacitor (the  is dropped for simplicity), Vo is V at t = 0 (try it), and  (tau) is a characteristic of the circuit called the time constant. a. Take the natural log of both sides and apply the addition rule for logarithms of a product on the right-hand side. b. Noting that the graph (Fig. 10) plots lnV versus t, arrange your equation in y = mx + b order, write y = mx + b under it, and circle the parts as in the Example. c. Write two new equations analogous to Equations 2 and 3 in the Example. Use the first of the two for calculating  and the second for finding lnVo and then Vo. d. Note that the units of lnV are the natural log of volts, lnV. As usual derive the units of the slope and interecept and use them to obtain the units of your experimental V and t. V V e (Eq. 9) t o    Figure 10. Graph of a logarithm versus time; data from Lab 33, a Physics II experiment. Discharge of a Capacitor y = -9.17E-03x + 2.00E+00 R2 = 9.98E-01 0.00 0.50 1.00 1.50 2.00 2.50

Morgan Extra Pages Graphing with Excel to be carried out in a computer lab, 3rd floor Calloway Hall or elsewhere The Excel spreadsheet consists of vertical columns and horizontal rows; a column and row intersect at a cell. A cell can contain data for use in calculations of all sorts. The Name Box shows the currently selected cell (Fig. 1). In the Excel 2007 and 2010 versions the drop-down menus familiar in most software screens have been replaced by tabs with horizontally-arranged command buttons of various categories (Fig. 2) ___________________________________________________________________ Open Excel, click on the Microsoft circle, upper left, and Save As your surname. xlsx on the desktop. Before leaving the lab e-mail the file to yourself and/or save to a flash drive. Also e-mail it to your instructor. Figure 1. Parts of an Excel spreadsheet. Name Box Figure 2. Tabs. Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com EXERCISE 1: BASIC OPERATIONS Click Save often as you work. 1. Type the heading “Edge Length” in Cell A1 and double click the crack between the A and B column heading for automatic widening of column A. Similarly, write headings for columns B and C and enter numbers in Cells A2 and A3 as in Fig. 3. Highlight Cells A2 and A3 by dragging the cursor (chunky plus-shape) over the two of them and letting go. 2. Note that there are three types of cursor crosses: chunky for selecting, barbed for moving entries or blocks of entries from cell to cell, and tiny (appearing only at the little square in the lower-right corner of a cell). Obtain a tiny arrow for Cell A3 and perform a plus-drag down Column A until the cells are filled up to 40 (in Cell A8). Note that the two highlighted cells set both the starting value of the fill and the intervals. 3. Click on Cell B2 and enter a formula for face area of a cube as follows: type =, click on Cell A2, type ^2, and press Enter (note the formula bar in Fig. 4). 4. Enter the formula for cube volume in Cell C2 (same procedure, but “=, click on A2, ^3, Enter”). 5. Highlight Cells B2 and C2; plus-drag down to Row 8 (Fig. 5). Do the numbers look correct? Click on some cells in the newly filled area and notice how Excel steps the row designations as it moves down the column (it can do it for horizontal plusdrags along rows also). This is the major programming development that has led to the popularity of spreadsheets. Figure 3. Entries. Figure 4. A formula. Figure 5. Plus-dragging formulas. Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com 6. Now let’s graph the Face Area versus Edge Length: select Cells A1 through B8, choose the Insert tab, and click the Scatter drop-down menu and select “Scatter with only Markers” (Fig. 6). 7. Move the graph (Excel calls it a “chart”) that appears up alongside your number table and dress it up as follows: a. Note that some Chart Layouts have appeared above. Click Layout 1 and alter each title to read Face Area for the vertical axis, Edge Length for the horizontal and Face Area vs. Edge Length for the Graph Title. b. Activate the Excel Least squares routine, called “fitting a trendline” in the program: right click any of the data markers and click Add Trendline. Choose Power and also check “Display equation on chart” and “Display R-squared value on chart.” Fig. 7 shows what the graph will look like at this point. c. The titles are explicit, so the legend is unnecessary. Click on it and press the delete button to remove it. Figure 6. Creating a scatter graph. Figure 7. A graph with a fitted curve. Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com 8. Now let’s overlay the Volume vs. Edge Length curve onto the same graph (optional for 203L/205L): Make a copy of your graph by clicking on the outer white area, clicking ctrl-c (or right click, copy), and pasting the copy somewhere else (ctrl-v). If you wish, delete the trendline as in Fig. 8. a. Right click on the outer white space, choose Select Data and click the Add button. b. You can type in the cell ranges by hand in the dialog box that comes up, but it is easier to click the red, white, and blue button on the right of each space and highlight what you want to go in. Click the red, white, and blue of the bar that has appeared, and you will bounce back to the Add dialog box. Use the Edge Length column for the x’s and Volume for the y’s. c. Right-click on any volume data point and choose Format Data Series. Clicking Secondary Axis will place its scale on the right of the graph as in Fig. 8. d. Dress up your graph with two axis titles (Layout-Labels-Axis Titles), etc. Figure 8. Adding a second curve and y-axis to the graph Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com EXERCISE 2: INTERPRETING A LINEAR GRAPH Introduction: Many experiments are repeated a number of times with one of the parameters involved varied from run to run. Often the goal is to measure the rate of change of a dependent variable, rather than a particular value. If the dependent variable can be expressed as a linear function of the independent parameter, then the slope and yintercept of an appropriate graph will give the rate of change and a particular value, respectively. An example of such an experiment in PHYS.203L/205L is the first part of Lab 20, in which weights are added to the bottom of a suspended spring (Figure 9). This experiment shows that a spring exerts a force Fs proportional to the distance stretched y = (y-yo), a relationship known as Hooke’s Law: Fs = – k(y – yo) (Eq. 1) where k is called the Hooke’s Law constant. The minus sign shows that the spring opposes any push or pull on it. In Lab 20 Fs is equal to (- Mg) and y is given by the reading on a meter stick. Masses were added to the bottom of the spring in 50-g increments giving weights in newtons of 0.49, 0.98, etc. The weight pan was used as the pointer for reading y and had a mass of 50 g, so yo could not be directly measured. For convenient graphing Equation 1 can be rewritten: -(Mg) = – ky + kyo Or (Mg) = ky – kyo (Eq. 1′) Procedure 1. On your spreadsheet note the tabs at the bottom left and double-click Sheet1. Type in “Basics,” and then click the Sheet2 tab to bring up a fresh worksheet. Change the sheet name to “Linear Fit” and fill in data as in this table. Hooke’s Law Experiment y (m) -Fs = Mg (N) 0.337 0.49 0.388 0.98 0.446 1.47 0.498 1.96 0.550 2.45 2. Highlight the cells with the numbers, and graph (Mg) versus y as in Steps 6 and 7 of the Basics section. Your Trendline this time will be Linear of course. If you are having trouble remembering what’s versus what, “y” looks like “v”, so what comes before the “v” of “versus” goes on the y (vertical) axis. Yes, this graph is confusing: the horizontal (“x”) axis is distance y, and the “y” axis is something else. 3. Click on the Equation/R2 box on the graph and highlight just the slope, that is, only the number that comes before the “x.” Copy it (control-c is a fast way to Figure 9. A spring with a weight stretching it Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com do it) and paste it (control-v) into an empty cell. Do likewise for the intercept (including the minus sign). SAVE YOUR FILE! 5. The next steps use the standard procedure for obtaining information from linear data. Write the general equation for a straight line immediately below a hand-written copy of Equation 1′ then circle matching items: (Mg) = k y + (- k yo) (Eq. 1′) y = m x + b Note the parentheses around the intercept term of Equation 1′ to emphasize that the minus sign is part of it. Equating above and below, you can create two useful new equations: slope m = k (Eq. 2) y-intercept b = -kyo (Eq. 3) 6. Solve Equation 2 for k, that is, rewrite left to right. Then substitute the value for slope m from your graph, and you have an experimental value for the Hooke’s Law constant k. Next solve Equation 3 for yo, substitute the value for intercept b from your graph and the value of k that you just found, and calculate yo. 7. Examine your linear graph for clues to finding the units of the slope and the yintercept. Use these units to find the units of k and yo. 8. Present your values of k and yo with their units neatly at the bottom of your spreadsheet. 9. R2 in Excel, like r in our lab manual and Corr. in the LoggerPro software, is a measure of how well the calculated line matches the data points. 1.00 would indicate a perfect match. State how good a match you think was made in this case? 10. Do the Homework, Further Exercises on Interpreting Linear Graphs, on the following pages. Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com Eq.1 M m f M a g               , (Eq.2) M slope m g       (Eq.3) M b f        Morgan Extra Pages Homework: Graph Interpretation Exercises EXAMPLE WITH COMPLETE SOLUTION In PHYS.203L and 205L we do Lab 9 Newton’s Second Law on Atwood’s Machine using a photogate sensor (Fig. 1). The Atwood’s apparatus can slow the rate of fall enough to be measured even with primitive timing devices. In our experiment LoggerPro software automatically collects and analyzes the data giving reliable measurements of g, the acceleration of gravity. The equation governing motion for Atwood’s Machine can be written: where a is the acceleration of the masses and string, g is the acceleration of gravity, M is the total mass at both ends of the string, m is the difference between the masses, and f is the frictional force at the hub of the pulley wheel. In this exercise you are given a graph of a vs. m obtained in this experiment with the values of M and the slope and intercept (Fig. 2). The goal is to extract values for acceleration of gravity g and frictional force f from this information. To analyze the graph we write y = mx + b, the general equation for a straight line, directly under Equation 1 and match up the various parameters: Equating above and below, you can create two new equations: and y m x b M m f M a g                Figure 1. The Atwood’s Machine setup (from the LoggerPro handout). Figure 2. Graph of acceleration versus mass difference; data from a Physics I experiment. Atwood’s Machine M = 0.400 kg a = 24.4 m – 0.018 R2 = 0.998 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 0.000 0.010 0.020 0.030 0.040 0.050 0.060  m (kg) a (m/s2) Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com 2 2 9.76 / 0.400 24.4 /( ) m s kg m kg s g Mm      To handle Equation 2 it pays to consider what the units of the slope are. A slope is “the rise over the run,“ so its units must be the units of the vertical axis divided by those of the horizontal axis. In this case: Now let’s solve Equation 2 for g and substitute the values of total mass M and of the slope m from the graph: Using 9.80 m/s2 as the Baltimore accepted value for g, we can calculate the percent error: A similar process with Equation 3 leads to a value for f, the frictional force at the hub of the pulley wheel. Note that the units of intercept b are simply whatever the vertical axis units are, m/s2 in this case. Solving Equation 3 for f: EXERCISE 1 The Picket Fence experiment makes use of LoggerPro software to calculate velocities at regular time intervals as the striped plate passes through the photogate (Fig. 3). The theoretical equation is v = vi + at (Eq. 4) where vi = 0 (the fence is dropped from rest) and a = g. a. Write Equation 4 with y = mx + b under it and circle matching factors as in the Example. b. What is the experimental value of the acceleration of gravity? What is its percent error from the accepted value for Baltimore, 9.80 m/s2? c. Does the value of the y-intercept make sense? d. How well did the straight Trendline match the data? 2 / 2 kg s m kg m s   0.4% 100 9.80 9.76 9.80 100 . . . %        Acc Exp Acc Error kg m s mN kg m s f Mb 7.2 10 / 7.2 0.400 ( 0.018 / ) 3 2 2           Figure 3. Graph of speed versus time as calculated by LoggerPro as a picket fence falls freely through a photogate. Picket Fence Drop y = 9.8224x + 0.0007 R2 = 0.9997 0 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 1.2 t (s) v (m/s) Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com EXERCISE 2 This is an electrical example from PHYS.204L/206L, potential difference, V, versus current, I (Fig. 4). The theoretical equation is V = IR (Eq. 5) and is known as “Ohm’s Law.” The unit symbols stand for volts, V, and Amperes, A. The factor R stands for resistance and is measured in units of ohms, symbol  (capital omega). The definition of the ohm is: V (Eq. 6) By coincidence the letter symbols for potential (a quantity ) and volts (its unit) are identical. Thus “voltage” has become the laboratory slang name for potential. a. Rearrange the Ohm’s Law equation to match y = mx + b.. b. What is the experimental resistance? c. Comment on the experimental intercept: is its value reasonable? EXERCISE 3 This graph (Fig. 5) also follows Ohm’s Law, but solved for current I. For this graph the experimenter held potential difference V constant at 15.0V and measured the current for resistances of 100, 50, 40, and 30  Solve Ohm’s Law for I and you will see that 1/R is the logical variable to use on the x axis. For units, someone once jokingly referred to a “reciprocal ohm” as a “mho,” and the name stuck. a. Rearrange Equation 5 solved for I to match y = mx + b. b. What is the experimental potential difference? c. Calculate the percent difference from the 15.0 V that the experimenter set on the power supply (the instrument used for such experiments). d. Comment on the experimental intercept: is its value reasonable? Figure 4. Graph of potential difference versus current; data from a Physics II experiment. The theoretical equation, V = IR, is known as “Ohm’s Law.” Ohm’s Law y = 0.628x – 0.0275 R2 = 0.9933 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 Current, I (A) Potential difference, V (V) Figure 5. Another application of Ohm’s Law: a graph of current versus the inverse of resistance, from a different electric circuit experiment. Current versus (1/Resistance) y = 14.727x – 0.2214 R2 = 0.9938 0 100 200 300 400 500 600 5 10 15 20 25 30 35 R-1 (millimhos) I (milliamperes) Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com EXERCISE 4 The Atwood’s Machine experiment (see the solved example above) can be done in another way: keep mass difference m the same and vary the total mass M (Fig. 6). a. Rewrite Equation 1 and factor out (1/M). b. Equate the coefficient of (1/M) with the experimental slope and solve for acceleration of gravity g. c. Substitute the values for slope, mass difference, and frictional force and calculate the experimental of g. d. Derive the units of the slope and show that the units of g come out as they should. e. Is the value of the experimental intercept reasonable? EXERCISE 5 In the previous two exercises the reciprocal of a variable was used to make the graph come out linear. In this one the trick will be to use the square root of a variable (Fig. 7). In PHYS.203L and 205L Lab 19 The Pendulum the theoretical equation is where the period T is the time per cycle, L is the length of the string, and g is the acceleration of gravity. a. Rewrite Equation 7 with the square root of L factored out and placed at the end. b. Equate the coefficient of √L with the experimental slope and solve for acceleration of gravity g. c. Substitute the value for slope and calculate the experimental of g. d. Derive the units of the slope and show that the units of g come out as they should. e. Is the value of the experimental intercept reasonable? 2 (Eq . 7) g T   L Figure 6. Graph of acceleration versus the reciprocal of total mass; data from a another Physics I experiment. Atwood’s Machine m = 0.020 kg f = 7.2 mN y = 0.1964x – 0.0735 R2 = 0.995 0.400 0.600 0.800 1.000 2.000 2.500 3.000 3.500 4.000 4.500 5.000 1/M (1/kg) a (m/s2) Effect of Pendulum Length on Period y = 2.0523x – 0.0331 R2 = 0.999 0.400 0.800 1.200 1.600 2.000 2.400 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 L1/2 (m1/2) T (s) Figure 7. Graph of period T versus the square root of pendulum length; data from a Physics I experiment. Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com EXERCISE 6 In Exercise 5 another approach would have been to square both sides of Equation 7 and plot T2 versus L. Lab 20 directs us to use that alternative. It involves another case of periodic or harmonic motion with a similar, but more complicated, equation for the period: where T is the period of the bobbing (Fig. 8), M is the suspended mass, ms is the mass of the spring, k is a measure of stiffness called the spring constant, and C is a dimensionless factor showing how much of the spring mass is effectively bobbing. a. Square both sides of Equation 8 and rearrange it to match y = mx + b. b. Write y = mx + b under your rearranged equation and circle matching factors as in the Example. c. Write two new equations analogous to Equations 2 and 3 in the Example. Use the first of the two for calculating k and the second for finding C from the data of Fig. 9. d. A theoretical analysis has shown that for most springs C = 1/3. Find the percent error from that value. e. Derive the units of the slope and intercept; show that the units of k come out as N/m and that C is dimensionless. 2 (Eq . 8) k T M Cm s    Figure 8. In Lab 20 mass M is suspended from a spring which is set to bobbing up and down, a good approximation to simple harmonic motion (SHM), described by Equation 8. Lab 20: SHM of a Spring Mass of the spring, ms = 25.1 g y = 3.0185x + 0.0197 R2 = 0.9965 0.0000 0.2000 0.4000 0.6000 0.8000 1.0000 0 0.05 0.1 0.15 0.2 0.25 0.3 M (kg) T 2 2 Figure 9. Graph of the square of the period T2 versus suspended mass M data from a Physics I experiment. Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com EXERCISE 7 This last exercise deals with an exponential equation, and the trick is to take the logarithm of both sides. In PHYS.204L/206L we do Lab 33 The RC Time Constant with theoretical equation: where V is the potential difference at time t across a circuit element called a capacitor (the  is dropped for simplicity), Vo is V at t = 0 (try it), and  (tau) is a characteristic of the circuit called the time constant. a. Take the natural log of both sides and apply the addition rule for logarithms of a product on the right-hand side. b. Noting that the graph (Fig. 10) plots lnV versus t, arrange your equation in y = mx + b order, write y = mx + b under it, and circle the parts as in the Example. c. Write two new equations analogous to Equations 2 and 3 in the Example. Use the first of the two for calculating  and the second for finding lnVo and then Vo. d. Note that the units of lnV are the natural log of volts, lnV. As usual derive the units of the slope and interecept and use them to obtain the units of your experimental V and t. V V e (Eq. 9) t o    Figure 10. Graph of a logarithm versus time; data from Lab 33, a Physics II experiment. Discharge of a Capacitor y = -9.17E-03x + 2.00E+00 R2 = 9.98E-01 0.00 0.50 1.00 1.50 2.00 2.50

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