Lab 5

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School

Pennsylvania State University *

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111

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Aerospace Engineering

Date

Oct 30, 2023

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docx

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HOOKE’S LAW – PART A INTRODUCTORY INSTRUCTIONS Lab goals: In this lab, you will 1) Investigate the nature of the spring force and how it depends on the stretching of a spring. Equipment needed: 1) IOLab with computer/software, 2) the long spring that comes with it 3) You will need to be able to attach the free end of the spring to a heavy book. We have found that binder clip works well. If you look carefully at the end of the spring you will see that it is not closed completely and you can slide the curve end of the string into the binder clip handles. General experimental setup. To measure position, we will need the wheel sensors. 1. In the first part of the lab (Part A), you will need to have the IOLAB horizontally on the table. 2. In the second part (Part B, next week), you will need the IOLAB to hang and to be able to oscillate freely back and forth. To do so, you will need a “ramp”. You can use any surface such as a book and you can choose any angle for the “ramp”. The image below shows an almost vertical ramp. Your angle can be much less steep. ** To prevent the spring from breaking, DO NOT, leave the spring stretched for long times. Phenomenon to Observe: In the horizontal case, we are directly measuring the force vs how far the spring stretches. In the angle case, we will have a dynamical situation where the IOLAB oscillates under the influence of gravity, the spring force and a little bit of rolling friction force. The two situations are very different but will the spring force behave differently?

Hooke’s Law – Part A Page 2 of 6 Primary Guiding Questions to think about: 1. Which way is the spring force acting? Is it always coming from stretching or do we have compression also happening. 2. Assuming Hooke’s law to be valid, what are the best fit value for the spring constant and for the equilibrium position? Where is the equilibrium position of the spring, does it make sense? 3. Is there a statistically significant difference between the fit for the spring constant k and for the equilibrium position y e in the horizontal or vertical setups? OBSERVING EXPERIMENT, MEASURING DIRECTLY FORCE AND POSITION First calibrate the force sensor and set up the IOLAB on the table horizontally as shown in the image below. Remember to regularly zero the force sensor and the position (wheel) sensor by hitting on the “zero sensor” button. You should zero the sensor right after your start recording. 1. Look carefully at your spring and try to pull a couple of time. Describe, without trying to explain , what you observe visually. Do not take data, just make notes on your spring and how it behaves. * * In this type of question, you should write everything that could be important for any future investigations, you are making a report on everything you see and notice. Write down even small seemingly unimportant details. Do not seek to explain yet! Inadequate Adequate Good Is able to describe what is A description is incomplete. Or, A description is complete but Clearly describes what happens in the

Hooke’s Law – Part A Page 3 of 6 observed without trying to explain. observations are adjusted to fit expectations. mixed up with explanations or pattern. experiments.  When pulling from the book the cart glides with easy with no noticeable stretch on the spring, when pulling from the cart the spring stretches quite a bit before the book begins to move. 2. We are interested in investigating any possible relationship between the force and the position of the IOLAB (which should be directly related to the stretching of the spring). We will use scatterplot on StatKey for two quantitative variables (position and force). Explain how you will use the equipment to make the scatterplot. How many data points do you want to record? For instance, if you wish to know both the force applied on the force sensor and the position of the IOLab at the same time, make sure that the statistical tool is selected, and hover at the time at which you are interested in these quantities. The value of that quantity will be indicated in the top right corner of the graph. Make sure that you record corresponding quantities at the same time . Inadequate Adequate Good Is able to describe how to use available equipment to make measurements All chosen measurements can be made, but no details are given about how it is done. All chosen measurements can be made, but the details of how it is done are vague or incomplete. All chosen measurements can be made and all details of how it is done are clearly provided.  I will be making a scatterplot measuring the force exerting on the IOLab cart against the location of the cart. The sensors I will be using to make these measurements will be the force sensor and the wheel sensor using only the position portion of the wheel sensor. I will start the recording holding both the book and the cart pulling the cart away from the book. I will then release the book and take measurement of the average force exerted by the spring and the spring length at release.

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Hooke’s Law – Part A Page 4 of 6 3. Collect your data and upload it to StatKey. Make sure that the range of positions you record is appropriate: the spring needs to be stretched at all times. Take a screenshot of the scatterplot and of the descriptive statistics (on the right of the scatterplot). Two different screenshots are easier to see. Graded on correctness. 4. Analyze your data and make sense of it. Write down any notes here. Inadequate Adequate Good Is able to identify a pattern in the data The pattern described is irrelevant or inconsistent The pattern has minor errors or omissions. The patterns represent the relevant trend in the data. When possible,

Hooke’s Law – Part A Page 5 of 6 with the data the trend is described in words.  The average force exerted was 2.993 N from an average spring length of 0.290 m. The data supports the equation F= -kx. Any data that lower than expected is most likely due to human error. 5. If we assume Hooke’s law is valid, what is the meaning of the slope and intercept of the regression line? Can you extract the spring constant k and the equilibrium position y e ? Write down any equations here that explains how you did the fit.  Assuming Hooke’s law is valid the intercept is the equilibrium position of regression line and the slope is the spring constant k.  The equations used were: Δs = s - s eq and k= F / x , where Δs = x when s eq is set to zero  When the intercept is set to 0 the slope reads a more accurate spring constant of 10.359 SPRING IN A DYNAMICAL SYSTEM. We will now do a little bit of preparation for the next lab by working a bit on the theory first. We are interested in investigating the property of the spring force on an inclined ramp. The goal here is to NOT use the force sensor but instead relate the acceleration of a moving IOLAB to its position. The question will be whether the properties of the spring (such as the spring constant k and the equilibrium value y e ) remains the same when it is oscillating at an angle rather than horizontally. Below, you will set up the IOLab on an angled surface, so that it can freely oscillate. We can neglect friction . 6. Draw a free-body diagram for the IOLab held by a spring on an incline ramp neglecting friction. (you can draw by hand and upload the picture or you can quickly create an image of the relevant forces).

Hooke’s Law – Part A Page 6 of 6 7. Using the coordinate system attached to the IOLab (with wheels down and spring attachment up), write Newton’s second law along the ramp.  F y = mgsinθ - F s 8. Write down an expression for a y as a function of y.  A y = gsinθ + k / m * y 9. If you were to write a y = c 1 y + c 2 , what are the expressions for c 1 and c 2 ? How do you relate those to the quantities appearing in a linear fit?  C 1 = k / m  C 2 = gsinθ Check your expressions above with your classmates or with the course instructor on the discussion board in order to ensure that your equations are correct for the next part of the lab.

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