Lab Hooke’s Discussion Assignment

Lab Hooke’s Discussion AssignmentPHY 105 Name____________________Hooke’s 1LAB: Hooke’s ObjectiveTo determine the direct relationship between force and displacement for ideal springs, thespring constant, and the work done to stretch the springs. You will compare data for twodifferent springsMaterials✔ Computer✔ PhET website✔ Google Sheets or MS ExcelTheoryIf a mass on an ideal massless spring is displaced from the equilibrium position(x0 = 0) to a new position x, Hooke’s law states that the spring will exert a restoring forceon the mass as follows.Eq. 1The “-” sign indicates that the direction of the spring force is in the direction opposite tothe direction of the displacement, or a restoring force. The value “k” (called the springconstant, with units of N/m) is a constant for a given spring, but different springs havedifferent “k-values.” Thus, the force exerted by a spring is variable, specifically thegreater distance it is stretched from equilibrium, the greater is the spring force attemptingto restore the spring to its equilibrium position. This relationship holds up to a pointcalled the elastic limit. Each spring has its own value of this limit. If you stretch a springbeyond its limit, then the spring will not return to its original shape, but will remainstretched out.Not all “springy” things obey Hooke’s . When a rubber band is stretched, therubber will exert a restoring force. The amount of this force depends on the amount thatthe rubber is stretched, but it is not a simple proportional relationship.Work is a measure of energy input to or energy output from a system. When theforce exerted on a system is constant, the quantity of work done by the force is easy todetermine,Eq. 2Work can have a negative value or a positive value. The value is negative when F pointsin the direction opposite to the displacement, and the value of work is positive when Fpoints in the same direction as the displacement.When, as in the case of a spring, the force is variable, the total work done by aforce cannot be simply calculated using this formula, but must instead be calculated by anintegral.Eq. 3Hooke’s 2For an ideal spring, this becomes as followsEq. 4If your force is always in the same direction as the displacement, the dot product may betreated as simple multiplication. In addition, for sufficiently closely-spaced data points,the integral can be treated as a summation and the total work can be calculated piecewise. If you measure the force at many positions, xn, between x0 = 0 and xf, you canestimate the amount of work done during any interval. The total work done in movingthe system from x0 to xf is then the sum of work done in the little intervals. The estimateof the work done in a small interval is the average force in that interval times the distancein that interval.Eq. 5Eq. 6In general, the area of a trapezoid is given by the formula:????????????? = 12 (????1 + ????2) ∗ ℎ???ℎ? Eq. 7Procedure1. Go to the following website: https://phet.colorado.edu/sims/html/masses-and- springs/latest/masses-and-springs_en.html2. Pick “Intro”. 3. Play around with this version for a bit. Please try the following:a. Put a large mass on one spring, and a small mass on the other spring. Which spring bounces faster (high frequency)? You can use thestopwatch (in the box on the right) if you want to time 10 bounces tocompare.b. Hit the stop button for both springs without removing the masses to stop the bouncing. Which mass has stretched the spring down farther (largedisplacement, x)? You can use the ruler (in the box on the right), or youcan turn on the Natural Length and Equilibrium Position lines, but youdon’t have to do this.c. Hit the reset button in the bottom right (orange with a circle arrow). Use the slider to change one the spring constant of one spring to large, and theother to small. Put the two 100g masses, one on each spring. Now whichspring bounces faster?https://phet.colorado.edu/sims/html/masses-and-springs/latest/masses-and-springs_en.html
https://phet.colorado.edu/sims/html/masses-and-springs/latest/masses-and-springs_en.html
Hooke’s 3d. Hit the stop button again. Which spring is stretched further?e. If you had an unknown weight, what are two ways you could determine its weight using the springs?f. Make a guess at the mass of the blue unknown weight. Explain your reasoning.4. Now switch to the “Lab” setup — look at the bottom edge, you should see four tabs, Intro, Vectors, Energy, and Lab. Pick Lab.5. In this version of the setup, you’ll see that you can change the orange mass using the slider at the top, or can type in different masses.6. Pick a value for the spring constant, and write down the position of the spring constant slider (you can number the tickmarks 1 at the left and 10 at the right).Your spring constant slider value is: __________7. In the upper right, click to turn on the “Displacement and Natural Length” checkbox. It may be useful to also turn on the “Movable Line” checkbox.8. Drag the ruler from the right to next to the spring.9. Select 8 different masses that you will investigate for your spring, within the range of masses available.10. For each mass, record the mass, and the displacement in Table 1.11. Calculate the force, making sure to use the mass in kg to do so.12. In MS Excel or Google Sheets, create a graph of F (vertical axis) vs. x (horizontal axis).13. As in the Graphing lab, fit a trendline to your graph, and display the equation and the R^2.14. Determine the slope of your trendline. This slope is the spring constant k in N/m.15. Using your trendline and either Equation 3 or Equation 4, determine a formula for the work. From this formula calculate the total work done to stretch the spring.Hooke’s 416. Using your data and Equations 5 and 6 (that is, area under the curve using trapezoids), calculate the total work done to stretch the spring.17. Determine the percent difference between this value and the one from Step 15.18. If your percent difference is over 20%, you probably have an error.19. Share data for steps 6-16 with a classmate who picked a different value for the slider. Make sure to use their name in your lab report.QuestionsIn your lab report, include the questions from Step 3 with letters A-F.1. Did you or your classmate (Step 19) have a stiffer spring (larger spring constant)?2. Which spring stretched farther, the stiffer spring or the less stiff spring?3. Which spring had a graph with a steeper slope?4. Why do Steps 15 and 16 give different values? Which do you think is closer to the actual value, and why?5. In the real world, what do you think would happen to the graphs for the springs if you continued to add mass until the metal started bending?6. In the real world, what do you think would happen to the graphs for the springs if you continued to add mass until the metal broke?7. Rubber bands do not quite follow the pattern established by your springs. If you had a rubber band, at first when you add just a small amount of mass, it would lengthen alot as it straightens out. Then it would behave similarly to a spring. And lastly itwould break. Sketch what you think this force vs. displacement graph would looklike.Hooke’s 5Data Table 1: Spring #1Spring Constant slider value: __________# Added mass ( ) Force applied to thespring ( )Displacement of springfrom equilibrium( )0 0 0 012345678Spring Constant #1 from slope of trendline = _______________(______)Work done according to Trendline andintegration (Step 15)_________________(_____)Work done according to data andsummation (Step 16)_________________(_____)Percent difference between the two values_________________(_____)Hooke’s 6Data Table 2: Spring #2Copy this data from a classmate.Spring Constant slider value: __________# Added mass ( ) Force applied to thespring ( )Displacement of springfrom equilibrium( )0 0 0 012345678Spring Constant #2 from slope of trendline = _______________(______)Work done according to Trendline andintegration (Step 15)_________________(_____)Work done according to data andsummation (Step 16)_________________(_____)Percent difference between the two values_________________(_____)Lab developed by Andria Schwortz from a lab by Ray Johnson (Quinsigamond Community College,Worcester, MA).

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