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3 Transformation Of Mechanical Energy Introduction and Theory One of the most powerful laws in physics is the Law of Conservation of Energy. Energy cannot be created or destroyed: it may be transformed from one form into another, but the total amount of energy never changes. Energy can exist in many forms, including heat, light, sound, electrical, nuclear and mechanical. When an object is lifted to a position of rest from the Earth's surface, work is done on the object. The force required to only lift the object is equal to the object's weight. The work done in lifting the object is equal to its weight multiplied by its height above the ground. The object is now said to possess a gravitational potential energy (GPE), which is equal to the work done in lifting it to that height: Gravitational Potential Energy = GPE = weight height = m g h If you let the object fall back toward the ground, it will accelerate downward. As seen in the above equation, as the object's height decreases, so does its potential energy. When the object's gravitational potential energy decreases, its kinetic energy (KE) will increase. The Law of Conservation of Energy says the loss of gravitational potential energy equals the gain in kinetic energy. The kinetic energy of a moving object is: Kinetic Energy = KE = ½ mass velocity 2 = ½ m v 2 This form of energy conservation is true, provided that none of the object's potential energy is transformed into some other form of energy, such as heat due to friction. Apparatus and Procedure In this experiment you will verify the Law of Conservation of Energy for a simple system using last week s Inclined Planes equipment. A cart will start from the top of the plane and then roll down. At the top, the cart will have its maximum GPE. As the cart travels down the plane it will lose GPE and gain KE. Ideally, the GPE lost will equal the KE gained. For a more detailed, step-by-step procedure, please turn to the page 26. For today s data sheet, either fill out the data sheet in the lab manual or use the Excel file that was ed to you in the beginning of the week. You are strongly encouraged to use the ed data sheet due to the number of calculations in this week s lab. Be sure to sign the attendance sheet before you leave class. 23

4 Relevant Data Sheet Information V!"#$% = 0.05m time Gravitational Potential Energy = GPE = mass of cart gravity height = m c g h Kinetic Energy = KE = ½ cart's mass Velocity Squared = ½ m c Velocity 2 Percent Error = (KE +GPEend) - GPE GPE 100% Questions Q1: (A) Looking at your data, what happens to the KE the longer you let the cart roll down the inclined plane. (B) Why is that? Q2: Make a graph, using the graph paper on page 27 to show what happens to the GPE and KE at distances of 0 cm, 50 cm, 100 cm, 150 cm, and 220 cm. Q3: Looking at your graph, does it support the hypothesis that the sum of the GPE and KE of a system always equals the total energy available. Q4: Evaluating the final column of your data table, do any of the values fall outside a 5% error margin? If any do, list the trials and explain why. Be specific in listing sources of human error. Q5: (A) Where could frictional effects have come in to "absorb" some of the gravitational potential energy put into the system? (You know there is friction in the system because the cart will eventually come to a stop on its own accord.) (B) How would this affect your timing data, your speed, and your KE? Q6: Based on the Conservation of Energy theorem, is it possible to get more kinetic energy out of your system than the gravitational potential energy you put into it? Explain why or why not. Q7: For all your trials where the margin of error is under 5%, where did the "missing" energy go and as what form? Q8: Under ideal circumstances if you were to let the cart roll to the bottom of the inclined plane (where the height = 0), what would be the cart s KE? 24

5 Q9: (A) If you were to do the experiment again, placing a 2.0 kg mass instead of a 0.5 kg mass on your cart, how would your timing data, velocity, and KE change? (B) Why or why not would energy still be conserved? 25

6 Mechanical Energy: Procedure Checklist Determining Heights of Plane and Mass of Cart o Place the plane on the 4x4 and 2x4 blocks of wood o Use the calipers to measure the heights at 20 cm, 70 cm, 120 cm, and 170 cm o Record the Angle of Inclination, using the Angle Indicator o Place the plane on the 4x4 block of wood o Use the calipers to measure the heights at 20 cm, 70 cm, 120 cm, and 170 cm o Record the Angle of Inclination, using the Angle Indicator o Use the electronic scale to find the mass of the cart and foam board o Use the electronic scale to find the mass of the cart, the added mass, and foam board Determining the Travel Time o Adjust the plane to its appropriate height o Place the photogate gate 5 cm in front of the appropriate end point o Keep photogate cables out of the way of the rolling cart o Place something to catch the cart at the end of the plane o Set the left-hand switch to "GATE" o Set the "MEMORY" switch to "OF" o Set the unit switch, if available, to "1 ms" o Place the cart front aligned with the 20 cm mark on the ramp o Press the "RESET" button and zero out the display o Release the cart o Record the timing data on your data sheet, being sure not to round off o Repeat until all timing data is collected o Pay close attention to the title of each data table to see how high up the plane should be and whether or not there is extra mass on the cart When finished collecting your timing data o Turn off the photogate timer o Place the equipment at your lab station as you found it o If time allows, review over the lab manual s questions o Next week turn in the completed data sheet and answers to all questions o Be sure to sign the attendance sheet before you leave lab today 26

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9 (GPE End + KE) Mass of Cart w/o Added Mass cm 70 cm cm cm Angle (mm) (mm) (mm) (mm) ( o ) (g) 4x4 and 2x4 Blocks (kg) 0 0 4x4 Block Mass of Cart w/ Added Mass 4x4 Block & 2x4 Block without Extra Mass on Cart Start Ht. Start Ht. End Ht. End Ht. Time V Final GPE Start GPE End KE GPE Start - (GPE End + KE) Percent Error GPE Start and # (mm) (m) (mm) (m) (s) (m/s) (J) (J) (J) (J)

10 (GPE End + KE) (GPE End + KE) 4x4 Block & 2x4 Block with Extra Mass on Cart Start Ht. Start Ht. End Ht. End Ht. Time V Final GPE Start GPE End KE GPE Start - (GPE End + KE) Percent Error GPE Start and # (mm) (m) (mm) (m) (s) (m/s) (J) (J) (J) (J) x4 Block without Extra Mass Start Ht. Start Ht. End Ht. End Ht. Time V Final GPE Start GPE End KE GPE Start - (GPE End + KE) Percent Error GPE Start and 30 # (mm) (m) (mm) (m) (s) (m/s) (J) (J) (J) (J)

11 (GPE End + KE) 4x4 Block with Extra Mass Start Ht. Start Ht. End Ht. End Ht. Time V Final GPE Start GPE End KE GPE Start - (GPE End + KE) Percent Error GPE Start and # (mm) (m) (mm) (m) (s) (m/s) (J) (J) (J) (J)

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13 Hooke's Law Introduction and Theory An important property of solids is their "stretchiness" or "squeeziness," which is called elasticity. In the case of many solids, the amount of stretch or squeeze, x, is proportional to the force causing the stretch or squeeze, F. This relationship can be expressed as: F x which is read as "force is proportional to stretch (or squeeze)". To change this expression into an equation, a constant of proportionality must be included. The expression now takes this form, which we call Hooke s Law: F = -kx where k is called the spring constant or stiffness constant. The value for k depends on the material being stretched or squeezed. Your goal in this experiment is to see if the spring on the apparatus obeys Hooke's Law and to find out how much potential energy is stored in your spring. Procedure The apparatus consists of a weight holder attached to a spring. A pointer that is attached to the weight holder enables you to mark the distance the spring moves when weights are placed on the weight holder. First, move the scale next to the weight holder to line up the pointer on the weight holder with the zero on the scale. Next hang a mass on the holder and record the distance the pointer moves. Look at the pointer horizontally to read the position of the pointer along the scale. (Please see Supplemental Appendix G on the course website for an illustration of parallax, a potential source of error in this lab.) Do this for 20 different masses, recording the information on your data table. Do not use 55 grams! Do not use more mass than is recordable on the spring s scale when collecting your data. When you have completed your measurements, be sure to remove all masses from the spring so as not to leave it stretched overnight. Be sure to sign the attendance sheet before you leave class. 33

14 Relevant Data Sheet Information 1,000 grams = kilograms 100 centimeters = 1.00 meters Force (in Newtons) = m a = m g In this case, g is the acceleration due to Earth's gravity and is 9.8 m/s 2. The m is the mass on the holder stretching the spring and should be in kilograms. Stored Elastic Potential Energy = 1/2 k x 2 Note: You will need to answer Q2 and Q3 before using this equation to answer Q8. Questions Q1. What error did the mirror-backed scale of the Hooke's Law apparatus help eliminate from your measurements? Q2. Make a graph of Force vs. stretch using the graph paper on either page 39 or page 41 of the manual based on the largest Force you have. If your spring obeys Hooke's Law, the points on your graph should lie along a straight line. Q3. On your graph, draw the "best fit" straight line, then compute the slope of your line using the formula: slope = change in Force / corresponding change in stretch = (F 2 F 1 ) (x 2 x 1 ) where F 1 is the force corresponding to the stretch distance x 1 and where F 2 corresponds to the stretch x 2. Choose your 2 data points so there are at least 10 data points between them. Show your work for the calculation. Q4. (A) Why is this slope your best experimental value for k, the spring constant for your spring? (B) What is the unit for k, your spring constant, in the metric system? Q5. Suppose you were doing the same experiment but started with the maximum weight suspended from the spring and then took additional readings as weight was removed. Would the slope of your graph still be the same? Explain why or why not. 34

15 Q6. (A) Use only your graph to extrapolate what the extension of your spring would be, if a 55 g mass is supported from the spring under theoretical conditions. Be sure to show the corresponding point on your graph. Do not do this using the spring in lab. You are to use your graph only. (B) Next, rearrange the equation F = kx to find the spring s stretch, if a 55 g mass is supported from the spring under theoretical conditions. Recall that your spring constant, k, was determined in Q3. (C) How close are your answers from Q6-A and Q6-B? (D) Which method do you think is more accurate for determining the spring s stretch under theoretical conditions? Explain why. Q7. (A) If a force of 50 N is applied to spring A with a spring constant of 2.5 N/m and to spring B with a spring constant of 1.0 N/m, which spring stretches the furthest? (B) How did you determine this? Q8. The final column of your data table shows the Elastic Potential Energy stored in your spring. What would happen if the potential energy was 4 times larger and your spring constant remained the same? Would the distance stretched remain the same; double; quadruple? Be sure to explain how you came up with your answer. 35

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17 Hooke's Law Data Sheet Trial Mass ( g ) Mass ( kg ) Force ( N ) Stretch ( cm ) Stretch ( m ) Elastic Potential Energy ( J ) Carry all calculations out to 3 decimal places only. 37

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