Conservation of Elastic Potential Energy
Inquiry Lab Kit for AP® Physics 1
Materials Included In Kit
Firing ramps, 6
Spheres, steel, 1" dia., 6
Additional Materials Required
(for each lab group) Balance, 0.01g precision (may be shared) Clamp holder Meter stick
Ruler, metric Support rod Support stand Textbooks, 3–4 (optional)
Safety Precautions
Projectiles may be inadvertently launched during this activity. Wear safety glasses. Follow all laboratory safety guidelines.
Disposal
All materials may be saved and stored for future use.
Lab Hints
 This laboratory activity can be completed in two 50minute class periods. It is important to allow time between the Introductory Activity and the GuidedInquiry Activity for students to discuss and design the guidedinquiry procedures. Also, all studentdesigned procedures must be approved for safety before students are allowed to implement them in the lab. Prelab Questions may be completed before lab begins the first day.
 This lab can be run with a springloaded cart and a flat inclined plane, though we found the data difficult to reproduce. A photogate timer may be used to more quantitatively assess the sphere’s motion.The noncompressed spring length, determined by disassembling the firing mechanism, was found to be 10.4 cm. When compressed to the first firing notch, the spring measures 7.1 cm. The difference between the noncompressed and compressed lengths should be taken as x (3.3 cm) for the first firing position. The x values for the remaining firing positions may be determined by measuring the distance between the midpoints of adjacent firing positions. We found a 1.1 cm distance between each firing position using this method. Thus, x for the second firing position may be determined by the following: 7.1 cm –1.1 cm = 6.0 cm, the length of the compressed spring at the second firing position; 10.4 cm – 6.0 cm = 4.4 cm, the compression distance at the second firing position.
 It is helpful to clamp the ends of the ramp to a metal support rod affixed to a support stand to keep the ramp steady during firings.
 The fourth firing position can propel the sphere with significant force, enough to reach the ramp’s end at high angles. If your support stand is not tall enough to elevate one end of the ramp to such a degree that the fourth firing position does not result in misfires (travel distances exceeding the ramp length), the support stand may be placed atop a book to achieve a greater slope.
 The exact angle the ramp makes with the table surface is neither easy to measure nor a necessary piece of data. It is important that a variety of slopes be used. The slope may be varied in a quantifiable way by simply measuring the distance between the ramp’s elevated end and the table surface.
 It is useful to conduct a large number of trials for each firing position and ramp slope to reduce the effects of error associated with catching the sphere at its apex. It is difficult to catch the sphere without impeding or promoting its movement. Recording data from a large number of trials will reveal significant outliers that can be eliminated from data sets.
 A good ramp slope for measuring the maximum sphere height at all firing positions can be obtained by elevating one end of the ramp to approximately 73 cm above the tabletop.
 Significant energy loss can be attributed to the shaking and rattling of the track when the spring is released, especially at the higher compression distances. Students should be aware of this limitation as they complete the activity and also take the limitation into account when discussing sources of error.
 To determine the effect of friction on the final height of the sphere, students can conduct more trials at various ramp elevations. In our trials, we found that the change in height of the sphere was less at smaller elevations than at larger elevations. This can be attributed to energy lost due to friction. Friction does work over the distance the sphere travels along the track. For greater distances the sphere travels along the track, more energy is lost as thermal energy and/or sound. For shorter travel distances, less energy is lost. Higher ramp elevations give more reliable changes in height of the sphere when using the two smallest spring compression distances. To achieve reliable change in height measurements for the two largest compression distances, the ramp needed to be nearly vertical. This is not an advisable procedure for students to follow because the sphere can take an unpredictable path when launched vertically.
 The experiment serves as a qualitative assessment of the law of conservation of energy but breaks down under rigorous mathematical treatment.
Teacher Tips
 Students may have difficulty coming to the conclusion that the ramp’s slope should not affect the sphere’s PE_{g}. This is the case because students will observe differences in maximum sphere height for the same firing position. Reassure students that the slope should not affect sphere height according to the mathematical equation for PE_{g}, which does not include a trigonometric term to account for slope.
 The change in potential energy of the spring may also be explained as the work done on or by the spring.
Further Extensions
Alignment to the Curriculum for AP® Physics 1
Enduring Understandings and Essential Knowledge Interactions with other objects or systems can change the total energy of a system. (4C) 4C1: The energy of a system includes its kinetic energy, potential energy, and microscopic internal energy. Examples should include gravitational potential energy, elastic potential energy, and kinetic energy.
Learning Objectives 4C1.1: The student is able to calculate the total energy of a system and justify the mathematical routines used in the calculation of component types of energy within the system whose sum is the total energy. 4C1.2: The student is able to predict changes in the total energy of a system due to changes in position and speed of objects or frictional interactions within the system.
Science Practices 2.2 The student can justify the selection of a mathematical routine to solve problems. 3.1 The student can pose scientific questions. 4.1 The student can justify the selection of the kind of data needed to answer a particular scientific question. 4.2 The student can design a plan for collecting data to answer a particular scientific question. 4.3 The student can collect data to answer a particular scientific question. 4.4 The student can evaluate sources of data to answer a particular scientific question. 5.1 The student can analyze data to identify patterns or relationships. 5.2 The student can refine observations and measurements based on data analysis. 5.3 The student can evaluate the evidence provided by data sets in relation to a particular scientific question. 6.1 The student can justify claims with evidence. 6.2 The student can construct explanations of phenomena based on evidence produced through scientific practices. 6.3 The student can articulate the reasons that scientific explanations and theories are refined or replaced. 6.4 The student can make claims and predictions about natural phenomena based on scientific theories and models. 7.2 The student can connect concepts in and across domain(s) to generalize or extrapolate in and/or across enduring understandings and/or big ideas.
Correlation to Next Generation Science Standards (NGSS)^{†}
Science & Engineering Practices
Planning and carrying out investigations Analyzing and interpreting data Using mathematics and computational thinking Constructing explanations and designing solutions
Disciplinary Core Ideas
HSPS3.A: Definitions of Energy HSPS3.B: Conservation of Energy and Energy Transfer
Crosscutting Concepts
Patterns Cause and effect Scale, proportion, and quantity Energy and matter
Performance Expectations
HSPS31. Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known. HSPS32. Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motion of particles (objects) and energy associated with the relative position of particles (objects). HSPS33. Design, build, and refine a device that works within given constraints to convert one form of energy into another form of energy.
Answers to Prelab Questions
 The unit for potential and kinetic energy is the joule, J. The joule is a derived unit and is equal to kg∙m^{2}/s^{2}. Use Equations 2, 3 or 4 from the Background to derive the joule.
From Equation 4: ΔPE_{g} = mgΔh Substitute in the units for mass (kg), acceleration due to gravity (m/s^{2}) and change in height (m): (kg)(m/s^{2})(m) = kg∙m^{2}/s^{2} = J
 Use the law of conservation of energy to write a mathematical equation describing how the change in potential of a spring is related to the change in gravitational potential energy.
–ΔPE_{s} = ΔPE_{g} –½k_{x}^{2} = mg(h_{f} – h_{i})
 Consider an experiment in which a box is propelled up a sloped ramp by a springloaded firing mechanism. Using relative terms, such as low, medium and high, and quantitative terms, such as zero, describe the amount of elastic potential energy (PE_{s}), kinetic energy (KE) and gravitational potential energy (PE_{g}), of the box at the following points on the ramp.
 When the box is in contact with the compressed spring.
PE_{s} = high, KE = zero, PE_{g} = zero
 Immediately after the box loses contact with the now uncompressed spring.
PE_{s} = zero, KE = high, PE_{g} = zero
 When the box is about halfway up the ramp.
PE_{s} = zero, KE = medium, PE_{g} = medium
 When the box is at its highest point on the ramp.
PE_{s} = zero, KE = zero, PE_{g} = high
Figure 1 shows the experiment described in the previous question. The center of the box is at a height of 0 m. The box has a mass of 2.20 kg. The spring constant of the spring is 400.0 N/m.
 Calculate the ΔPE_{s} when the spring is compressed by 0.15 m from its equilibrium state.
ΔPE_{s} = –½kx^{2} ΔPE_{s} = –½ x 400.0 N/m x (0.15 m)^{2} ΔPE_{s} = –4.5 J
 The spring is released and propels the box up the ramp. Determine the maximum height the box will reach using the principle of conservation of energy.
–ΔPE_{s} = ΔPE_{g} = mg(h_{f} – h_{i}) –(–4.5 J) = (2.20 kg) x (9.81 m/s^{2}) x (h_{f} – 0 m) h_{f} = 0.21 m
The spring in Figure 1 is removed and a new spring is put in its place. When the spring is compressed by 0.15 m, the box reaches a maximum height of 0.61 m.
 Calculate the spring constant of the new spring.
–ΔPES = ΔPE_{g} –½kx^{2} = mg(h_{f} – h_{i}) k = 2mg(h_{f} – 0 m)/x^{2} k = 2 x (2.20 kg) x (9.81 m/s^{2}) x (0.61 m)/(0.15 m)^{2} k = 1170 N/m
 Discuss sources of error in the design of the experiment that might account for the experimental value of the spring constant being less than the actual or true value.
As the box moves across the surface of the ramp, some energy may be lost due to friction. It is not stated that the ramp is frictionless. Therefore, as the box slides, some energy will be transformed into heat and/or sound due to friction. If energy is lost while the box moves up the ramp, then it will not attain its predicted height based on the compression of the spring. Similarly, using the final height as a measure of the spring constant will result in the spring constant being less if the box goes to a lower height than expected.
Sample Data
Introductory Activity
Sample Data for Notch 1
{13787_Data_Table_1}
Analyze the Results
When the ramp was at an elevation of 45.5 cm, the distance the sphere traveled along the track increased as the compression distance increased. It appeared as though the sphere traveled approximately four times as far on the second notch when compared to the first notch. The fourth compression distance (notch 4) propelled the sphere all the way to the top of the ramp, hitting the bumper. GuidedInquiry Activity
{13787_Data_Table_2}
{13787_Data_Figure_3}
Slope = 544.71 m ^{–1}Slope = k/2mgk = 2mg x slope = 2(0.067 kg)(9.81 m/s ^{2})(544.71 m ^{–1}) k = 716 N/m For Notch 4:
Answers to Questions
GuidedInquiry
 Did the sphere always return to the same place on the ramp when the compression distance and angle remained constant? If no, explain a possible cause for the variation.
The sphere did not always return to the same place on the ramp when the compression distance and angle remained constant. There were some instances of the launch handle getting caught on a finger and having an impeded release. By not having a smooth release, some of the potential energy of the spring was put into the finger. In all cases, there was friction between the sphere and the track. This is evidenced by the sound of the sphere going over the track. The sound of the sphere traveling over the track was not the same between each trial. In some trials, the sound was louder indicating that friction was not the same between each trial.
 How does the angle of the ramp affect the distance travelled by the sphere when the compression distance remains constant?
When the ramp is at a smaller angle, the sphere travels a greater overall distance along the track. When the ramp is at a greater angle, the sphere travels a shorter overall distance.
 How does the angle of the ramp affect the change in height of the sphere when the compression distance remains constant? Justify your answer with data collected from the Introductory Activity.
The angle of the ramp does not have a great effect on the change in height of the sphere. Although the maximum heights for the different angles varied, the change in height remained constant. Between the three ramp angles in the Introductory Activity, the change in heights were fairly close, ranging from 2.5 cm to 3.1 cm. When the ramp elevations were 45.5 cm and 60.0 cm, the change in heights were nearly equal at 3.0 cm and 3.1 cm, respectively. According to the law of conservation of energy, when the energy stored in the spring is released, it will result in a change in the gravitational potential energy of the sphere. These two energies must be equal, –ΔPE_{s} = ΔPE_{g}, that is –½kx^{2} = mg(h_{f} – h_{i}). There is no angle variable in the equation.
 Consider the height measurements made in Part B of the Introductory Activity. Which ramp angle allowed for greater reproducibility in the measurements of height? Explain.
It was easier to measure the final height of the sphere when the ramp was at a smaller angle. At the larger angles, it was more difficult to determine exactly how high the sphere was traveling. The change in motion of the sphere at its peak position occurred too quickly to measure the height. Even after launching the sphere numerous times at the same angle, the recorded height is an assumed average of all of the trials. At the smaller angle, the change in motion of the sphere at the peak position occurred more slowly. It was easier to make minor adjustments to the position of the meter stick as the sphere ascended the ramp.
 Given the difficulty of determining the exact final height of the sphere, what could be changed in the experimental design to increase the reproducibility of your measurements?
It was more difficult to determine the final position of the sphere when it travelled shorter distances on the ramp. To ensure the final height is reproducible at shorter distances, more trials should be conducted so the meter stick can be adjusted to the leading edge. It would also be advantageous to have both partners measure the height to confirm agreement between the measurements.
 Consider the following statement: Energy lost due to friction has a significant influence on the final height achieved by the sphere on the ramp. The force due to friction and the distance over which friction acts determine the amount of energy lost. The greater the distance over which the friction acts, the more energy is lost as thermal energy.
 Compare, in qualitative terms, the amount of energy lost due to friction for the three ramp angles used in the Introductory Activity. Explain how you made your determinations.
The most energy was lost due to friction when the ramp elevation was 30.0 cm. At this elevation, the sphere traveled the farthest distance along the ramp. The least amount of energy was lost when the ramp elevation was 60.0 cm. The sphere traveled the shortest distance along the ramp.
 Which ramp angle provided the most reliable change in height data? Explain.
When the ramp was at greater angles, the change in height data was more reliable. At greater angles, the sphere travelled shorter distances along the ramp and lost less energy due to friction. At greater angles, the transformation of elastic potential energy to gravitational potential energy was more complete.
 Based on the observations made in the Introductory Activity, how many trials should be conducted for each notch to ensure reliable data are collected?
It was somewhat difficult to be sure the sphere always reached the same height. Overall it appeared as though the sphere reached the same average height for the separate notches. At least 10 trials should be conducted for each notch to be sure the height at each notch is consistent.
 Based on the derived equation from Prelab Question 2, draw a graph predicting the relationship between Δh in terms of Δx^{2}. How could the value of the spring constant, k, be calculated from the slope of the graph?
{13787_Answers_Figure_4}
The slope of the graph of Δh versus Δx^{2} is equal to k/2mg. The value of k can be found by multiplying the slope by 2mg, where m is the mass of the sphere and g is the acceleration due to gravity.
 What values or measurements are needed in order to support the relationship between the change in elastic potential energy and the change in gravitational potential energy following the law of conservation of energy?
In order to confirm the relationship between elastic potential energy and gravitational potential energy, the compression distance of the spring, x, must be measured. The mass of the sphere, m, and the initial and final heights, h_{i} and h_{f}, respectively, of the sphere need to be measured as well. The spring constant, k, is needed but must be calculated. The acceleration due to gravity, g, is 9.81 m/s^{2}.
 Using the Introductory Activity procedure and the discussion questions, design an experiment to verify the quantitative relationship between spring potential energy and gravitational potential energy. Identify the independent and dependent variables, as well as any constants.
The ramp must be elevated so that all four notches (compression distances) can be utilized. The ramp height must be greater than 60 cm. Measure and record the height of the ramp. Place the launch handle into the first notch. Place the sphere on the ramp and measure the height of its leading edge. When ready, launch the sphere up the ramp. Repeatedly launch the sphere until it returns to the same position on the ramp. Measure and record the height of the leading edge of the sphere when it stops momentarily on the ramp. Repeat the same steps for the remaining three notches. When finished, place the launch handle in its uncompressed position.
Review Questions for AP^{®} Physics 1 A spring loaded launch mechanism is used to propel a box up a frictionless ramp. The box has a mass of 2.20 kg and an initial height of 0 m. The spring constant of the launch mechanism is 1776 N/m.
 The spring is compressed by 0.20 m and released. Determine the maximum height of the box.
ΔPE_{s} = –½kx^{2} ΔPE_{g} = mg(h_{f} – h_{i}) –ΔPE_{s} = ΔPE_{g} _{½kx2 = mgh} h = kx^{2}/2mg = 1776 N/m (0.20 m)^{2}/2 x 2.20 kg x 9.81 m/s^{2} h = 1.6 m
 The box then slides back down the ramp from its maximum height into the spring. What is the compression distance of the spring? Explain.
The compression distance of the spring will be 0.20 m. According to the law of conservation of energy, energy can only be transformed. The gravitational potential energy of the box must be equal to the elastic potential energy of the spring. Because the spring propelled the box to a height of 1.6 m, when the box returns to the spring, it will compress the spring by 0.20 m.
 Predict how the following independent changes to the springbox system would affect the maximum height of the box.
 Doubling the mass of the box.
If the mass of the box is doubled, then the maximum height will decrease by half. The mass of the box and final height are indirectly proportional. If the mass increases, then the height must decrease.
 Doubling the compression distance of the spring.
The box will travel four times higher by doubling the compression distance. The compression distance variable is squared, so when the distance is doubled, the variable increases by a factor of four.
 Decreasing the angle of the ramp.
The angle of the ramp will have no effect on the final height of the box.
 Assume the spring is now attached to the box and has a mass of 0.150 kg. The spring is compressed by 0.20 m and released. Will the boxspring object travel to the same height as the box in Question 1? If not, will it go higher or lower? Justify your answer.
The boxspring object will not travel to the same height as the lone box. The elastic potential energy remains the same because the spring is unchanged and it is compressed the same distance. Following the principle of the law of conservation of energy, the amount of gravitational potential energy is also the same. Gravitational potential energy depends on the mass of the object, its height and the acceleration due to gravity. The mass of the object and its height are inversely related. The boxspring object has a greater mass than the box. Therefore, the more massive boxspring object will travel to a lower height while still possessing the same gravitational potential energy as the lone box.
References
AP Physics 1: AlgebraBased and Physics 2: AlgebraBased Curriculum Framework; The College Board: New York, NY, 2014.
