You have been given access to all the materials we have used so far in class: - probes: - metersticks - motion detectors - pulleys - force sensors - springs masses - cars Using this equipment (along with any other equipment that we may have, with reasonable request).you must design an experiment that will test how changes in the kinetic and/or potential energy of system and work done on that system are related.As part of your experiment, you should think carefully about the capabilities of this equipment and how well you can use them to make useful measurements of mass, distance and time. To state your experimental goal explicitly, you will test whether either the work-energy theorem: W=Fdcosphi =Delta K+Delta U or the energy conservation law: U_(1)+K_(i)=U_(f)+K_(f) are actually supported by data gathered from your experiments. To do this you will either make a graph of Delta K+Delta Uvs W,or Delta K=Delta U When setting up your experiment, you must carefully define your system and consider whether any external forces will do work on your system. If yes, you will test the work-energy theorem; if no, you will test the energy conservation law. In either case, your measurements of mass, distance and time must allow you to calculate the work done or the kinds of energy being studied in your experiment. 1 Guiding Questions These guiding questions are provided to help you think about how to design an experiment and take measurements in that experiment. These questions are only meant to guide you; they are not meant to be answered formally. 1. What object(s) are in your system? (e.g. if you are setting up masses on a pulley is the system just the masses on a pulley? or will you also include the Earth in your system?) 2. What external forces are acting on your system?In which direction is the displacement of your system? Are these forces doing work on your system? 3. What have you learned so far about how these quantities (work, potential energy kinetic energy) are related? 4. What factors will you be ignoring as you model your results in a set of tables and graphs? Why? (For example: will you ignore air resistance or friction? Why? What might be the net impact on your results for ignoring such forces, or other effects?) Physics - Ernst

## Experiment: Testing the Work-Energy Theorem with a Cart and Spring<br /><br />This experiment will test the work-energy theorem, W = ΔK + ΔU, by analyzing the motion of a cart attached to a spring on a relatively frictionless track.<br /><br />**1. System:** The system consists of the cart, the spring, and the Earth.<br /><br />**2. External Forces and Work:** The primary external force doing work on the system is the force applied by hand to initially stretch the spring. Friction between the cart and the track and air resistance will be considered negligible for the purposes of this experiment, though they will introduce a small error. Gravity is part of the system (due to the inclusion of Earth) and thus the gravitational potential energy is *internal* to the system.<br /><br />**3. Relationship between Work, KE, and PE:** The work-energy theorem states that the net work done on a system is equal to the change in its kinetic and potential energies. In this experiment, the work done by the hand stretching the spring will be converted into elastic potential energy stored in the spring. As the spring pulls the cart, this potential energy will be converted into the cart's kinetic energy.<br /><br />**4. Ignored Factors:** We will ignore friction between the cart and the track and air resistance. These forces will cause a small amount of energy dissipation, leading to a slightly lower final kinetic energy than predicted by the work-energy theorem. We will also assume the spring obeys Hooke's Law perfectly within the range of extensions used.<br /><br />**Experimental Procedure:**<br /><br />1. **Setup:** Attach a spring to the front of the cart and secure the other end of the spring to a fixed point on the track. Place a motion detector at the end of the track to measure the cart's velocity.<br /><br />2. **Calibration:** Zero the force sensor and measure the mass of the cart using a balance. Determine the spring constant (k) of the spring. This can be done by hanging known masses from the spring and measuring the extension, then plotting Force vs. extension. The slope of this graph will be the spring constant.<br /><br />3. **Data Collection:**<br /> a. Pull the cart back a distance 'x' from its equilibrium position, stretching the spring. Measure this displacement with a meterstick. Record the force required to hold the cart in this position using a force sensor. This force will be equal to kx.<br /> b. Release the cart and let it oscillate. The motion detector will record the velocity of the cart as a function of time.<br /><br />4. **Data Analysis:**<br /> a. **Calculate Work:** The work done in stretching the spring is W = (1/2)kx². Since the force changes as the spring stretches, we use the average force (kx/2) multiplied by the displacement x, or integrate the force over the displacement.<br /> b. **Calculate Kinetic Energy:** Determine the maximum velocity of the cart from the velocity-time graph. Calculate the maximum kinetic energy using K = (1/2)mv². This maximum kinetic energy occurs when the spring is at its equilibrium position and all the potential energy has been converted.<br /> c. **Calculate Potential Energy:** Calculate the initial potential energy stored in the spring using U = (1/2)kx².<br /> d. **Compare W and ΔK + ΔU:** Since the initial kinetic and potential energies are zero (cart at rest, spring unstretched), ΔK + ΔU is simply the final kinetic energy plus the final potential energy (which is zero at the point of maximum velocity). Compare the calculated work done (W) with the maximum kinetic energy (K).<br /><br />5. **Graphing:** Plot the maximum kinetic energy (K) against the work done (W) for several different initial displacements (x). If the work-energy theorem holds, the graph should be a straight line with a slope of approximately 1, passing through the origin.<br /><br />**Expected Results:**<br /><br />The experiment should demonstrate that the work done on the system by stretching the spring is equal to the change in the system's kinetic energy (since the final potential energy is zero at the point of maximum velocity). The graph of K vs. W should confirm this relationship.<br /><br />**Sources of Error:**<br /><br />* Friction between the cart and the track and air resistance will dissipate some energy, leading to a slightly lower final kinetic energy than predicted.<br />* Errors in measuring the displacement, mass, and velocity will affect the calculated values of work and kinetic energy.<br />* The spring may not perfectly obey Hooke's Law, especially for large extensions.<br /><br /><br />This experimental design allows for a clear test of the work-energy theorem using readily available equipment and provides a method for quantifying and analyzing potential sources of error.<br />