Phys 244 Conservation of Energy F2F Fall 2022(1)

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Apr 3, 2024

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George Mason University Physics 244 (F2F) Study of Motion on an Inclined Plane Learning Goals: 1. The main goal of this lab is to examine motion in terms of energy. 2. Students will be able to identify if energy is conserved within measurement precision or if energy loss is apparent due to frictional heat loss. Materials: Motion track, cart system, a motion sensor, a spring system to initiate motion of the cart up the track, Capstone software, MS Excel. References: Giancoli, Physics 7th Edition: Chapter 6, Sections: 3, 4, 6, and 7 OpenStax, College Physics, Chapter 7 Introduction: Conservation principles are essential tools to model physical phenomena. One example of such conservation principle is “Conservation of Energy”. As seen in the cartoon in figure 1, energy is neither created nor destroyed. We must be careful with this statement as energy can be transferred to another form of energy or lost from a system due to frictional heat. If the value of a physical quantity is conserved, then this value remains constant. Finding the equation for the total energy differs from situation to situation. In this experiment you will try to determine if the energy equation is sufficient if you just consider the kinetic energy and the potential energy of the cart moving up and down the track. Figure 1: Energy conservation 1
Total Energy = Potential Energy + Kinetic Energy (1) If we want to examine a system at two different times - we need to examine if the system’s total energy remains constant. Thus, one must know initial and final energies. Etot i = P E i + K E i = Et ot f = P E f + K E f (2) In equation 2, i - subscripts represent the initial energies and f-subscripts the final energies. Remember that energy conversion cannot only occur from potential energy transferred into motion (kinetic energy) but also into heat that removes energy from the cart and is transferred into the surrounding environment. Friction we know from life experience causes heat, thus frictional forces may also play a role in today’s experiment. Your group’s goal is to determine if the total energy in the system is sufficiently explained by just examining potential and kinetic energies or if you must invoke frictional forces. The potential energy of the cart is gravitational potential energy given by PE = mg Δ h. (3) Here the change in height ( Δ h) is measured from a zero - level your group decides upon. (Remember as a scalar value – potential energy is simply a ‘value’ and not a vector and the zero therefore can be set where convenient for the experiment.) A convenient location would be the bottom of the track where the cart starts its ascent while no longer in contact with the spring providing an external force. Remember, the cart does not reach the actual bottom of the track - so take the height measurements with respect to the cart’s position – not the track. The difference is the change in height of the cart and therefore ( Δ h). The kinetic energy of the cart is given by KE = 1 2 m v 2 (4) Kinetic energy (4) is called ‘the energy of motion’ as it depends upon the velocity of the object. In fact, the growth of the kinetic energy depends upon the square of the velocity of the cart. This means it grows much faster than the velocity itself. If the total energy is conserved, a graph of “total energy vs. time” should be a horizontal line. Gravity is a conservative force, so if it is the only force involved we expect the total energy to be conserved. Ideally the motion would be measured at the center of mass. In this case, the measurement taken by the motion sensor will be from the front of the cart. 2
Experiment: In this lab, you will determine if energy is conserved during motion of a cart down an incline plane. You can use this knowledge to examine any graphs of energy transfers from potential energy to kinetic energy for energy losses once the lab is complete. You are also seeing an alternative approach to solving motion problems, when possible, without the use of force vectors but using energy equations. This energy approach has the advantage of combining scalars (energies) rather than forces (vectors) which must be added using their x and y components if the motion takes place in 2 dimensions. Your group will be acquiring position vs. time data from Capstone and graphing the position, velocity and acceleration of the cart on an inclined track as it moves up and back down a dynamics track. The initial force supplied to the cart will be from a launcher attached to the end of the track. Finally, you will analyze energy graphs created from the gathered data. The motion sensor is mounted on the track so the position it records is the distance from the top of the track. The position data are therefore taken along the track. This makes the calculation of the velocity of the cart straightforward. However, it complicates the calculation of the vertical displacement, h, which is necessary to calculate the potential energy. Notice in the diagram how the two heights are taken so that you may determine the change in height of the cart during this motion. Also note how the motion sensor will record the value P in the image. This value will be the largest when the cart is furthest away from the motion sensor. Figure 2: Side view with labeled variables - track/cart system 3
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In the figure 2, P is the distance to the cart as measured by the position sensor in Capstone and Z is the distance the front of the cart has moved up the track. You can determine P 0 from your Capstone data and then subtract P from P 0 to get Z. The cart launcher which sets the cart into motion (giving it a big push up the incline) restricts your choice of h o since the cart is not moving freely up the incline until it leaves the launcher. You can set h 0 = 0. It will be you reference level for calculating the potential energy. Finally, your group needs to determine the angle, , of the track's incline. There will be an angle hanger on your track system that will allow you to estimate the angle of incline. Make sure the string/mass is hanging freely before measuring. Each student should read the value and if there is any discrepancy in measurements, use the average angle value for calculations. The graph of position vs. time collected will be parabolic. We know this already since the equation for motion in gravity is: y – y o = v o t -½at 2 . Think carefully about this motion and the way the motion sensor collects data. The closer the cart is to the motion sensor the higher up the track the cart is located. In the graphical representation of the position data in Capstone, the minimum value of the curve corresponds to the maximum height of the cart, where potential energy is at a maximum. Also note when the distance from the sensor P is equal to P o the cart has just left the launcher and therefore has its maximum kinetic energy . We know as the cart moves up the track kinetic energy is being changed into potential energy and therefore the cart must slow because of this energy transfer. To find the P o value data can be collected when the cart is motionless on the extended spring. Data Collection and Analysis: 1. First determine the mass of the cart. 2. To take data set up the Pasco 850 Universal Interface with a motion sensor set to take data at 40 Hz and the switch at the top of the motion sensor set to record motion at short distances. Make sure the surface of the motion sensor is perpendicular to the track. Hold the track firmly so it will not recoil when the cart is launched. Be sure not to have any fingers wrapped around the track that could be hit by the cart. 3. In Capstone, choose “table and graph” from the display templates. Set it up to show position and velocity graphs. In the table create a time column, a position column, and a velocity column. 4
4. Press the Record button and then launch the dynamics cart up the track so that it will reverse its direction of motion before getting too close to the motion sensor. You may need to do this a couple of times for practice. If it goes too high the data will not be parabolic and it is necessary to adjust the angle of incline or the compression of the spring in the launcher. Note that your group should not use an angle higher than 20 degrees. 5. Press the Stop button after the cart has come to rest. 6. Inspect the data. Your group must be able to interpret the value for P o from this graph (or create a new graph with the spring sprung and take a separate data set to determine P o ). This is the P o value you are interested in to adjust all measurements of P (read from the motion sensor) into measurements of Z, the height the cart is at. 7. After you have found P o , highlight the data points of one position parabola in the graph for all points below P 0 , giving you a cut-off parabola (showing the cart rise and fall but never in contact with the spring). Make sure you don't choose any points that are larger than P o on either side of the parabola. 8. Copy the data for position vs. time, and velocity from Capstone to Excel and create a table with the following format. Time (s) P (m) Z (m) Δh (m) v (m/s) PE (J) KE (J) E tot (J) Use the following equations for Z and Δ h to calculate the values in the table: Z = P o –P Δ h = Zsin θ - h o (with h 0 =0) Make sure that all measurements are properly converted to the units shown in the table. 9. Create a graph of PE, KE, and E tot on a single plot. What can you say about the PE and KE as time evolves? 10. Does the graph show that energy is conserved or do frictional losses exist? (Remember E tot being a horizontal line in indicative of the sum of PE+KE being conserved.) 11. You will likely find that the E tot graph will usually show a slope. If this is seen in your data, what could be the cause of this loss of energy? 5
12. What was assumed at the start of this laboratory that resulted in an energy equation that had only potential and kinetic energies? References: Image on Page 1 from: http://www.edu.helsinki.fi/astel-ope/energy/energy_is_stored.htm 6
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