Purpose of the experiment

Conservation of Energy PES 116 Advanced Physics Lab I Purpose of the experiment Learn about different forms of energy. Learn how to use the Conservation of Energy to solve complicated dynamics problems. FYI FYI The early bird gets the worm, but the second mouse gets the cheese. Conservation of Energy - 1

Table of Contents Background 3 Equipment Bouncing Ball 6 Pendulum 7 Lab Procedure Ball 8 New Column 10 Coef. of Restitution 13 Pendulum 14 Equipment List LabPro Interface Motion Sensor Part A - Large playground ball Playground ball Ring stand clamp Ring stand (small) Threaded rod Part B - Tennis ball pendulum Pendulum Ring stand clamp Ring stand (large) Conservation of Energy - 2

Background Conservation of energy equation Let s take a look at a simple case of an object held up in the air. An object suspended above the ground has potential energy. When the object is released the object moves toward the ground and will begin to gain speed, due to gravity. This motion means an increase in kinetic energy. As the height decreases the energy associated with this height (Potential energy) will decrease. Since energy is conserved, the sum of the kinetic and potential energies must remain constant. As one of these energies is increased or decreased the other must change in order to keep the total energy of the system the same. The mathematic equation is quite simple, all the energy present initially in a system must be the same as the energy in the final state. TE i (initial total energy) = TE f (final total energy) Using the three types of energy described above, we can expand this expression a little. KE i + PE i + E i = KE f + PE f + E f For most labs this semester I will try to minimize energy losses so the equation becomes: KE i + PE i = KE f + PE f Never forget that energy loss can be present and will affect the conservation of energy. If we assume that the potential energy is gravitational potential energy, which it is for this lab, the equation can be further expanded. Where the KE is equal to ½ mv 2. 1 2 mv 2 i + mgh i = 1 2 mv 2 f + mgh f This makes analyzing many otherwise complicated physical systems a one-liner. This is because an important quantity in most experiments is velocity. Velocity can be calculated from the equations of motion as long as the acceleration is known. The acceleration is found using Newton s Laws. The conservation of energy cuts through all this red tape and finds the velocity directly! Conservation of Energy - 3

The Equipment Bouncing Ball For this part of the lab you will be verifying the conservation of energy equation as it is applied to a bouncing ball. ring stand Starting point should be about 3 to 4 inches above the floor. table sonic ranger, pointed down. play-ground ball Note: Bad data could be from interference from objects within the cone of the ranger. Table supports, feet, hands, books, table top, can all be forms of interference. floor Figure 1: Diagram of the experimental setup. Something to think about: When you release a ball from rest and let it bounce off the ground does it return to the same height you dropped it from? What prevents the ball from returning to your hand? Conservation of Energy - 4

Simple Pendulum Everyone should be familiar with the pendulum, so any explanation on what it is seems like a waste of time. The physics behind the pendulum is another matter. At this point you have two possible math techniques you can use to analyze the motion of the pendulum: Newton s Laws or the Conservation of Energy. Considering the title of this lab, it is not hard to figure out which technique we will end up using. To use Newton s Laws we first need to switch to polar coordinates. This simplifies matters a bit but you can imagine that the mathematics will still get hairy! I will leave these math steps up to you, if you wish to work them out. This is going to be a lot of work! On the other hand, if we use the conservation of energy, this becomes a one-liner. Using conservation of energy: KE i + PE i = KE f + PE f θ L m h PE = 0 Substituting the expression for kinetic and potential energy: 1 2 mv i 2 + mgh i = 1 2 mv f2 + mgh f That is it! Now you can solve for any velocity at any position you desire. The potential energy was picked to be zero at the bottom of the swing. However, I could as easily defined the zero point anywhere. But this selection makes the math easier. This is because at the top of the swing the ball will have a PE but the KE will be zero (velocity = 0). At the bottom of the swing PE will equal zero (h = 0, defined that way) and there will be a KE. Therefore, rewriting the simplified conservation of energy equation: 2 mgh = 1 2 mv f Simple! (you can even cancel out the mass if you like!) Conservation of Energy - 5

The Lab The goal of this lab is to test the usefulness of the conservation of energy equation. Part A Playground Ball Measure and record the mass of the ball you plan to use in this experiment. Connect the Motion Detector to the DIG/SONIC 1 channel of the interface. Start Logger Pro program. Getting today s data from the sonic ranger will be a little tricky. Be patient, and be careful to remove any interference. 1. Load the file PES 116/Cons of E 1/Ball.cmbl. Run the program in the Repeating mode (instead of a single sweep) and test that the ranger is giving a reasonable and clean signal by moving the ball in front of the ranger. Return the setting back to Single before beginning the experiment. 2. Start the data collection and release the ball (be careful and take your hands out of the way). You only need small bounces (3 to 4 above the ground). The ranger is setup to record the distance from the ranger to an object. This will result in data values that will be low when the ball is close to the top of its bounce, and give high values when the ball strikes the floor. We will fix this later, just be aware of it for now. Once you think you have some good data, about 3 to 4 smooth curves, you will want to isolate it from the rest of the noise. Drag a box 1 around the good data. Select the Zoom In button for a closer look at what is within the Drag box (refer to Computer Software Module for more information). 1 Click and hold down the mouse button at the top left corner of the area you want to expand. Drag the mouse cursor to the lower right corner and release the mouse button. Conservation of Energy - 6

3. The problem we have to deal with now is how to zero-out the data. That is, to set the zero position value to the point when the ball hits the floor. position position This is what your data will look like. time This is what you want the data to look like! time Use the Examine tool located in the toolbar. Move the mouse across the graph. Move the mouse to each of the places where the ball hits the floor and record the distance values. Average all the points together to get a single measurement of the ranger-to-floor distance. We need to do two things. First, shift these peak values to zero to represent the floor. And second, flip the graph so that an increase in the position data will mean an increase in height off the floor. We need to subtract the ranger-to-floor distance value from all the position data! To flip the data all we need to do is multiply all the position data by negative one. The data table operates like a little spread sheet program, so this is not as difficult as you might think. However, it will require you to learn some more features of the Logger Pro software. Follow these steps to create the new column Distance from floor. Conservation of Energy - 7

We need to create a new data column! Use the New Column command under the Data menu. Change the name of the new column to Distance above ground. Change short name to height and the units to meters m. Type in the following equation. The ( 1) flips the graph. Position is the raw data from the motion sensor. Substitute the calculated average value of the distance to floor (described above) for the Averaged distance to floor. Conservation of Energy - 8

You should now notice our new column in the data table window. You might have to scroll to the right to see it. 4. This does not display the graph of this column. We need to add it to the graph window. Click on the y-axis label in the graph window. This will bring up a popup window. Select the new data set or go to the More option to find the new column. You should now see the graph of position vs. time you expected to see in the beginning. Finally we have a graph of the height vs. time for the bouncing ball! This is only the beginning of the data analysis. The rest is more straightforward! Wait until the end of the lab to print the graph! 5. Let s also get a plot of the potential energy. We need to add another new column. Enter Potential Energy as the name of your new column. The units of energy are joules, so enter J as the units. Potential energy = mgh ( the mass, times the acceleration of gravity, times the height of the object from the zero point, in this case the floor). In the Equation text box type: (Enter here: the mass of the ball) * 9.81 * Distance above ground measure the mass of the ball with the 1.2kg scale located by the sink. This will create the new column for the potential energy. You will want to plot this in the graph window. If the graph of the potential energy goes off scale, you will need to rescale your graph. The easiest way is to use the Auto scale toolbar button. Check in Computer Software Module for more options and information on rescaling the graph. 6. The next graph we should look at is that of kinetic energy. Once again add a new column called kinetic energy. Use the definition of kinetic energy : KE = 1 2 m v2. Enter: 0.5 * (the mass of the ball) * derivative( Distance above ground )^2 Conservation of Energy - 9

Notes: the velocity of an object is the first time derivative of position. The symbol ^ represents the value on the left of this symbol raised to power of the number on the right of the symbol. Example: x 2 = x^2. Display the kinetic energy on the same graph for comparison. 7. According to the conservation of energy, the sum of the kinetic and potential energies should be constant. Test this to see if it is true. Add a new column called total energy. Repeat the steps above so we can see it on the graph window. 8. Get a print out of this final graph with position, potential energy, kinetic energy, and the total energy displayed. Change the title to include your name so you can find it at the printer, double-click on the title to make these changes. Make sure you use the Print Graph command! Write on the graph what each of the lines represent (position, KE, PE, TE). 9. Point out specific areas where the ball is either at rest, changing direction, or at a maximum height. Comment on what each of the energies are at each of these points and explain why this is so. This is the main question of this lab, so be complete. How do these graphs prove or disprove the conservation of energy? The total energy of the system must remain constant, Right?! Looking at your graph, what do you notice happens to the value of the total energy when the ball hits the ground? Do you think that the ball violates the conservation of energy? (Hint: NO!) Since the ball does not violate the conservation of energy why do we see this discrepancy? Think about all the other forms of energy we are measuring and how many forms of energy we are not measuring. What happens to the total energy after each bounce? How do you account for this? What would change in this experiment if you used a very light ball, like a beach ball? Conservation of Energy - 10

What would happen to your experimental results if you entered the wrong mass for the ball in this experiment? A little something Extra The coefficient of Restitution (C R ) is a measure of the loss in energy from a collision. More precisely it is the ratio of the exit velocity (v e ) divided by the approach velocity (v o ) in a collision. ve CR vo For an ideal elastic collision, C R = 1 (which means that the exit velocity equals the approach velocity). Some examples: A basketball has C R 0.6, and a baseball C R 0.55. If a ball with coefficient C R is dropped from a height h, it will reach the ground with ½ mv 2 = mgh the approach velocity is therefore vo = 2gh So the exit velocity after rebounding once is ve = CRvo = CR 2gh and therefore the ball will reach a new height of 2 2 v CR ( 2gh e ) 2 h = e CRh 2g = 2g = Similarly, the height after rebounding twice will be 4 h = C h ee R Use the data you just collected and calculate a value for the coefficient of restitution. Describe the procedure you used and show all work. Use your value for C R and calculate the height of the second bounce. Compare this to what measured. The coefficient of restitution is a good measure of the energy lost from the bouncing ball, but what form do you think this energy took. Think about the soft, rubber ball and what happens to its shape as it hits the ground. Conservation of Energy - 11

Part B - Simple pendulum Load the file: PES116/Cons of E/Pendulum.cmbl. Examine the pendulum setup and notice that there are only a few things we can measure directly which may affect the pendulum s behavior. The mass of the pendulums ball. The length of the pendulum (from the pivot point to the center of the ball). The height of the swing (tricky to measure). Using these measurements and the conservation of energy we can find out anything we want about the motion of the pendulum. Getting the measurements You have learned about many techniques to make measurements this semester, as well as problems associated with each type of measurement. I m going to leave the decision on how to measure the height of the pendulum s swing up to you, but I will give you a few suggestions. You must defend your decision, explaining why you think your measurement technique will give the most accurate measurement. To measure the height you could use a: 1. Ruler. Simple but remember that the height is measured from bottom of the swing not the height from the table top! 2. Sonic ranger positioned under the pendulum. A more high tech version of the ruler. 3. Protractor. Use this to measure the angle at which you swing the pendulum. Find the height in terms of the angle and the length of the pendulum. 4. There is yet another way to find the maximum height of the pendulum. This is a math trick using geometry and the position vs. time data you can collect using the sonic ranger. Study the following drawings and see if you can figure it out: Conservation of Energy - 12

θ θ L Note: The length of the pendulum (L) is the distance from the pivot point to the center of the ball. y Ball is farthest from the range finder. position x Ball is at the middle of oscillation. h Ball is closest to the range finder. Farthest from the range finder. Motion Sensor Graph of the Motion sensor data: distance vs. time Closest to the range finder. time Answer these questions; they will help you see the math trick. What does the amplitude of the oscillations represent? Find an equation for L as a function of y and h? Find an equation for y as a function of x and L? You should now have enough information to solve for h. The equation for h should contain the length of the pendulum (L) and the value x. What is x? Look at the graph of position vs. time. Conservation of Energy - 13

Measuring the velocity of the tennis ball Before you release the tennis ball, we need to find a way to measure the velocity at the bottom of the swing. Use the sonic ranger. Make a new column called velocity. The units will be, of course be, m/s. In the Equation text box enter: derivative( Latest:Position ). Make sure you display the velocity data in the graph window. Sketch/Printout the position vs. time and velocity vs. time graphs. On the graph, indicate where the ball reaches: the top of its swing the bottom of its swing Record the velocity (use the data table and the Examine button): at the top of the swing at the bottom of the swing Test your results using the conservation of energy Use the height you measured and calculate the expected velocity at the bottom of the swing. Compare this velocity to the one you measured. Explain any differences between the two numbers. What could cause this difference? Think about the experimental setup and the measurements you made. Identify problems you see. Explain how your experimental results could be improved. What about the pendulum, over a short amount of time the energy loss is small, but pendulum do not keep swinging forever. What do you think happens to all the energy? Conservation of Energy - 14