MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.01 Fall Term 2010 Experiment 03: Work and Energy Purpose of the Experiment: In this experiment you allow a cart to roll down an inclined ramp and run into a spring that is attached to a force sensor. You will measure the position of the cart and the force exerted on it by the spring while they are in contact. It is a real world experiment, which means that there are non-conservative forces: friction as the cart rolls up and down the track, and dissipation (internal friction?) in the spring. The goals of the experiment are: To investigate experimentally the work kinetic energy theorem, how potential energy in a gravity field converts to kinetic energy which is then converted into the potential energy of a compressed spring. To observe and quantify the effect of non-conservative forces and estimate the work done by these forces at various stages of the cart s motion up and down the ramp. Experimental Materials: Logger Lite Software Vernier LabPro Interface PASCO cart with 250 gm weights PASCO track Vernier Motion Detector Vernier Dual-Range Force Sensor with spring and securing device small 2 4 block Setting Up the Experiment: Refer to the photo to the right and the figure at the top of the next page. The motion detector should be placed at one end of the track, as in the photo at the right. The motion sensor works best if it is aimed slightly above the center of the cart rather than pointing directly at it. (That reduces the effect of sound waves that bounce off the track before hitting the cart.) The slide switch on top of the motion sensor should be set to the narrow beam position. Place the force sensor at the other end of the track as shown in following figure. Experiment 03 1 October 28/29, 2010
Raise the track by placing a short piece of 2 4 under the motion sensor where it clips onto the track, as you can see in the photo to the right. This should raise the end of the track about 4.2 cm above the table; as the track is 122 cm long, you can calculate the slope θ = 1.97 and sin θ = 0.0344. y x 4.2 cm θ Place a cart on the track with the end having the Velcro TM patches facing the motion sensor. Put two 250 gm weights in the cart, which will bring its total mass to 750 gm. (The extra mass reduces vibrations and gives less noisy measurements.) Place the cart about 30 cm up the track from the force sensor and release it. It will roll down the track, bounce most of the way back up, and repeat that several times. You will notice the track slides when the cart runs into the spring; this is an example of conservation of momentum. To prevent the track from sliding, place your thumb on the end of the track resting on the table and press it firmly against the table. If you don t do this, when the cart runs into the spring some of its kinetic energy will be dissipated by friction of the track on the table which will introduce an unknown error in your analysis. Connect the Vernier Motion Detector (DIG/SONIC 1) and the Dual-Range Force Sensor (channel 1) to the Vernier LabPro interface. Set the Force Sensor to the ± 10 N setting, as this is the range of force applied in this experiment. Be sure to tare (zero) the force sensor before each measurement. This can be done by navigating to the Experiment menu and selecting Zero. You may choose which device to zero. The Logger Lite Program WorkEnergy: The Logger Lite program you will use in this experiment is called WorkEnergy.gmbl. Start the program, go to the Experiment menu, and select Data Collection. You may set parameters, including collection length and sampling rate, to your liking (default values are 15 sec and 100 samples/sec). The rate that you choose is the rate at which the force is measured; the position will be measured 1/4 as often. The maximum sample rate for the Force Sensor is 400 samples/sec. In the experiment you will let the cart roll into the spring starting from rest about 30 cm 2
up the track from the point where it first touches the spring. Try this to see how things behave. The Collect option (green play button) will start the program measuring the voltage on channel 1, alternatively, you can hit the space bar to commence measuring. The program will start to record the voltage immediately and will continue for the length of time set in the Data Collection window (1 sec or less is usually enough). You should then see a plot like the one in the graph below. When you are ready to measure, hold the cart in position, and click Collect (green play button). Alternatively, press the space bar. Be sure to hold the track so it will not slide and release the cart at about the same time. After the 15 s have elapsed, you should see a graph of raw data something like this one. The top curve is the position of the cart with respect to the motion detector and the bottom curve is the voltage output of the force sensor. You can see from the peaks in the lower curve the times when the cart bounces off the spring and you can see from the upper curve that the cart bounces back up to a lower height each time. You should repeat this process several times until you are sure the track is not moving when the cart collides with the spring, the raw data looks clean and smooth, and you have five good looking bounces (and six force peaks). When you have achieved a run you are satisfied with, you may save your data as a.csv file by selecting Export from the File menu. Open your data in your Excel spread sheet. Now you can proceed to analyze your results just as you did in the prelab homework problem. Experiment 03 3 October 28/29, 2010
Processing the Data: For neatness and clarity you may want to insert a blank column between the position column and the next column representing the times relevant to the force sensor.we will not use the values of the force itself in the subsequent analysis. Rather we will use the force data to determine the times when the cart first touches the spring and when it leaves it. Use the Chart Wizard (choose XY scatter charts with no connecting lines) to plot position verses time and force verses time on two new pages. Note that force and its associated times comprise a very long data set. When selecting it for charting, keep track of how far you have yet to go the reach the end by watching the scroll button on the right hand side of the Excel window (on my version at least). Make sure the data is represented by the smallest points possible on your charts. The figure on the next page shows what a portion of your position plot might look like, with some reference information superimposed on it. 0.9 t 0 t 1 t 3 t 4 0.8 Position(m) 0.7 0.6 t 2 t 5 0.5 0.4 A B C 0 2 4 6 8 10 12 Time(s) The horizontal dashed line indicates the position of the cart when it just touches the spring. t 1 is the time at which the cart just leaves the spring, rebounding from it for the first time. In the region A, between t 1 and t 2, the cart is moving up the track. At t 2 the cart comes to rest at the top of its first rebound. In the region B, between t 2 and t 3, the cart is moving down the track. At t 3 it just touches the spring and begins to compress it. At t 4 the cart leaves the spring again. In the region C, between t 4 and t 5, the cart is moving back up the track. At t 5 the cart comes to rest at the top of its second rebound. This experiment is capable of producing satisfyingly good results, but only if reasonable care is taken to determine these times precisely. You will use the data you have downloaded to study energy loss in this system in three stages. Experiment 03 4 October 28/29, 2010
Rolling Friction The magnitude of the deceleration in region A is larger than the magnitude of the acceleration in region B. In the first case friction and a component of gravity act in the same direction, in the second they act in opposite directions. The friction force can be obtained by taking half of the difference in the two accelerations and multiplying it by the mass of the cart. The energy lost should be the product of the friction force and the distance traveled. To check the reliability of this model you will compare this energy loss with the difference in kinetic energies at t 1 and t 3. Collision Loss This can be found by computing the difference in kinetic energies at t 3 and t 4. It will be interesting to see how this loss compares to the loss due to rolling friction. Consistency Check You will compute the fraction of the total mechanical energy remaining after the cart has executed the first full cycle of its motion. It is reasonable to assume that this fraction remains constant during subsequent cycles. You will check this by comparing the predicted energy loss with the measured decrease in maximum excursion in the subsequent cycles. Keep track of your progress in the data analysis by making entries on two pages, a Scratch Sheet and a Results Page. Copies of these two pages are attached to the end of these instructions and will be be available in class. Turn in these two pages at the end of the experiment, together with your prelab results. Determining Times and Positions: Begin by looking at the figure on the previous page, then go to your own plot of position verses time and make rough visual estimates of the following times: the location of the first maximum in the position data (t 0 +t 1 )/2, t 2, the location of the second maximum (t 3 +t 4 )/2, and t 5. Enter these values on the Scratch Sheet. They will be helpful as you try to locate specific features in the position and force data. It is important to get precise values of t 1, t 3 and t 4, so you should work on these first. Unfortunately, for reasons having to do with the the apparatus, the absolute values of the times recorded for the force data are not reliable. However, over short time periods, the relative times for the force data are reasonably accurate. There is a simple procedure for getting around this problem. For example, you will use the force data to find the difference between t 0 and t 1. You will combine this with an accurate value of t 0 + t 1 to find t 1 itself. Examine point 1 first. Scroll down the force column until you see the voltages begin to rise from their no contact values. This is where the cart hits the spring for the first time. Continue down until the voltage appears to return to its no contact values. Use the cursor to highlight the time and force cells from about 10 rows before this region to about 10 rows afterward. Use Chart Wizard to graph the force verses time in this region. Simply keep pressing next in the dialog box until you get to finish and the graph appears on your worksheet. You should get something similar the to following. 5
Inspect the graph. Find the point (on the right) where the force first appears to have returned to its no contact value. Write down that time. Find the point (on the left) where the force first appears to have deviated from its no contact value. Write down that time. Calculate the difference between these two times and enter it as t 1 t 0 on your Scratch Sheet. Use a similar procedure to find t 4 t 3. Enter it on your Scratch Sheet. Now turn your attention to the position data. Scroll down the position column. Select about 20 rows in the time and position columns on either side of the time you estimated to be the location of the first maximum in the position data, (t 0 + t 1 )/2. Use Chart Wizard to graph the position verses time in this region. Simply keep pressing next in the dialog box until you get to finish and the graph appears on your worksheet. You should get something similar the to following. From this curve, make the best estimate you can of the time at which the position reaches a maximum. Enter this value as (t 0 + t 1 )/2 on the Scratch Sheet. Repeat the process to find the value of (t 3 + t 4 )/2. Go to your Scratch Sheet and use the data you have entered to compute the times t 1, t 2 and t 4. Enter these values in Table 1 on the Results Page. Scan down the time and position 6
columns and find the positions associated with these three times. Enter them in Table 1. These three positions should be identical to within experimental error. Compute the average of the three position values and enter it on the Scratch Sheet. The time and position for points 2 and 5 are both found from the time and position columns. Scan down the columns to find the local minima in the position, graph the posititon verses time in that region, and estimate when the actual minima occur. These are broad minima, so some care is necessary in making the estimate. Enter the times and positions in the Table 1 on the results page. Also enter the two positions as the positions of the first and second minima on the Scratch Sheet. Delete the graph. This is a good time to fill in Table 2 on the Results Page. Point 2 corresponds to the top of the first rebound. The excursion at that point is the difference between the position when the cart just touches the spring and the position at t 2 from Table 1. Compute this from the data on the Scratch Sheet and enter the excursion in the Table 2. Likewise, compute and enter the excursion for rebound 2 using the position you measured at t 5. Find the values of the position at the other local minima, enter them on the Scratch Sheet, and enter the corresponding excursions in Table 2 of the Results Page. Velocities and Accelerations: Since the acceleration is expected to be a constant in each of the regions A, B and C, the position should be parabolic in time. You will fit the data in each of the three regions to determine the accelerations and the values of the velocity when when the car begins and ends its spring-free motion. Consider region A first. In the time and position columns, select the rows corresponding to the times from t 1 to t 2, inclusive. Use the Chart Wizard to plot position verses time on a separate sheet. Highlight the data points on the graph and Click on Add Trendline in the Chart menu. Choose a polynomial fit of order 2 (that is, a parabola). Don t forget to check Display equation on chart from the Options panel. From the fitted equation determine the magnitude of the upward acceleration and enter it in the first box in Table 3. Likewise, determine from the fitted equation the velocity at the beginning of the A region (for best results, you must evaluate the velocity at the precise time for t 1 from Table 1) and enter it in the line for Point 1 in Table 1. Use the same procedure for region B to find the magnitude of the downward acceleration and the velocity at t 3. Use the procedure again in region C, but there you need only determine the velocity at t 4. Studying the Energy Losses: Rolling Friction You have already filled out the first two lines in Table 3. Use that information and the total mass of the cart plus weights (0.750 kg) to find the friction force. Enter it in Table 3. Find the distance traveled (twice the excursion) using Table 2 and enter that. Compute the energy lost to friction using the rolling friction model and enter that. Go to table 1. Compute and enter the kinetic energies at t 1, t 3 and t 4. Find the difference between the kinetic energies at t 3 and t 1. Enter that in Table 3. This value should be fairly close to the energy loss you computed due to friction. A difference of 10% or less could be attributed to noise in the data and imitations of the data fitting procedure. Experiment 03 7 October 28/29, 2010
Collision Loss Use the data from Table 1 to determine the energy lost by the cart during its first collision with the spring. Enter it in Table 3. How does it compare with the loss due to friction? Consistency Check Your first look at the full data record showed clearly that mechanical energy was being lost in the system because the cart rebounded to successively smaller peak excursions as time when on. The peak excursion is proportional to the total mechanical energy of the cart. You can now check if the detailed energy losses you have found are consistent with this gross behavior. Fill out the last two entries in Table 3. We will refer to the fraction of the initial mechanical energy remaining after the first rebound as f. Reproduce Table 2 in your Excel worksheet. Add a third column and label it Model. The model result should be the excursion from the first rebound multiplied by f n 1 where n is the rebound number. Here is how to get this value into the cells of the third column. Assume you are using columns K, L and M, row 1 has the titles of the columns, row 2 has the first data. Assume also that the first excursion has a value 0.306 (stored in L2) and the fraction f has the value 0.79. Enter in cell M2 the expression = 0.306 P OW ER(0.79, K2 1) and press enter. The value in the cell should read 0.306. Select the cell and choose copy. Select the remaining cells in that column and choose paste. The values appearing in those cells should decrease as one goes down the column. Select the entire table you have just created and use Chart Wizard to see graphically how the model excursions agree with the actual values. 8
Scratch Sheet for Experiment 3 Rough Time Estimates (t 0 + t 1 )/2 t 2 (t 3 + t 4 )/2 Precise Time Values t 1 - t 0 (t 0 + t 1 )/2 t 1 t 5 t 4 - t 3 (t 3 + t 4 )/2 t 3 t 4 Average of Positions at t 1, t 3 and t 4 (from Table 1 on Results Page) Positions at Top of Rebound Rebound Position 1 (at t 2 ) 2 (at t 5 ) 3 4 5 6 9
Results Page for Experiment 3 Point Time Position Velocity KE 1 Table 1 Table 2 Rebound Excursion 1 2 0 0 2 3 3 4 4 5 0 0 5 6 Table 3 Magnitude of acceleration upward (m s -2 ) Magnitude of acceleration downward (m s -2 ) Friction force (Newtons) Distance traveled (m) Energy lost due to friction (Joules) Measured change in KE (Joules) KE lost during collision (Joules) Total mechanical energy lost during first rebound (Joules) Fraction of initial mechanical energy remaining 10
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.01 Physics I Fall Term 2009 Experiment 3: Material to be Handed In CLASS TABLE GROUP NAME NAME NAME Please fill out the attached Scratch Sheet and Results Page representing your joint efforts in class. Each person should also attach the required materials from PreLab done out of class. 11 1