18 18 Atwood’s Machine Physics Experiment Manual 012-09233 012-09233 Physics Experiment Manual 18 Atwood’s Machine Newton’s Second Law – Atwood’s Machine (Photogate/Pulley System) Mechanics: Archimedes’ Principle, buoyancy DataStudio file: 18 Atwood’s Machine.ds Equipment List Qty Items Part Numbers 1 PASCO Interface (for one sensor) 1 Photogate/Pulley System ME-6838 1 Mass and Hanger Set ME-9348 1 Universal Table Clamp ME-9376B 1 m String SE-8050 Introduction The purpose of this activity is to study the relationship among force, mass, and acceleration using an Atwood’s Machine apparatus. Use the Photogate/Pulley System to measure the motion of both masses strung over a pulley as one moves up and the other moves down. Use DataStudio to record the changing speed of the masses as they move. The slope of the graph of velocity vs. time is the acceleration of the system. Background The acceleration of an object depends on the net applied force and the object’s mass. In an Atwood's Machine, the difference in weight between two hanging masses determines the net force acting on the system of both masses. This net force accelerates both of the hanging masses; the heavier mass is accelerated downward, and the lighter mass is accelerated upward. Based on the above free body diagram, T is the tension in the string, M2 > M1, and g is the acceleration due to gravity. Taking the convention that up is positive and down is negative, the net force equations for M1 and M2 are: Assuming that the pulley is massless and frictionless, and the string has no mass and doesn’t stretch, let T 1 = T 2. Solving for a, the acceleration of the system of both masses, the theoretical acceleration is g times the difference in mass divided by the total mass: EMBED "Equation" \* mergeformat SAFETY REMINDER Follow directions for using the equipment. Setup Set up the PASCO Interface and computer and start DataStudio. Connect the Photogate to the interface. Open the DataStudio file: 18 Atwood’s.ds The DataStudio file has a Graph display of Velocity versus Time. Mount the clamp to the edge of a table. Use the Pulley Mounting Rod to attach the Pulley to the tab on the Photogate. Place Photogate/Pulley in the clamp so that the rod is horizontal. Use a piece of string about 10 cm longer than the distance from the top of the pulley to the floor. Place the string in the groove of the pulley. Fasten mass hangers to each end of the string. To attach the string to the hanger, form a loop at the end of the string and tie it back on itself to ‘knot it’ or wrap the string about 5 times around the notch in the mass hanger. Place five to six masses from your mass set, totaling (roughly) 100 grams of mass on one mass hanger and record the total mass as M1 in the Data Table in the Lab Report section. Be sure to include the 5 grams from the mass hanger in the total mass. On the other mass hanger, place about six masses, totaling slightly more mass than M1. Record this total mass of the masses and mass hanger as M2 in the Data Table in the Lab Report section. Move the M2 mass hanger of upward until the M1 mass hanger almost touches the floor. Hold the M2 mass hanger to keep it from falling. Turn the pulley so that the Photogate beam is unblocked (the red light-emitting diode (LED) on the Photogate does not light). Procedure Part 1: Constant Total Mass NOTE: The procedure is easier if one person handles the apparatus and a second person handles the computer. Release the M2 mass hanger and let it fall. Click ‘Start’ to begin recording data. Stop recording just before the M2 mass hanger reaches the floor. • Do not let the upward moving mass hit the Pulley. For Run #2, move a mass from the M2 mass hanger to the M1 mass hanger. This process changes the net force without changing the total system mass. Record the new total mass for each hanger with masses in the Data Table in the Lab Report section. Allow the mass to fall. Begin data recording. Stop recording data just before the hanger reaches the floor. Repeat the above step to create three more mass combinations. For each run, the net force changes but the total mass of the system remains constant. Part 1: Constant Net Force Arrange the masses as they were for Run #1. Now, change the total mass of the system but keep the net force the same. To do this, add exactly the same amount of additional mass to both mass hangers. • Make sure that the difference in mass is the same as it was for the beginning of Part 1. Add approximately 10 grams to each mass hanger. Record the new total mass for each hanger with masses in the Data Table in the Lab Report section. Release the M2 mass hanger and let it fall. Start data recording. Stop recording just before the M2 mass hanger reaches the floor. Repeat the above step to create three more data runs. For each data run, the net force remains the same, but the total mass of the system changes. Analyze Determine the experimental acceleration for each of the data runs. Find the slope of the velocity vs. time plot, the average acceleration of the masses. Select Run #1 from the Data Menu in the Graph display. (If multiple data runs are showing, first select No Data from the Data Menu and then select Run #1.) Click the “Scale to fit” button to rescale the Graph axes to fit the data. Next, click the ‘Fit’ menu button and select ‘Linear’. The slope “m” is the average acceleration. Record the slope of the linear fit in the Data Table in the Lab Report section. Repeat the above procedure for each of the remaining data runs. Calculations For each of the data runs, using the measured mass values, calculate and record the net force in the Data Table in the Lab Report section. EMBED Equation.3 Calculate and record the total mass in the Data Table. Using the total mass and net force, calculate the theoretical acceleration using: EMBED "Equation" \* mergeformat For each data run, calculate and record the percent difference between the experimental acceleration and the theoretical acceleration. Lab Report: Newton’s Second Law – Atwood’s Machine Name: ________________________________________________________________ Data Table: Constant Total Mass Run M1 (kg) M2 (kg) aexp (m/s2) Fnet (N) M1+ M2 (kg) atheory (m/s2) Percent difference Run #1 Run #2 Run #3 Run #4 Run #5 Data Table: Constant Net Force Run M1 (kg) M2 (kg) aexp (m/s2) Fnet (N) M1+ M2 (kg) atheory (m/s2) Percent difference Run #6 Run #7 Run #8 Run #9 Run #10 Questions Compare the experimental acceleration with the theoretical acceleration by determining the percentage difference. What are some reasons that would account for this percent difference? For the Constant Total Mass data, plot a graph of Fnet vs. aexp. Note: Include a negative sign for acceleration values when M1 > M2. Attach your plot to the Lab Report. Draw the best-fit line on your plot. What does the slope of the best-fit line represent? How does the Force vs. Acceleration plot relate to Newton’s Second Law? Teacher Notes Time Estimates Preparation: 15 min Activity: 30 min Objectives Students will be able to… use the Photogate/Pulley System to measure the velocity of accelerating masses on an Atwood’s machine. use the graph of velocity versus time to determine the acceleration of the masses. calculate the theoretical value of the acceleration. compare the theoretical value of acceleration to the measured value of acceleration. Notes As one mass descends, the other rises. It is possible for the masses to collide. Advise students that if the masses collide, they can record another data run for that trial. Check to make sure that the spoke arc length is correctly set for the pulley you are using. In the setup window, double click on the Smart Pulley icon make sure that the value is 0.015 m for the 10 spoke pulley or 0.05 m for the 3 spoke pulley. Make sure that the string remains in contact with the pulley throughout the experiment and that the masses fall vertically. If you have erratic data, you may wish select a portion of the graph to analyze. Click and hold down on the mouse button while dragging to form a box around the region of the graph you want. When you release the mouse button, the graph region will be highlighted and only highlighted data points will be used in the linear fit. Data Table: Constant Total Mass Run M1 (kg) M2 (kg) aexp (m/s2) Fnet (N) M1+ M2 (kg) atheory (m/s2) Percent difference Run #1 0.110 0.130 0.735 0.196 0.24 0.817 10.0 % Run #2 0.130 0.110 -0.392 -0.196 0.24 -0.817 52.0 % Run #3 0.085 0.155 2.539 0.686 0.24 2.858 11.2 % Run #4 0.125 0.115 -0.342 -0.098 0.24 -0.408 16.2 % Run #5 0.095 0.145 1.765 0.490 0.24 2.042 13.6 % Data Table: Constant Net Force Run M1 (kg) M2 (kg) aexp (m/s2) Fnet (N) M1+ M2 (kg) atheory (m/s2) Percent difference Run #6 0.110 0.130 0.780 0.196 0.24 0.817 4.5 % Run #7 0.120 0.140 0.642 0.196 0.26 0.754 14.9 % Run #8 0.140 0.160 0.491 0.196 0.30 0.653 24.8 % Run #9 0.150 0.170 0.417 0.196 0.32 0.613 32.0 % Run #10 0.160 0.180 0.440 0.196 0.34 0.576 23.6 % Questions Compare the experimental acceleration with the theoretical acceleration by determining the percentage difference. What are some reasons that would account for this percent difference? For Part 1, where the total mass is constant but the net force changes, the percentage differences range from 10.0% to 52.0%. For Part 2, where the total mass changes but the net force is constant, the percentage differences range from 4.5% to 31.9%. Several reasons account for the differences. One is that the string and pulley are not massless or frictionless. For the Constant Total Mass data, plot a graph of Fnet vs. aexp. Note: Include a negative sign for acceleration values when M1 > M2. Attach your plot to the Lab Report. Students can use DataStudio to graph the data. From the Experiment Menu, select New Empty Data Table… Type in the data values and drag a Graph icon to the Data. From the Fit menu, select ‘Linear’. Draw the best-fit line on your plot. What does the slope of the best-fit line represent? The slope represents the total mass of the system. How does the Force vs. Acceleration plot relate to Newton’s Second Law? In the Graph display, force vs. acceleration is plotted to yield mass. This relationship is the same as Newton’s Second Law. Sample Data PASCO © 2004 18 - PAGE 1 of NUMPAGES 8 18 - PAGE 2 of NUMPAGES 8 © 2004 PASCO PASCO © 2004 18 - PAGE 3 of NUMPAGES 8