Experiment P29: Transforming Gravitational Potential Energy to Kinetic Energy (Smart Pulley)

Transcription

1 PASCO scientific Physics Lab Manual: P29-1 Experiment P29: Transforming Gravitational Potential Energy to Kinetic Energy (Smart Pulley) Concept Time SW Interface Macintosh File Windows File energy 45 m 300 _ 700 P27 Grav PE to Total KE P29_GRAV.SWS EQUIPMENT NEEDED Interface meter stick Smart Pulley paper clips (for masses < 1 g) balance (for measuring mass) rotational apparatus calipers spirit level mass and hanger set table clamp PURPOSE In this laboratory activity, you will use a falling object to apply a constant net torque to a rotating disk. As the object falls, its gravitational potential energy decreases and it gains translational kinetic energy. At the same time, the rotational kinetic energy of the disk increases. By measuring the decrease in the falling object s gravitational potential energy and the increase in the object s translational kinetic energy and the disk s rotational kinetic energy, you will determine whether energy is conserved. THEORY The gravitational potential energy of an object depends on its weight and its vertical distance, h, relative to a reference point (usually the Earth s surface). The gravitational potential energy is: P. E. grav = mgh where m is the mass of the object and g is the acceleration due to gravity. The kinetic energy of a rotating object depends on its rotational inertia, I, and its angular speed, ω. The rotational kinetic energy is: K. E. rotational = 1 Iω 2. 2 As the object falls, it has translational kinetic energy: K. E.= 1 2 mv2 object, and v is its speed. where m is the mass of the INTRODUCTION In the Pre-Lab for this activity, you will determine the rotational inertia of the main platter of the Rotational Apparatus. The Smart Pulley measures the motion of the platter as it is accelerated by a net torque. The program calculates and displays the speed of the system. The slope of the best fit line of speed versus time is the tangential acceleration of the rotating system. If you know the torque that was applied to the rotating system, you can use the torque and the tangential acceleration to calculate the rotational inertia of the system. Finding the Rotational Inertia To find the rotational inertia of the main platter, a known torque is applied to the main platter and the resulting motion is measured. jhn 1996, PASCO scientific P29-1

2 P29-2: Physics Lab Manual PASCO scientific Since τ = Ια, solving for the rotational inertia, I, gives I = τ α. where α is the angular acceleration and τ is the torque. Now, τ = r F, where r is the distance from the center of the main platter to the point where a force is applied, and F is the applied force. In this case, the applied force is the tension (T) in a string that is tied to a step pulley that is part of the main platter. The string is pulled by a hanging mass m. The value of r is the radius of the step pulley on the platter. The radius is perpendicular to the applied force (T). Therefore, for this activity, the torque is: τ = rt. Applying Newton s Second Law for the hanging mass, m, results in: F = ma = mg T Solving for the tension in the string gives: T = m(g a). The expression for torque becomes: τ = rt = rm(g a) The linear acceleration a of the hanging mass is the tangential acceleration, a T, of the main platter. The angular acceleration is related to the tangential acceleration as follows: α = a T r Substituting for the torque and angular acceleration gives: I = τ α = rm(g a) a T r r = rm(g a) = mgr 2 g mr 2 = mr 2 1 a T a T a T The rotational inertia, I, can be calculated from the tangential acceleration, a T., the hanging mass, m, and the radius of the step pulley, r. Procedure In the Procedure for this activity, a hanging mass that is attached to the step pulley on the main platter is allowed to fall, causing the main platter to accelerate. The Smart Pulley measures the motion of the main platter and the motion of the falling mass. The program calculates and displays the speed, v, of the falling mass. You will use the program to calculate the angular speed of the rotating main platter and the change in rotational kinetic energy of the platter. You will need to calculate the following: change in gravitational potential energy of the falling mass change in kinetic energy of the falling mass total kinetic energy of the falling mass plus the rotating main platter The angular speed, ω, of the main platter is related to the linear speed, v, of the falling object: ω = v r where r is the radius of the step pulley on the main platter. P , PASCO scientific jhn

3 PASCO scientific Physics Lab Manual: P29-3 The change in gravitational potential energy is compared to the sum of the rotational kinetic energy of the falling mass and the rotational kinetic energy of the main platter. PRE-LAB Pre-Lab Part A: Computer Setup Determining the Rotational Inertia of the Main Platter 1. Connect the interface to the computer, turn on the interface, and turn on the computer. 2. Connect the Smart Pulley s stereo phone plug into Digital Channel 1 of the interface. 3. Open the document titled as shown: Macintosh X29 Inertia Windows X29_INRT.SWS The document will open with a Graph display of Velocity (m/sec) versus Time (sec) and a Digits display of Velocity. Note: For quick reference, see the Experiment Notes window. To bring a display to the top, click on its window or select the name of the display from the list at the end of the Display menu. Change the Experiment Setup window by clicking on the Zoom box or the Restore or Maximize button in the upper right hand corner of that window. jhn 1996, PASCO scientific P29-3

4 P29-4: Physics Lab Manual PASCO scientific Pre-Lab Part B: Equipment Setup Determining the Rotational Inertia of the Main Platter 1. Put the Rotational Apparatus base on a sturdy table near the edge of the table. 2. Put the spindle into the base of the apparatus. 3. Place the main platter on the spindle with the step pulleys on the top side of the platter. 4. Place the spirit level on the main platter. Use the leveling feet to level the apparatus. 5. Attach the table clamp to the edge of the table near the apparatus. 6. Mount the Smart Pulley s rod vertically in the table clamp. 7. Use a piece of string that is about 50 centimeters longer than the distance from the top of the Smart Pulley to the floor. Put one end of the string through the hole in the edge of the smallest step pulley on the main platter. Thread the string upward through the hole. 8. Tie the string around the round-head screw on the step pulley. Spindle Step pulley String Screw Hole 9. Put the string in the groove on the Smart Pulley. Step pulley Rotational Apparatus String Smart Pulley 10. Attach a mass hanger to the other end of the string. 11. Adjust the Smart Pulley so the string is parallel to the main platter. Object of known mass If you are using a PASCO mass hanger, you can attach the string by wrapping it three or four times around the notch at the top of the mass hanger. P , PASCO scientific jhn

5 PASCO scientific Physics Lab Manual: P29-5 Pre-Lab Part C: Data Recording Determining the Frictional Mass 1. To compensate for friction, find out how much mass must be put on the end of the string to overcome kinetic friction and allow the main platter to rotate at a constant speed. This friction mass will be subtracted from the total mass used to accelerate the main platter in the Procedure part of this activity. 2. Wind the string around the platter s step pulley by rotating the platter until the mass hanger is raised almost to the Smart Pulley. Hold the platter in place. 3. Click the Velocity Digits display to make it active. Move it so you can see the display. 4. Click the MON button ( ) to measure speed. Release the platter so it can begin rotating. Watch the speed in the Digits display. 5. Add or subtract mass from the hanger until the speed displayed in the Digits display is nearly constant. NOTE: You can use individual paper clips to change the mass by small amounts. When the hanger reaches the floor, rewind the string around the step pulley by rotating the platter until the mass hanger is almost to the Smart Pulley, and then release the platter again. Continue adjusting the mass on the hanger until the platter rotates with a nearly constant speed. 6. Click the STOP button ( ) to end the measurement of friction. 7. Carefully measure and record the total mass on the end of the string in the Data section. frictional mass = kg Pre-Lab Part D: Data Recording Determining the Rotational Inertia 1. Put a total of 50 g (0.050 kg) on the end of the string (don t forget to account for the mass hanger itself). Measure and record the total mass in the Data section. hanging mass = kg 2. Wind the string around the platter s step pulley by rotating the platter until the mass hanger is raised almost to the Smart Pulley. Hold the platter. 3. Click the REC button ( ) to begin recording data and release the platter. Click the STOP button just before the falling mass reaches the floor. Run #1 will appear in the Data list in the Experiment Setup window. jhn 1996, PASCO scientific P29-5

6 P29-6: Physics Lab Manual PASCO scientific 4. Use calipers to measure the diameter of the step pulley on the rotating platform. Calculate and record the radius of the step pulley in the Data section. radius of step pulley = m Pre-Lab Part D: Analyzing the Data Determining the Rotational Inertia 1. Click the Graph display to make it active. Click the Statistics button ( ) to open the Statistics area on the right hand part of the graph. 2. Click the Statistics Menu button ( ). Select Curve Fit, Linear Fit from the Statistics Menu. The coefficient a2 in the Statistics area is the slope of the best fit line for the data. This is the acceleration, at. Record the value of a2 as the tangential acceleration in the Data section. tangential acceleration = m/sec/sec 3. Calculate the rotational inertia of the rotating main platter. Record the value in the Data section. Remember, I = τ α = mr g 2 1 where m is the mass on the end of the string a T minus the frictional mass. DATA TABLE rotational inertia of main platter = kg m 2 Item Value Frictional mass Hanging mass Mass, m (hanging - frictional mass) Radius of step pulley Tangential acceleration kg kg kg m m/sec/sec Rotational inertia of main platter kg m 2 P , PASCO scientific jhn

7 PASCO scientific Physics Lab Manual: P29-7 PROCEDURE In the Procedure for this activity, a hanging mass that is attached to the step pulley on the main platter is allowed to fall, causing the main platter to accelerate. The Smart Pulley measures the motion of the main platter and the motion of the falling mass. The program calculates and displays the speed, v, of the falling mass. You can use the program to calculate the rotational kinetic energy of the rotating main platter The angular speed, ω, of the main platter is related to the linear speed, v, of the falling object: ω = v r where r is the radius of the step pulley on the main platter. PART I: Computer Setup 1. Leave the Smart Pulley s stereo phone plug connected into Digital Channel 1 of the interface. 3. Open the document titled as shown: Macintosh P29 Grav PE to Total KE Windows P29_GRAV.SWS An alert window appears when you select Open from the File menu. 4. Click Don t Save or OK, and then find the document. jhn 1996, PASCO scientific P29-7

8 P29-8: Physics Lab Manual PASCO scientific The document will open with a Graph display and a Table display of Velocity (m/sec). The velocity data tell how fast the falling mass is dropping. The velocity will be used later to determine platter s angular speed and the object s translational kinetic energy. Note: For quick reference, see the Experiment Notes window. To bring a display to the top, click on its window or select the name of the display from the list at the end of the Display menu. Change the Experiment Setup window by clicking on the Zoom box or the Restore or Maximize button in the upper right hand corner of that window. PART II: Sensor Calibration and Equipment Setup You do not need to calibrate the Smart Pulley. P , PASCO scientific jhn

9 PASCO scientific Physics Lab Manual: P Use the setup from the Pre- Lab. Step pulley Rotational Apparatus String Smart Pulley 2. Adjust the position of the mass hanger on the string. Attach the hanger so it is high enough on the string that it will not hit the floor at its lowest position. 3. Let the mass hanger fall to its lowest position. Measure the distance between the bottom of the mass hanger and the floor, and record this value as h2 in the Data section. Upper Position (h 1 ) Lower Position (h2) lower position, h2 = m 4. Rotate the platter to wind the string around the step pulley until the mass hanger is almost up to the Smart Pulley. Hold the platter at this position. Measure h1, the distance of the hanging mass from the floor. Record h1 in the Data section. upper position, h1 = m PART IIIA: Data Recording Hanging Mass = kg 1. Turn the pulley on the Smart Pulley so the photogate s beam is UNblocked (the red lightemitting diode is OFF). Make sure the bottom of the mass hanger is at its measured upper position, h1. jhn 1996, PASCO scientific P29-9

10 P29-10: Physics Lab Manual PASCO scientific 2. Click the REC button ( ) to begin recording data. Release the platter and let the mass fall while the Smart Pulley measures the motion 3. Allow the mass to reach its lower position and then begin to rise. Click the STOP button ( ) to end data recording. Run #1 will appear in the Data list in the Experiment Setup window. 4. Repeat the data recording process two more times, keeping h1 constant. PART IIIB: Data Recording Different Hanging Masses 1. Repeat the data recording process but change the hanging mass to 100, 150, and then 200 grams (0.100, 0.150, and kg respectively). 2. Keep h1 constant. Because the string may stretch, measure h2 after each run of data is recorded. ANALYSIS 1. Click the Experiment menu in the menu bar. Select Run #1 from the Experiment menu. The Graph and Table will display data from Run #1. 2. Click the Table to make it active. Click the Statistics button ( ) to open the Table s Statistics area. The Statistics area at the bottom of the Table shows Min (minimum), Max (maximum), Mean, and Std. Deviation (standard deviation) 3. The maximum speed corresponds to the point where the gravitational potential energy of the falling object reached a minimum and the rotational kinetic energy of the platter reached a maximum. 4. Use the program s Experiment Calculator to calculate the maximum rotational kinetic energy of the main platter in Joules. Remember, the rotational kinetic energy depends on the rotational inertia, I, and the angular speed, ω, of the main platter. The angular speed is related to the linear speed, v, of the falling object: ω = v r pulley on the main platter. where r is the radius of the step P , PASCO scientific jhn

11 PASCO scientific Physics Lab Manual: P29-11 a. Click the Calculator button ( ) in the Experiment Setup window to open the Experiment Calculator window. b. In the formula area, type 0.5 and then click the multiplication button ( ). c. Type in your value of the rotational inertia of the main platter. (In this example, the value is ) Click the multiplication button again. Type a left parentheses ( from the keyboard. d. Click the INPUT Menu button. Select Digital 1, Velocity (v) from the INPUT menu. The formula area will is the symbol for Digital Channel 1, and v is the symbol for velocity. e. Click the division button ( ) on the keypad. Type in your value of the radius of the step pulley. (In this example, the radius is ) Type a right parentheses ) from the keyboard. f. Type ^2 from the keyboard. This will square the value of the angular speed. g. Type Rotational KE in the Calculation Name area. Type RKE in the Short Name area. Type J in the Units area. h. Click the equals button ( ) in the keypad, or press <enter> or <return> on the keyboard to store your formula. jhn 1996, PASCO scientific P29-11

12 P29-12: Physics Lab Manual PASCO scientific 5. Click the Add Column Menu button ( ) in the Table. Select Calculations, Rotational KE from the Add Column Menu. The new column will display the calculated values of the rotational kinetic energy. The Table will have two columns; one for the linear speed of the falling object, and one for the rotational kinetic energy of the main platter. 6. Record the maximum linear speed, v, for Run #1 in the Data section. 7. Record the maximum rotational kinetic energy, RKE, for Run #1 in the Data section. 8. Repeat the analysis process for the other runs of data. Use the Experiment menu to select each run of data. Record the maximum linear speed and maximum rotational kinetic energy for each run in the Data section. 9. For each run of data, calculate and record the following in the Data section: h, the distance dropped by the falling object the maximum linear kinetic energy, 1 2 mv max2, of the falling object total KE, the sum of the maximum rotational kinetic energy and the maximum linear kinetic energy GPE (mg h), the change of gravitational potential energy of the falling object percent difference between the Total KE and the gravitational potential energy P , PASCO scientific jhn

13 PASCO scientific Physics Lab Manual: P29-13 DATA TABLE Item Value rotational inertial of main platter, I upper position, h1 kg m2 m Run mass (kg) h2 (m) h (m) v max (m/s) 1 Iω 2 2 max (J) 1 mv 2 2 max (J) Total KE (J) mg h (J) % diff QUESTIONS 1. Is the rotational kinetic energy equal to the gravitational potential energy of the falling object? 2. How does the total kinetic energy compare to the gravitational potential energy of the falling object? jhn 1996, PASCO scientific P29-13