Goals: Understand how to get acceleration from 3 graphs, and obtain acceleration due to gravity from an incline. We will use motion detectors to determine the position of a cart on an incline track. From this we can generate distance-time, velocity-time, and acceleration-time graphs. Acceleration and its error can be determined from each one of these. Using this, and the an indirect measure of the angle of the track, we can measure the acceleration due to gravity g. PROCEDURE: 1. Level an aluminum dynamics track on a table so that a dynamics cart will not roll without a push. Use a bubble level and the adjusting knobs at the bottom of the ends of the track to level the track both side-toside and front-to-back. 2. Attach a motion detector to the U-shaped metal bar protruding from one end of the track, which is meter away from the end of the track. Plug the motion detector into an interface box, and the interface box into the USB port of a laptop computer containing LoggerPro software. Make sure to plug the interface box into a working outlet, and that the laptop computer has sufficient charge for the experiment. Attach the motion detector to the Dig/Sonic1 port. 3. Elevate the end of the track (containing the motion detector) using wooden blocks. Start with the smallest angle, and later progressively larger angles, for a total of 4 different progressively larger angles. 4. Measure the center-to-center distance along the track between the track feet (using the built in tape measure), and the vertical distance from the table top to the feet of the elevated end of the track. Be careful as the small plastic rulers have a roughly 0.5 cm dead space at the end of the ruler. When you make a measurement, you need to include the length of this blank portion of the ruler. 5. Start LoggerPro, and go to the menu File -> Open - >_Physics With Vernier -> 06 Ball Toss. This will open a file containing distance-time, velocity-time, and acceleration-time graphs. Physics 140 Lab Manual Page 6 6. Allow a dynamics cart, with an attached cardboard reflector, to accelerate from rest down the track. Make sure to activate the motion detector through LoggerPro when this occurs. And make sure someone stops the cart so that it doesnt jump the track, fall on the floor, and break its fragile wheels. The motion detector needs to be aimed slightly higher that one would expect to work properly. 7. Double check the graphs before continuing. The distancetime graph should be a parabolic curve without any jumps (it should be smooth). The velocity-time graph should look linear (a line). And the acceleration-time graph should look like a horizontal line but most likely will look fairly jagged for numerical reasons associated with how the software generates the graphs. 8. Select the distance-time graph and select a large portion of your data (click and drag over it). Click the curvefit button which looks like a button with f(x) on the top tool bar. Choose a quadratic fit (At2 +Bt+C), then press Try Fit, and OK. Record A ! A . For constant acceleration, the relationship for distance is d = vot + at2 . The coefficient A = a, where a is the acceleration. Thus a = 2A, and ! a =2! A . 9. Select the velocity-time graph and select the same portion of data as in step 8. Click the curve-fit button, and choose a linear fit (mt+b). Click Try Fit, and OK. Record m ! m . Since you just measured the slope of the velocity-time graph, you measured the acceleration. 10. Select the acceleration-time graph and select the same portion of data as in step 8. Click the stats button, and use the mean and standard deviation for a ! a . 11. Repeat steps 4 through 10 for a larger elevation, until you have 4 different elevations. Analysis: 12. Using the data from the 4 different elevations of the track, calculate sin! " sin! . Since sin! = opp hyp , Physics 140 Lab Manual Page 7 ! sin" = (sin" )max ! sin! , where (sin! )max = oppmax hypmin and oppmax = opp +! opp , hypmin = hyp !! hyp 13. For the accelerations obtained from the position-time graphs for the four angles, plot a versus sin! . Make sure to include the errors in each variable, which will turn each data point into a box. For example if a = 2 1 m/s2 and sin! = 0.5 0.1, there would be points at (0.4,3),(0.4,1),(0.6,3),(0.6,1). 14. Find the slope of steepest line that goes through the most boxes. If there is a line that can go through all 4 boxes, find the steepest line that goes through all 4 boxes. Otherwise, find the steepest line that goes through any 3 of the 4 boxes. Worst case, find the steepest line that goes through any 2 of the 4 boxes. Repeat this for the line with the smallest slope (least steep). The graph may look similar to this (make sure to include axes, labels, units, etc): 15. For the accelerations obtained from the velocity-time graphs for the four angles, plot a versus sin! . Make sure to include the errors in each variable, which will turn each data point into a box. 16. Find the slope of steepest line that goes through the most boxes. If there is a line that can go through all 4 Physics 140 Lab Manual Page 8 boxes, find the steepest line that goes through all 4 boxes. Otherwise, find the steepest line that goes through any 3 of the 4 boxes. Worst case, find the steepest line that goes through any 2 of the 4 boxes. Repeat this for the line with the smallest slope (least steep). 17. For the accelerations obtained from the 4 accelerationtime graphs, plot acceleration versus sin! . Make sure to include the errors in each variable, which will turn each data point into a box. 18. Find the slope of steepest line that goes through the most boxes. If there is a line that can go through all 4 boxes, find the steepest line that goes through all 4 boxes. Otherwise, find the steepest line that goes through any 3 of the 4 boxes. Worst case, find the steepest line that goes through any 2 of the 4 boxes. Repeat this for the line with the smallest slope (least steep). 19. The slope of the acceleration-sin! graph is g. This comes from Newtons laws and analysis of the forces acting on the dynamics cart. Find g for each of the 3 graphs you produced using: g = slopemax + slopemin 2 and ! g = slopemax ! slopemin 2 20. Do any of the results of g ! g overlap the actual value of g (9.81 m/s2 )?

hyp_right(cm): 0.1 0.1 0.1 0.1 hyp(cm): 61.9 61.9 M 61.9 61.9 Chyp(cm): 0.2 0.2 V 0.2 0.2 opp(cm): 4.5 6.2 6.9 9.0 Oopp(cm): 0.1 0.1 0.1 0.1 sin(e): 0.0727 0.1002 0.1115 0.1454 0.0019 V 0.0020 0.0021 Osin(e): 0.0019 From position vs time graph: A (from At?): 0.3242 0.5025 0.5428 0.6768 04 (from At2): 0.00366 V 0.00702 0.00245 0.00245 a(m/s2): 0.6484 1.0050 1.0856 1.3536 (m/s2); 0.0073 0.0140 0.0049 0.0049 From velocity vs time graph: a(m/s): 0.6445 0.9445 0.9745 1.3723 (m/s): 0.01045 v 0.05452 0.03452 0.04226 From acceleration vs time graph: a(m/s): 0.6590 0.8590 v 1.0136 1.3863 Oa(m/s2): 0.25452 0.24226 0.28452 0.32904 g(m/s): 9.81 Results: Position Velocity Acceleration From Graph: Min slope, Smin(m/s): X X 1x Max slope, Smax(m/s2): x X g(m/s2); 0(m/s2); Was this method successful? hyp_right(cm): 0.1 0.1 0.1 0.1 hyp(cm): 61.9 61.9 M 61.9 61.9 Chyp(cm): 0.2 0.2 V 0.2 0.2 opp(cm): 4.5 6.2 6.9 9.0 Oopp(cm): 0.1 0.1 0.1 0.1 sin(e): 0.0727 0.1002 0.1115 0.1454 0.0019 V 0.0020 0.0021 Osin(e): 0.0019 From position vs time graph: A (from At?): 0.3242 0.5025 0.5428 0.6768 04 (from At2): 0.00366 V 0.00702 0.00245 0.00245 a(m/s2): 0.6484 1.0050 1.0856 1.3536 (m/s2); 0.0073 0.0140 0.0049 0.0049 From velocity vs time graph: a(m/s): 0.6445 0.9445 0.9745 1.3723 (m/s): 0.01045 v 0.05452 0.03452 0.04226 From acceleration vs time graph: a(m/s): 0.6590 0.8590 v 1.0136 1.3863 Oa(m/s2): 0.25452 0.24226 0.28452 0.32904 g(m/s): 9.81 Results: Position Velocity Acceleration From Graph: Min slope, Smin(m/s): X X 1x Max slope, Smax(m/s2): x X g(m/s2); 0(m/s2); Was this method successful