Experiment 3. Atwood's Machine Purpose To study Newton's second law of motion by examining the behavior of an Atwood's machine. Apparatus Pasco Atwood's Machine apparatus, photogate head, mounting stand, string, two weight hangers, slotted masses of various sizes, rubber bands, Pasco LXI Data Logger, soft floor pad, electronic balance, pan balance Theory Newton's second law of motion is expressed by the equation: Fnet = Ma Equation (3-1) where Fnet is the net force acting on an object or system, M is the mass of the object or system, and a is the acceleration of the object or system. We want to apply this law to Atwood's machine, the standard form of which is a system of two masses connected by a string suspended from a pulley as shown in Figure 1 below. 2 Figure 1 The apparatus used in this experiment differs from the standard one only in the fact that it uses two pulleys rather than one. This difference does not alter the analysis of the experiment. The force of gravity on an individual object of mass m is given by: W = mg . Equation (3-2) When we analyze the motion of the Atwood's machine we have to choose a "direction" to call positive. Of course, we do that in every problem, but we are usually making the choice for a single object. Here, 3-129. Find the product of the system mass M and your experimental acceleration a. The proportional error in this product is the sum of the proportional errors in these two terms. The uncertainty in the product Ma is that proportional error multiplied by the calculated value of Ma. Record all of these results on your data sheet. 30. From Equation (3-5), we see that: f = Amg - Ma Calculate this value and record it on your data sheet. The uncertainty in the result, because this is a difference calculation, is the sum of the uncertainties in the two terms on the right side of the equation. Calculate this uncertainty in the friction and record it on your data sheet. [Remember to be mindful of significant figures as you write these results.] Procedure B 31. Move an additional small amount of mass from the ascending hanger to the descending hanger. Again, record the mass on each side (including the hangers) and the mass difference on the data sheet. For this procedure simply determine the new mass difference by adding twice the mass moved to the previous mass difference. Note that the system mass is not changed by moving mass from one hanger to the other. Do not do anything that does change the total system mass. 32. Repeat steps 18 through 24. 33. Repeat steps 31 and 32 three more times, until measurements have been made for a total of 5 different mass differences (including the original set from procedure A). For each mass difference, find an experimental acceleration. [Note: you only need to make one determination of the acceleration for each mass difference. For the part-A result use the mean acceleration that you found there.] 34. Following the instructions given in the section on Graphs and Graphing Techniques in the Introduction to this manual, plot a graph of the experimental acceleration vs. the mass difference. Plot the experimental acceleration on the vertical axis and the mass difference on the horizontal axis. [Concerning item #4 in the general instructions for graphing: for this graph, do begin the graph at (0,0). This will allow you to graphically determine the intercept of the best-fit line with the horizontal axis.] 35. Plot the best-fit straight line. Determine its slope and record it on your data sheet. Also, extend the best-fit line to the mass-difference axis and determine the value of Am at which the line intersects the horizontal axis. a exp (mis? ) Record this value on your data sheet as the "horizontal intercept". [See the figure at right. Note: the graph in Figure 2 does not follow the graphing format instructions; it is software generated for easier pasting into a word am (kg) processor document.] Figure 2the situation is different. If object . I' moves up then object .2' moves down, and vice-versa. We choose to let the situation in which object 'l' moves downward while object '2' moves upward be the positive "direction" for the system. We also stipulate that we will arrange the experiment so that mass ' I' is the heavier mass. We will use the symbols Am for the difference in the masses m, - m, and f for the force of friction in the pulleys. [Note that this designation for Am implies that m, > m,. Mass I is therefore the descending mass.] The net force on the system is then: Fnet = Amg - S The system mass is the sum of the two masses: Equation (3-3) M = mi + mz Equation (3-4) Applying Newton's second law [Equation (2-1)] to this situation produces: Amg - f = Ma. Equation (3-5) Note that Equation (2-5) does not take into account the inertia of the pulleys. The pulleys used in this experiment have very low masses and small radii; thus, their inertia contribution may be safely ignored in order to simplify the analysis. We also want to obtain the tension in the string that joins the two objects. We can do this by considering either object individually. Newton's second law applies to each object. The forces on each object are its own weight, directed downward, and the tension in the string, directed upward. Recall that for the two objects we have opposite signs for up and down. The results from this analysis are displayed below: Ti = m (g - a) (a) , and 12 = my (8 + a) (b). Equation (3-6) The two values of the tension should be very close to one another since the force of friction in the pulley is quite small. In the ideal case, T, - T2 would be equal to the force of friction. A large difference will generally indicate either an experimental error or a problem with some part of the apparatus. Note on the Apparatus The Atwood's machine is a pair of pulleys on a bracket. There should be a photogate head attached to the bracket so that it "looks" at the spokes of one of the pulleys. The photogate head should be connected to the digital adapter, which is inserted into one of the ports on the LXI data logger. One side of the photogate head emits a beam of light and the other side detects that light. The spokes of the spinning pulley produce interruptions in that light beam, and the photogate head sends electrical signals to the Data Logger that result from the flashes of light reaching the photo-detector. The LXI converts the signals into information about the velocity of the objects attached to the LXI.Procedure A Measured system mass M 2103 _kg + 0.0001 kg Proportional error in M.0OO.476 Initial value of m, .|103 kg Initial value of m2 21003 kg Initial mass difference Am 0.106 kg + 0.0002 kg Proportional error in Am 0.02 t, (s ) v, (m/s) t, (s) v2 (m/s) a (m/s2) 1.862 0. 41 2. 000 0 0 46 0: 3623 0801 0. 46 1 129 0 . 256 :3 354 0.403 0.40 1.011 0.59 . 3125 0.604 0.59 1 . 014 0 0 72 - 3171 mean value of a . 3318 m/s2 sample standard deviation in a .02260 m/$2 standard error in the mean for a 201 13 m/s2 Value of n from Table I-1 for 90 % confidence level and 4 data 2.4 Uncertainty in a .02712 m / $ 2 proportional error in a .8174 Amg .098 N uncertainty in Amg ,0096 N Ma . 6978 N proportional error in Ma 20822 uncertainty in Ma :005737 IN calculated friction force f .02822 uncertainty in f .000462 N 3-9Procedure B Table for Additional Measurements m, (kg) m2 (kg) Am (kg) t , ( s ) v, (m/s) t , ( s ) v, (m/s) derp (m/s? ) . 1103 . 1003 0160 1.862 . 41 2.000 . 46 - 3623 - 1203 -0903 . 0200 1. 077 -41 10276 . 65 1206 . 1303 0803 1.0300 : 375 1. 5 52 1.5 4 2.090 . 1408 0 0703 . 0400 571 0.63 10-577 1 = 16 2.849 1303 . 0603 16500 2.806 1 41 3.18 1061 3-846 Note: Transcribe the data from Part A into the first row of the table above and include that point in your graph. Note: Include appropriate units in your answers below. slope of graph (2) 1 (2 mm) calculated system mass (from slope) percent error in calculated system mass horizontal intercept of graph O calculated friction force (from intercept) O Does the value of the friction force resulting from this calculation lie within the range (calculated value plus or minus uncertainty) for the friction force that you found in Procedure A? 3-10Additional Questions: 1. For any one mass difference use Equations (3-6) (a&b) to find (a) the tension in the string above the descending mass m, and (b) , and (b) the tension in the string above the ascending mass my. [Remember to include units.] Record those tensions as answers 'a' and "b' in the blanks below. Indicate which mass difference is being used and show your work in the space below the answer blanks. a) b ) with An 'n' cobut it note the 2. One more calculation of the friction force: f= T, - T2 . Display your calculation and results below. Again, include units. Wwe know both the nor nacur When an object meal fores 3. How do your three values of the friction compare with one another? If the three values differ by "much", why do you think that happened? [Note that all three values should be very small. The differences, ideally, should be even smaller.] 3-11