PHYSICS 83 - LAB Experiment 4 Fall 004 ATWOOD S MACHINE: NEWTON S SECOND LAW th In this experiment we will use a machine, used by George Atwood in the 8 century, to measure the gravitational acceleration, g, and to test Newton s second law: F =ma. There are two versions of pulley wheel for this machine. In one version the pulley wheel is solid, and in the other it is in the form of a wheel with spokes. You will get one version or the other with which to take data. THE THEORY Two unequal masses, m and m hang from the ends of the string, which can be considered to be massless (compared to the other masses in the setup). Assume that m is larger than m. It can be shown that m will accelerate downward with an acceleration a which can be gotten from the equation: where I and r are the moment of inertia and radius of the pulley, respectively, and f is the frictional force calculated as if it acts at the pulley rim. For the solid pulley, the moment of inertia is For the wheel pulley, the calculation of the moment of inertia is a bit more complicated and is given in the appendix. A good approximation is to use the value In this experiment, the sum ( m + m ) is kept constant. A graph of the acceleration a as a function of (m m ) should be a straight line with a slope given by Figure : wheel pulley Figure : solid pulley The intercept with the vertical axis can be used to derive f. Derivation of the first equation above:
PHYSICS 83 - LAB Experiment 4 Fall 004 DATA COLLECTION. Measure the distance y between the upper platform and the floor. Estimate the error in this measurement.. Hang the following masses, m =0 grams and m =0 grams, at the ends of the string. Set m on top of the hinged platform. (m m ) = 0 grams and (m + m ) = 30 grams. 3. Measure the time taken for the mass m to reach the floor when the platform is released. Repeat this until you have taken at least 4 time measurements. Compute the average and the standard deviation (see sample calculation below). 4. Calculate the acceleration, where y is the distance traveled and <t> is the average time. Calculate the uncertainty in a, Äa in steps: i) calculate a second acceleration, a using <t>+ó t (ó t was calculated in 3.) ii) take Äa as the absolute value of the difference between a and a 5. Transfer gram to the other side, so that m is now 9 grams and m is now grams. The mass difference, (m m ), is now 8 grams, but the sum, (m + m ), remains constant. Repeat steps 3. and 4. 6. Repeat steps 3. and 4. for the difference (m m ) equal to 6 grams, 4 grams, and grams. Calculation of average and standard deviation Given the following four measured times 4.48, 4.50, 4.43, 4.63, the average is defined as The sample standard deviation [which is different than the standard deviation for a very large number of measurements] is given by The average time is thus <t> = 4.5 ±0.09 s.
PHYSICS 83 - LAB Experiment 4 Fall 004 DATA ANALYSIS. Prepare a graph on which you will plot your data. On the graph paper provided, draw a horizontal axis labeled mass difference (grams) with a scale running from 0 to 0 grams. Label the vertical axis acceleration (cm/s ) with the scale appropriate to your data.. On your graph, show uncertainty estimates for the accelerations, Äa, calculated in step 4. of the Data Collection section. 3. Draw the best straight line through the data points. From the slope of the line, calculate g. From its intercept, calculate f (see worksheet). 4. Draw the worst two lines compatible with your error bars and use their slopes and intercepts to estimate the uncertainty in your measured value of g (see worksheet). 3
PHYSICS 83 - LAB Expt 4 Worksheet Fall 004 ATWOOD S MACHINE: NEWTON S SECOND LAW STUDENT NAME DATE PARTNER S NAME LAB SECT Pulley type: solid spoked wheel M = grams y = cm m m (m m ) t <t> a a i grams grams grams s s s s cm/s cm/s t
PHYSICS 83 - LAB Expt 4 Worksheet Fall 004
PHYSICS 83 - LAB Expt 4 Worksheet Fall 004 QUESTIONS. Determine the value of g from the slope of your graph of acceleration vs. (m - m ). Show your work.. Determine the value of f from the intercept between your line of best fit and the vertical axis (i.e. at (m - m ) = 0. Show your work. 3. Explain why, in the calculation of uncertainty in the acceleration, we can safely ignore the uncertainty in the measurement of the distance traveled. (Hint: estimate the uncertainty in the distance measurement and compare the relative uncertainties in the distance and time measurements.) Show your calculations. 4. Compute the uncertainty in your measured value of g. Show your work. 3
PHYSICS 83 - LAB Expt 4 Worksheet Fall 004 At the end of the lab, turn in your worksheet with the data and answered questions as well as your graph. 4
PHYSICS 83 - LAB Experiment 4 Appendix Fall 004 CALCULATION OF MOMENT OF INERTIA WHEEL WITH MASSIVE SPOKES AND HEXAGONAL SYMMETRY This appendix will outline the calculation of the moment of inertia, I, for a wheel where the spokes have mass. In addition, hexagonal symmetry will be assumed, so we can do the calculation for /6 of the wheel. A diagram of /6 of the wheel is shown in the figure. The wheel is solid for 0 r fr (where R is the outer diameter of the wheel and f is a fraction between 0 and ). The spokes are in the space fr r fr. The wheel is solid again in the region f R r R. At a radius r = fr the spoke covers an angular range 0, and at radius r=fr it covers an angular range 0. For convenience, we set = a and = b. We also assume that the range of goes linearly from a to b as r goes from f R to f R. This means that at some r in the space fr r fr the range of is given by The angular range is calculated from The scheme is to first calculate I, then calculate M, and finally divide I by MR so we can express I as a fraction of MR. We assume a uniform density,. We will evaluate I in sections: calculate I for 0 r fr calculate I for fr r fr calculate I for f R r R 3
PHYSICS 83 - LAB Experiment 4 Appendix Fall 004 Then total I = I + I + I 3 is given by We now calculate the quantity MR, and as with I, evaluate this in sections. Calculate: M = MR for 0 r fr M = MR for fr r fr M = MR for f R r R 3
PHYSICS 83 - LAB Experiment 4 Appendix Fall 004 Then total MR = M + M + M 3 is given by Now take the ratio I/MR, it is independent of and R: Two trivial checks: a) For a solid wheel, we expect I/MR = ½. For solid wheel set a = b = /3. Then b a = 0 and we have: b) For a rim only, we expect I/MR =. In this case we have a = b = 0, f = 0 and f = and 3