A Determination of g, the Acceleration Due to Gravity, from Newton's Laws of Motion


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1 A Determination of g, the Acceleration Due to Gravity, from Newton's Laws of Motion Objective In the experiment you will determine the cart acceleration, a, and the friction force, f, experimentally for different known masses m 1 and m 2. You will then use these results to obtain g, the acceleration due to gravity. You will also learn how to deal with random errors and systematic errors in experiments. Equipment Cart, pulley with mounting clamps, string, spark timer, timing tape, wood stop, table clamp, weight hanger, weights, sand bag, masking tape, ruler and triple beam balance. Introduction In beginning physics, students test their ability to understand and apply Newton's second law by doing homework problems involving the constant acceleration of masses subject to constant net forces. The masses may be moving vertically or on surfaces that are horizontal or inclined, with or without friction. Often there are two or more masses involved connected by ropes with possibly one or more of the masses hanging from pulleys. Generally the mass of each object is given but the gravity force of the earth on it, i.e., its weight is not. To obtain the weight it is necessary to use the relation where g is called the acceleration due to gravity and must be given. Usually, the student is told that, although g varies with location, a good average value of g is 9.80 m/s 2 for motion near the earth's surface. In this experiment you will use an arrangement of the type described above to actually determine g at your laboratory location given some known masses, an understanding of Newton's second law and a record of the motion. Mass m 2 is a cart, able to roll (with some friction) on your lab table. m 2 is connected to the hanging mass m 1 by a light (negligible mass) string passing over a pulley. When the system is released from rest with the spark timer activated, the position of m 2 (and m 1 ) is recorded
2 as dots on the tape every 1/60 th of a second. It is clear that the acceleration of m 2 (and m 1 ) and the resulting record will depend on the value of g. (If you could do this experiment on the moon for instance, would you expect the acceleration of m 2 to be greater or less than that on earth? Why?) Theory Let T be the tension in the string while m 1 is moving downward. Then the string exerts an upward force T on m 1 and a force T to the right on m 2. Let f be the kinetic friction force opposing the motion of m 2. is the weight of m 1. These individual forces are shown in Figure 2a.2 below. (Note that the vertical forces on m 2 have been omitted as they will not be needed in this particular analysis.) The net force on m 1 is downward and equals m 1 g  T. The net force on m 2 is to the right and equals T  f. Since the accelerations of m 1 and m 2 must have the same magnitude, we can designate this magnitude by the letter a. Newton's second law when applied to m l and m 2 then gives: for m 1 and for m 2. On adding these two equations, the result is:. On solving for g, (2a.1)
3 Brief Comments on the Treatment of Experimental Errors Experimental error or uncertainty is inherent in any experimental result at some level, however small. It is set by a combination of the design of the experiment, the quality of the apparatus and the skill of the experimenter. Generally it is possible to separate the sources of experimental errors into two categories: random and systematic. A. Random Errors The existence of random errors in a measurement can be inferred if repetition of the measurement does not give the same result each time. By definition, random errors are those that tend to average out upon repetitions of the measurement. For random errors then, the more repetitions of a measurement the less the uncertainty in the resulting average. From the mathematics of statistics it can be shown that the uncertainty due to random error in the average of N measurements decreases as when N is "large". Thus 400 measurements should give an average with half the uncertainty due to random error as compared to 100 measurements. The first step in treating the random error in a large number N of repeated measurements is to calculate the average. The average is the desired result and it is the uncertainty of the average due to random errors which must then be determined. If the N measurements are labeled y 1, y 2, y 3,... y N the average as denoted by defined as: is (The average is also sometimes called the mean in statistics.) To establish the uncertainty in the average, the usual procedure is to first calculate what is called σ (sigma), the standard deviation of measurements. If it is believed that no one measurement is more accurate that any of the others (as we will assume) then σ is defined by: The significance of σ is that any one additional y measurement has about a 2 in 3 chance of falling between ± σ of the average,. Statistical analysis further shows that the average, has about a 2 in 3 chance of falling within ± σ of the true value. Thus, after specifying the average of N repeated measurements, it is common to append ± σ as a measure of the uncertainty due to the randomness of the measurements.
4 Example Suppose you have the following 5 measurements of the same quantity. Note: One or two more decimal points must be kept in the calculations than are in the data until the calculations are complete. The standard deviation of these measurements is: The uncertainty in the average is = The result is then ± At this point some rounding off is appropriate and the result would be reported as 1.04 ± 0.01 or possibly as ± It should be pointed out that the number 5 is stretching the concept of large N beyond reasonable bounds but that this is common practice when a large number of measurements are not available. B. Systematic Errors A systematic error is one which tends to repeat and thus create a shift in the average from the true value. Systematic errors may be provided by the experimenter, the apparatus or by poor design of the experiment. Because they are not revealed by repeated measurements, care must be exercised to investigate and account for all possible sources of systematic errors. This can be very difficult to do. Such errors sometimes remain unknown until other experimenters with other apparatus obtain convincing evidence that a previous result is off by more than the originally reported uncertainty.
5 Error Analysis in 2048C Laboratory Experiments A. Treatment of Random Errors Generally in your lab experiments in 2048C you will not measure the same quantity over and over. Often, however, you will have a moderate number of data points which are used to determine a single experimental quantity. Under such circumstances the data can be treated in two ways. The first way is the graphical method such as you used in the g experiment where the data are used to create a straight line plot the slope of which is the desired experimental quantity. The second way is to convert the data to a form equivalent to a series of repeated measurements. In the case of the g experiment, this would correspond to determining the accelerator between every pair of data points on the tape and creating a whole set of values for acceleration a, which you would then average. In general the graphical method is preferable and it is the only one you will use. It has the advantage that it gives a useful visual picture of the data. It allows you to see whether, in fact, the data exhibit the expected linear behavior and it also makes it easy to see the scatter or random errors in the data. To treat the random errors in your data using the second (nongraphical) approach you would use the results presented earlier for repeated measurements to calculate the uncertainty in the average. In the case of the graphical approach, the random errors in the data points produce a statistical error or uncertainty in the slope. The statistical treatment of random data errors in the graphical approach involves the same principles, but somewhat different mathematics, as the treatment for the effect of random errors in repeated measurements. These mathematics will not be discussed in this course in any detail but have been embodied in a computer program which you will use inside Excel. This program will analyze and plot your data and give you the uncertainty in the slope produced by random errors in your data. B. Treatment of Systematic Errors You will be expected to discuss for each experiment possible sources of systematic errors. In general, this means giving plausible reasons as to why your results might differ from the expected result by more than the uncertainty due to random error. Procedure Set up the equipment as shown in Figure 2a.1. Set the length of the string so that the hanging mass m 1 will not hit the floor when the cart reaches the edge of the table. Use about 30 cm of timer tape and adjust for at least 40 cm travel of m l and the cart. Fasten the tape to the bottom of the cart with a masking tape.
6 The amount of data which you will take in this experiment and the time required to obtain it are both small. Whether you obtain good data or not hinges on your experimental skills and attention to instructions. You must observe the following precautions while setting up. 1. Be sure that the pulley height is adjusted so that the string from the pulley to the cart is horizontal. 2. Be sure that the pulley, the cart, and the timer are all in one line with the cart aimed at the pulley. To take data, hold the cart in a steady position with m 1, just below the pulley. Have your partner take any slack out of the timing tape and then have him or her start the timer. Wait an instant for the timer to come up to speed and then abruptly release the cart. Practice the above once or twice before taking any data. Take at least three motion records with the following values of m 1 and m 2 : 1) m 1 = kg, m 2 = cart 2) m l = kg, m 2 = cart 3) m 1 = kg, m 2 = cart plus sandbag. As you obtain each tape, write on it the values for m l and m 2 so you do not get the tapes mixed up later. Inspect your tapes to see if the timer marks appear consistent with motion that starts from rest and increases in speed as time goes on. When you are convinced that you have reasonable looking tapes, ask your instructor to look at them to see if he or she concurs. Determine the force of friction using the following method: Since friction is not negligible in this experiment, the kinetic friction force, f, opposing the motion must be measured. To do this we use Newton's first law. That is, if you hang a weight on the string that produces a force equal and opposite to the kinetic friction, then the cart once set into motion ought to move at constant velocity. With m 2 = cart, start by using a mass m = 25 g initially. Give the cart a nudge and observe the motion. Increase or decrease m as needed to produce a constant velocity. You can determine the needed value of m to produce a constant velocity by direct observation. Be sure to determine the friction force for all values of m 2 that you use. Because the friction force is now given by mg and you are to determine g in this experiment, you must replace f by mg in Equation (2a.1) and solve for g. Show that the result is:
7 (2a.2) Data Analysis The assumption in this experiment (which your graphs should verify) is that the friction force is constant. If so, it follows from Equation 2a.1 or 2a.2 that the acceleration a is constant. In this case the distance, D, that the cart travels in time, t, is given by: (2a.3), and the instantaneous velocity, v, of the cart after time, t, is: (2a.4), Since the cart is released from rest, it is true that v o = 0 at the instant of release. However, your timer may not have made a mark at just that instant or you may not have released the cart cleanly. In any case, the likely uncertainty as to which tape mark represents t = 0 makes Equation 2a.4 rather than 2a.3 the best choice for analyzing the data. You should recognize that Equation 2a.4, is of the form so that a graph of v versus t on linear paper should be a straight line whose slope is the acceleration a. Figure 2a.3 illustrates this. Note that if you were to change the instant of time that you label t = 0, the graph would be shifted horizontally. v o, the intercept on the v axis, would change but the slope representing
8 the acceleration a would remain the same. Hence, such a graph will give a value for a unaffected by any initial timing uncertainty. Since the marks on the tape represent distances traveled, you will have to use these distances to determine instantaneous velocities in order to plot Equation 2a.4. To do this, measure and record the distances ΔD in cm between the centers of successive timing marks. Pick a convenient point to start measuring and try to measure as many distances as you can. Carefully measure to at least the nearest 0.05 cm (half a millimeter). Use your clear plastic ruler to do this. Omit measuring values of ΔD that are less than 3 or 4 millimeters. Also do not include the last two or three points on the tape as they may have been recorded when the cart was stopped by the wood block. Run Excel and enter in separate columns the measured distances ΔD and the corresponding time t in seconds for each of your tapes. Identify each run and record the corresponding cart mass m 2, mass of the pulling weight m 1, and the mass of the hanging weight m used to determine friction force. As discussed, the time you choose to call t = 0 doesn't matter in determining the acceleration from the graph of Equation 2a.4. It is convenient to let t = 1/60 s = s correspond to your first value of t. m 1 = m 2 = m = Time in seconds Measured ΔD in cm v = ΔD/Δt in cm/s In another column, calculate the velocity v (by using equations in Excel, do not use your calculator to do this) for each time interval by dividing ΔD by 1/60 sec which is the length of each time interval corresponding to ΔD. Graph v versus t using the Regression feature of Excel which can be obtained from Tools Data Analysis menu. The time t should be plotted on the xaxis and the measured distances ΔD on the yaxis. Make sure to click on
9 the boxes for Residuals, Residual Plots, Line Fit Plots and New Worksheet Ply to produce the information and graphs that you need for analysis. Go to your Line Fit Plot. Add a trend line and display the equation of the best fit line for v versus t. Get the slope of this line (what does this slope represent?). Also determine the uncertainty in the slope by inspecting the resulting tables. Repeat the calculation of velocity v and the graphing instructions on this paragraph for the other 2 runs. Because of the linear relation between v and t in Equation 2a.4, the average velocities you have determined are also instantaneous velocities midway through the time interval between marks. Hence a plot of your versus t on linear graph paper will give you a plot whose slope is the acceleration a. Calculate the value for the acceleration due to gravity g for each of your runs using Equation 2a.2 and your values for a, m 1, m 2 and m. Tabulate the 3 values of a and their uncertainties, and also the 3 values of g that you got. Calculate the standard deviation σ of the three measurements of g as a measure of its uncertainty due to the randomness of the measurements. Calculate the percentage difference between each of your results and the value of g = 980 cm/s 2 (9.80 m/s 2 ) expected at this location. NOTE: Include printouts of your data tables and graphs from Excel in your report. Also include the tabulation of the 3 values of a and the corresponding uncertainties, and also the 3 values of g, σ and the calculated percentage differences.
10 Questions: a. Inspect the Line Fit Plot and check how well the measured values of the velocity v plotted on the y axis coincide with the Predicted Y on the computer fitted line. Inspect the Residuals and the Residual Plot. Are your residuals reasonable or are there obvious outliers or data points that are possibly entered in error? b. By how much do each of your values for g differ from 980 cm/s 2? How does each difference compare with the corresponding uncertainty Δg due to the random errors in the data? Does it appear that there is a systematic error in either or both of your results? c. If you worked carefully and your equipment worked properly, you may see the effect of a small systematic error inherent in the design of this experiment. In order to get the cart wheels to spin faster and faster as the cart accelerates, the table must exert a static friction force on the wheel rim in a direction opposite to the cart motion. This is not the friction force you measured when the cart was moving at constant velocity and so it has not been accounted for. A calculation of this retarding force on the cart based on the mass and geometry of the wheels and the physics of circular motion shows that it should cause the measured value of g to come out about 2% low for the values of m 1, and m 2 used here. What result do you obtain if your g values are each increased by 2% (about 20 cm/s 2 ) to account for the systematic errors above? Are your data accurate enough to see this effect? Do you still have systematic errors larger than the above? d. List a few other possible sources of systematic errors that should be checked if time allowed. (Hint: Think about some of the assumptions you likely have made about the equipment that may or may not be valid.) Because each run was made under different conditions, i.e., with different masses, they do not represent an exact repetition of a single measurement. Each run may be subject therefore, to different systematic errors. Until such errors are accounted for, averaging of your results is not strictly valid Dr. H. K. Ng. All Rights Reserved. Revised Fall 2013 B. R. Reyes
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