Physic Lab Report

Hooke’s Law Experiment Objective: To measure the spring constant of a spring using two different methods. Background: If a weight, , is hung from one end of an ordinary spring, causing it to stretch a distance , then an equal and opposite force, is created in the spring which opposes the pull of the weight. If is not so large as to permanently distort the spring, then this force, will restore the spring to its original length after the load is removed. The magnitude of this restoring force is directly proportional to the stretch, The constant is called the spring constant. To emphasize that length of the spring we write refers to the change in (1) In this form it is apparent that if a plot of as a function of has a linear portion, this provides confirmation that the spring follows Hooke's Law and enables us to find . An additional approach is possible. One definition of simple harmonic motion is that it is motion under a linear, “Hooke's Law” restoring force. Note that for simple harmonic motion, the period does not depend upon the amplitude of the oscillation. For such a motion, we have 2 4 2 / (2) Thus, the mass includes the mass of the spring itself. However, the entire spring does not vibrate with the same amplitude as the load (the attached mass) and therefore it is reasonable to assume that the effective load (m) is the mass hung from the end of the spring plus some fraction of the mass of the spring. Based on similar experiments, one third of the mass of the spring is a good estimation of the effective load due to the spring, thus 1 3 where is the effective load of the spring. Using this in Eq. (2), we find 4 2 { 1/ 3( 2 )} (3) 4 Eq (3) uses an approximation for the contribution of the mass of the spring to the oscillation. If we rewrite Eq (2) as the effective mass of the spring and hanging mass (load), then T2 4 2 m load m ES k 4 2 m load k 4 m2 k ES (4) where mload is the hanging mass and mES is the effective mass of the spring. If we assume that the effective spring mass is the same for all loads, then a graph of period squared (T2) vs. hanging mass (mload) is a straight line, where 4 2/k is the slope and 4 2mES/k is the intercept. Meter Stick Procedure: 1. Hang a spring from a horizontal metal rod. 2. Attach a mass hanger directly to the bottom of the hanging spring and record the position of the bottom of the mass hanger relative to a meter stick. 3. Add masses to the spring and record the position of the bottom of the mass hanger. 1. Hang a mass from the spring and use a stopwatch to time 15 oscillations of the mass and spring. 2. Repeat for other masses. 5 Physics Section A1 Your Name Partner(s): Full Name(s) Date Performed: TA: Full Name Two experiments were performed to find the spring constant of a steel spring. The spring constant was determined statically, by measuring its elongation when subjected to loading, and dynamically, by measuring the period of a mass hung from one end and set into vertical oscillation. The resulting values of 2.94 ± 0.01 N/m and 2.98 ± 0.02 N/m, respectively. Our spring's behavior followed Hooke's law to within the limits of accuracy of the two experiments. (76 words) The purpose of this experiment was to measure and compare the spring constant of a steel spring using two different procedures. First we investigated the relationship between the force applied to a spring and the displacement of the spring from its rest length. We hung various masses from the springs, and measured the vertical displacement. We found a spring constant of 2.94 ± 0.01 N/m. Our results confirmed Hooke’s Law, Fs = -kx. In the second procedure, we set the spring into vertical oscillation with a suspended mass and measured the period of oscillation. Using this method, we found a spring constant of 2.98 ± 0.02 N/m. Our results verified that the period of oscillation depended on the effective mass of the spring and the period of oscillation. (128 words) 6 A Poor Abstract – Too long because it has too much detail and unnecessary information. The purpose of this experiment was to determine the spring constant k of a steel spring using two different methods. First we investigated the relationship between the force applied to a spring and the displacement of the spring from its rest length in order to verify Hooke’s law. We found a spring constant of k = 2.94 ± 0.01 N/m. Our results confirmed Hooke’s Law, Next we measured the period of a mass hung from one end of a spring and set into vertical oscillation. We found that the spring constant depended on the effective mass of the spring and the period of oscillation. The period of the motion was the same whether the amplitude of the oscillation is large or small. Using this method we found a spring constant of 2.98 ± 0.02 N/m. This value is consistent with the result obtained using the first method. (291 words) Remember to be concise. 7 Name: ___ Partner: ___ ________ TA’s Initials on data sheet Date:__Date Exp. Performed You can include uncertainty for a tool such as a meter stick in the top of a data table. Hooke’s Law and a Simple Spring Part 1 Position Uncertainty can be written as a percentage. Reference 1 2 3 4 5 6 7 8 9 10 11 For the subtraction all the uncertainties were the same; thus it was put in the top of a column; that was not possible for the average. Do you know why there are only 2 sig figs here? Table 1 Location of the Mass Hanger Reference in cm Mass ( ) ±1% Trial 1 0 1 3 5 10 20 40 60 80 100 120 140 69.55 69.27 68.61 67.95 66.42 62.90 56.32 49.65 42.97 36.32 29.63 23.07 Force Trial 1 0.00981 0.0294 0.0491 0.0981 0.196 0.392 0.589 0.785 0.981 1.18 1.37 -0.28 -0.94 -1.60 -3.13 -6.65 -13.23 -19.90 -26.58 -33.23 -39.92 -46.48 Trial 3 69.50 69.19 68.50 67.87 66.20 62.89 56.22 49.60 42.97 36.30 29.70 23.05 69.50 69.18 68.53 67.88 66.21 62.90 56.30 49.61 42.95 36.32 29.72 23.10 Table 2 Displacement x Trial 2 Trial 3 -0.36 -1.05 -1.68 -3.35 -6.66 -13.33 -19.95 -26.58 -33.25 -39.85 -46.50 Average Spring Constant k Trial2 -0.37 -1.02 -1.67 -3.34 -6.65 -13.25 -19.94 -26.60 -33.23 -39.83 -46.45 Trial 4 69.50 If the value is four sig figs then include the trailing 0. 69.17 68.52 67.86 66.20 62.93 56.23 49.6 42.95 36.32 29.72 23.12 Trial 4 Average -0.38 -1.03 -1.69 -3.35 -6.62 -13.32 -19.95 -26.60 -33.23 -39.83 -46.43 -0.35 ± 0.02 -1.01± 0.02 -1.66± 0.02 -3.29± 0.05 -6.65± 0.01 -13.28± 0.02 -19.94± 0.01 -26.59± 0.01 -33.24± 0.01 -39.86± 0.02 -46.47± 0.02 If you drop that zero then it is 3 sig figs! This should have been 49.60! Spring Constant 2.8 ± 0.2 2.91± 0.08 2.96± 0.05 2.98± 0.05 2.95± 0.03 2.95± 0.03 2.95± 0.03 2.95± 0.03 2.95± 0.03 2.95± 0.03 2.96± 0.03 = __2.94 ± 0.01 N/m _ Spring Constant k from Graph = __ 2.95 N/m ________ Calculate the standard error. There is a way to find the uncertainty for slopes of graphs but we will not do that in this course. Remember to include units and uncertainty. 8 Name: ___ ________ With 1% uncertainty in the slotted masses, the uncertainty for 145g is ±1g and for 55g, it is ±0.06g; thus for the smallest two loads, can be written in the form below. Part 2: Mass of spring = Cross out mistakes with a single line; do not use white-out 10.19 ± 0.02 x 10-3 kg____ Table 3 Load (x10-3 kg) 145 (mass of spring) ±1% Time for Trial 1 27.94 20 (small) oscillations Trial 2 27.79 (medium) (s) Trial 3 27.90 (large) ±0.08 Average 27.88 ± 0.04 20 oscillations (s) 105 55.0 23.98 17.60 24.06 17.44 23.95 17.34 25.0 12.32 12.56 12.40 12.03 12.34 24.00± 0.03 17.46± 0.08 1.394± 0.002 1.200± 0.002 0.873± 0.004 (s ) 1.943 ± 0.006 1.440 ± 0.004 0.762± 0.007 0.382 ± 0.001 k from Eq. (3) (N/m) 3.01 ± 0.03 3.02± 0.04 2.94± 0.03 Period (s) Period 2 2 Average Spring Constant k = 2.97± 0.03 2.98± 0.02 N/m_ Spring Constant k from Graph = __ 3.01 N/m 12.35± 0.02 Do not cross out mistakes this way! Sometimes what you had was correct. You may want to be able to read it. 0.618± 0.001 Period was 12.35s/20= 0.6175s Uncertainty was 0.001; thus we write the period with 3 sig figs (0.618) Spring’s Effective Mass from Graph = _ % Difference between k in Part 1 and Part 2 = _ k in parts 1 and 2 was 3 sig figs; but the difference between the two values is one sig fig!!!! Remember to include an uncertainty! Guessing and then seeing what the value is for the average of a small number of trials helps build your ability to predict uncertainty. Why 0.08 in this case? One way to measure your reaction time is to have a friend hold a meter stick vertically between your thumb and first finger; note the cm mark between your fingers; the friend drops the meter stick, and you catch it. Determine how far it fell before your caught it. Since it had no initial velocity and the only force acting on the meter stick is gravity, then d=½gt2. For most people, the distance the meter stick falls is about 15 cm. However, there is a difference between someone dropping a meter stick and timing an oscillating object. When timing an object, we can observe the motion and use the rhythm to reduce reaction time error. This might reduce the timing error to a third of the original value: treaction time/3 = sqrt(2d/g)/3 = 0.06 s However, there are two uncertainties – starting the stopwatch and stopping the stopwatch; thus, we need to propagate the error. t = sqrt( tstart + tstart) = treaction time sqrt(2) = 0.08s We might have made a different assumption, then we might have used had a value or 0.05 to 1.0 s. Remember to include units and uncertainty 9 Restoring Force vs. Displacement Magnitude 1.4 Force = 2.95x - 0.0001 x = displacemnt 1.2 1.0 Not all graphs will start at zero as this one does. If your data range for the yaxis is from 6 to 20 newtons, then use a graph that starts at 5 and ends on 20 or 25 newtons. Never force a line to go through the origin. NEVER!! Your TA may have other information that he/she wants on a graph. Then be sure to include that. Force (N) 0.8 0.6 Graphs must conform to all the rules given in lab handbook. Each graph should be on a separate page. This graph was inserted into this Word document by a simple “copy and paste” from Excel into Word. 0.4 0.2 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Displacement Magnitude (m) 0.40 0.45 0.50 10 This title is incorrect! Do you know why? Mass vs. Time Squared 3 y = 13.09x + 0.052 If you use Excel then edit x and y so it looks like this T2 = 13.09M +.052 2.5 y = 13.566x Dashed curve forced to intercept the origin Do not force a curve! Error bars were added to this graph to show that when the curve was forced to go through the origin it then did not honor the data collected. Time2 2 1.5 Do not use the default backgound color, use white. It can be difficult to see data points when printed. 1 Add gridlines for both axes, not just the default as in this plot. Scale the graph so that the data is spread across the graph. This graph has too much unused space. 0.5 Since there are two graphs - each is worth 5 points. I would give this graph only 2 points out of 5. 0 0 0.02 0.04 0.06 0.08 0.1 Mass 0.12 0.14 0.16 0.18 0.2 11 Restoring Force vs. Displacement 1.6 Force = 2.9541x - 0.0001 1.4 1.2 Force (N) 1.0 0.8 0.6 0.4 0.2 0.0 0.00 0.05 0.10 0.45 0.50 2.5 T2 = 13.09M + 0.052 2 Time2 (s2) 1.5 1 0.5 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Mass (kg) 12 1. Displacement: the length that the spring is stretched Displacement Location with Mass 1 (0.010kg) - Reference Location 66.42 10 2 69.55 10 2 3.13 10 2 ): Propagation of error for addition and subtraction 2. Uncertainty of displacement ( 2 x = (uncertainty in reference) + (uncertainty in location 1) -2 2 -2 x = (0.05 10 m ) + (0.05 10 m) 2 2 x = 0.07 10 -2m 3. Force on spring from the hanging mass (10.0 )(1 /1000 )(9.81 / 2 ) 0.0981 4. Standard Error for Average Displacement for 0.9811N force (xi -x) 2 Standard Error = ( 1) x = {[( 3.13 10 -2m ( 3.29 10 -2m)) 2 ( 3.35 10 -2m ( 3.29 10 -2m)) 2 ( 3.34 10-2 m ( 3.29 10-2 m))2 ( 3.35 10-2 m ( 3.29 10-2 m))2 ] /[(4 1)4]}1/ 2 2 x = 0.05 10 m 5. Using Hooke’s Law (F = - kx) to find the spring constant / 2 0.0981/( 3.29 10 ) No units! No credit would be given for this sample calculation. 2.98 6. Spring constant uncertainty: Propagation of error for multiplication and division k = k ( F/F) 2 + ( x/x) 2 0.01 0.0981 0.0981 k = 2.98N/m 2 + 0.05 10 2 3.29 10 2 2 = 0.05 N/m 7. Spring constant from period of oscillation 4 2 { 1/ 3( )} 2 4(3.14) 2 (145 10 3 1 10.19 10 3 3 ) / 1.392 2 3.01 N/m 13 8. Spring constant uncertainty: propagation of error for T2 A = nA( T/T) where A= T n 2 T = 2(1.394s) 2(.002s/1.394s)= 0.006 s 2 9. Spring constant from the slope from T2 vs. Mload graph (2 ) 2 (2 ) (2 3.14)2 13.09s2 /kg 2 3.01 / 10. Spring’s Effective Mass, MES, from the intercept (2 )2 /(2 )2 3.01N/m 0.052 2 /(2 3.14)2 0.0040 11. % difference for spring constant k % % Measured Value1 - Measured Value 2 Measured Value1+Measured Value 2 / 2 2.94N/m - 2.98N/m 2.94N/m + 2.98N/m / 2 100% 100% 1% 14 In part 1, a spring was hung vertically with a mass hanger attached to the lower end of the spring, and masses from 1g to 140g were added. The downward location of the spring was measured once it came to rest. In this configuration, two equal and opposite forces acted on the hanging mass: gravity directed downward and the spring’s elastic restoring force directed upward, in the opposite direction of displacement. Using Hooke’s Law (F = -kx), a spring constant was calculated for each measurement. The spring constants for each value of displacement are the same, within experimental uncertainty (Table 2), which verifies Hooke’s law. The average spring constant is 2.94±0.01 N/m. A graph of force versus the magnitude of displacement resulted in the expected straight line in the range of forces examined and is consistent with Hooke’s law. The slope of this line, 2.95 N/m, is the spring constant, which agrees with value found by taking the average of the calculated spring constant (2.94±0.01 N/m). The intercept for the best fit straight line intersects close to the origin, which is also consistent with Hooke’s law. The sources of error in this part of the experiment are due to the precision of the location measurement using the meter stick and the accuracy of the slotted masses. The meter stick was mounted vertically and behind the spring. The location was measured relative to the base of the mass hanger. Effort was made to sight the measurements directly; however, because of the location of the meter stick it was necessary to view the meter stick at a slight angle. However, this sighting was required for each measurement, and the displacement was the difference between the location and the reference. Thus, this systematic error due to parallax should be minimal. However the random error of measurement precision remains. For displacements 20 cm or more, the uncertainty of the displacement of the spring is 0.5 % or less and has little impact on the uncertainty of k; in those cases the 1% uncertainty in the slotted masses has the greatest contribution to the uncertainty of k. However, for small displacements the displacement uncertainty has the largest impact on the uncertainty in k. For example, the 1.0 g mass displaced the spring by -0.0035 ± 0.0002 m, a relative uncertainty of 6%. Using a motion sensor to measure distance would increase the precision for small displacements. In Part 2, we determined k dynamically using the period of an oscillating mass. The time for twenty oscillations was measured for five different masses; for each mass the period of oscillation was measured three times using different oscillation amplitudes, as suggested by out lab instructor. 15 The period of a mass oscillating vertically on a spring depends on the spring constant and the mass of the oscillating object, but not on the amplitude of the oscillation. Our measurements confirmed that the amplitude of oscillation, within experimental uncertainty, did not affect period (Table 3), To reduce the reaction time, we observed the motion and used the rhythm to start and stop the stopwatch. ( For small masses, the period of the oscillation is shorter; this is consistent with Eq (2). These shorter periods for the 55g and 25g masses made accuracy in the timing both critical and difficult. The measured times for 20 oscillations of the 55g mass are not as consistent as for the other masses. This was the result of reaction time random error. Two measurements for 20 oscillations of the 25g mass were so different from the other measurements that we made additional measurements and replaced those data points. There was another complication for these smaller mass, large amplitude oscillations caused the slotted masses to bounce on the mass hanger. This meant that we had to use smaller amplitude differences between the large and small amplitude oscillations for the smaller masses. (You may need to combine two equations.) Using Eq (3), we found k for the four different loads added to the spring. The four values of k for the four different masses were in agreement (Table 3). The average value of k is 2.98±0.02 N/m. ( A graph of T2 vs. Mload is a straight line and consistent with the theory that the period is a function of the effective mass of the spring and the spring constant of the spring, Eq, (4). The spring constant k from the slope is 3.01 N/m; the effective mass of the spring MES from the intercept of the best fit line is 4.0g, which is approximately 40% of the mass of the spring, which is somewhat higher than the fraction used in Eq. (3). The sources of error in this part of the experiment are due to the accuracy of the slotted weights and the accuracy of the time measurements. As mentioned previously, the reaction time uncertainty is greater for the smaller loads. However, due to the care that taken in the time measurements and the fact that 20 different oscillations were measured, the uncertainty in the time measurements was not as important in this experiment as the uncertainty in the slotted masses. There is uncertainty (1%) in the mass of the slotted weights. It would have been prudent to have measured the masses on the triple beam balance so that we would have less uncertainty in the mass of the oscillating weights; however, we did not make those measurements. The value of the spring constant found in Part 1 (2.94±0.01 N/m) and Part 2 (2.98± 0.02 N/m) do not agree. However, the percent difference between the two values is only 1%. One possible explanation for the small discrepancy may be that the time measurements were precise, but not accurate due to a systematic error in the timing. If our time measurements of the twenty oscillations were low by as little as 0.05s, then the spring constant values would agree. 16 Besides measuring the spring constant using two very different methods, we verified Hooke’s law, verified the linear relationship between period squared and load for a vertically oscillating spring, and observed that the amplitude of the oscillations did not affect the period. sentences. Answer questions in complete, grammatically correct 17