Professor Line PHYS 163 A Albert Square April 1, 2024 Experiment #0 Free Fall My Partners: Emmy Noether Billy Cobham Activity Goals: The goal of this activity is to determine if free-fall motion correctly describes the motion of a 500-g steel ball falling vertically through air. Procedure: The procedure used for this activity can be found on pp. 300-305 of the Phys 163 Fall 2024 Lab Manual. 1 Data: A 500-g steel ball was dropped from four different measured heights, and the time to reach the ground was measured. Ball mass: 500 ± 1 g Initial Height 1 y (m) 3.5 ± 0.1 Measure ± 1 2 3 Fall time t (s) 0.82 0.88 0.86 Avg t Fall Time 0.85 0.85 0.02 0.03 Initial Height 3 y (m) 10.5 ± 0.1 Measure ± 1 2 3 Fall time t (s) 1.48 1.50 1.45 Avg t Fall Time 1.48 1.48 0.02 0.03 Trial Trial 0.02 0.02 0.02 0.02 0.02 0.02 Initial Height 2 y (m) 7 Deviation from average 0.03 0.03 0.01 Initial Height 4 Deviation from average 0.00 0.02 0.03 ± 0.1 Measure ± 1 2 3 Fall time t (s) 1.19 1.20 1.19 Avg t Fall Time 1.19 1.19 0.02 0.02 Measure ± 1 2 3 Fall time t (s) 1.64 1.66 1.63 Avg t Fall Time 1.64 1.64 0.02 0.02 Trial y (m) 13.5 0.02 0.02 0.02 Deviation from average 0.00 0.01 0.00 ± 0.1 Trial 0.02 0.02 0.02 Deviation from average 0.00 0.02 0.01 Measurement Uncertainties: The following quantities were measured: • Initial height, H (m) • Time of fall, t (s) 2 Uncertainty in initial height: 10 cm The ball was dropped by hand from an open window located on each floor of Kirkbride Hall and its roof. The distance from the location of release to the ground was determined from the length of a string, weighted at one end and lowered until it reached the ground. Using a 3meter ruler, with reading uncertainty of 0.001 m, the reading uncertainty associated with the largest distance was estimated at 0.005 m. However, because the ball was dropped by hand reaching out the open window, an initial height uncertainty of 0.1 m was chosen to account for a lack of precision in locating the hand at the same height as the bottom of the window. Uncertainty in fall time: 0.02 s A digital stopwatch, which displayed time to 0.01 seconds, was used. It was assumed that the stopwatch worked correctly, so the reading uncertainty was taken to be 0.005 s. However, to account for reaction times (starting and stopping the stop watch just at the beginning and end of the fall) an uncertainty of 0.02 s was used. This was determined by making several trials of timing a second hand on the lab’s wall clock as it ticked-off 2 seconds. Analysis of Data: Sample Calculations: The following calculations were made: • Initial height, using a 3-meter ruler. As an example, the following are the calculations used in finding height 4, along with its uncertainty: H4 = (3±.001) + (3±.001) + (3±.001) + (3±.001) + (1.5±.001) =13.5± 0.005 m δH4 = 0.001+ 0.001+ 0.001+ 0.001+ 0.001= 0.005 m • Absolute difference from the average – the absolute deviation For example, the deviation associated with gcalc for the height 4 (H4) data: 10.0 −9.7 = 0.3 ms 2 • Average value 3 The average value of N measured or calculated quantities, as well as the average uncertainty for the average value, is determined using the formula xavg 1 N xi For example, to determine the average measured value of t (for Initial Height 1 data): tavg = (0.82 + 0.88+ 0.86) s = 0.85 s 3 For repeated measurements or calculations, the reported uncertainty for the average is determined by the larger of (the average of the individual uncertainties) and (the largest absolute difference between the individual quantities and the average of those quantities). The absolute difference from the average is given by xi − xavg . • Uncertainty for repeated measurements The largest of the deviation values was compared to the average reading uncertainty. The larger of these two was the reported uncertainty: For the Initial Height 1 data, 0.03 > 0.02, and the reported uncertainty is 0.03 s. • δ tavg = 0.03 s Calculation of the value for g The value of g was calculated using the constant acceleration ∆y = vo,y∆t − g(∆t)2 For example, using the height 4 data: ( ) ∆ g =−(2∆ t)y2 =− 2×(1 .−6413s.)52m =10.0ms2 δg = g δ∆(∆yy)+δ(∆∆tt)+δ(∆∆tt) 10..6402 =10.0 130..15 + 10..6402 + =10.0(0.0074 + 0.012 + 0.012) 4 =10.0(0.031)= 0.3 ms 2 • Comparison of gcalc to gtheory by % difference 9.7 −9.8 ×100% = 0.8% 9.8 • Comparison of gcalc to gtheory by agreement within uncertainty For Initial Height 3 data, g = 9.6 ± 0.4 m/s2. This means the experimental value falls somewhere within the range 9.2ms 2 ≤ gexperimental ≤10.2ms 2 The accepted value for g is 9.81 ± 0.01 m/s2, which falls within the experimentally-determined range. Results: • 2H Calculation of g, assuming free-fall analysis: g = t 2 Table R1 – Calculated values of g • H (m) ±H t (s) ±t 3.5 0.1 0.85 0.03 9.6 1.0 7.0 10.5 13.5 0.1 0.1 0.1 1.19 1.48 1.64 0.02 0.03 0.02 9.8 9.6 10.0 0.5 0.4 0.3 gcalc (m/s2) ± gcalc Summary of calculations of g from data: gtheory given in lab manual Table R2 – Comparison of gexperimental to gtheoretical 5 9.6 ± gexp 1.0 9.8 9.6 10.0 0.5 0.4 0.3 gexp (m/s2) 9.81 ± gtheory 0.01 9.81 9.81 9.81 0.01 0.01 0.01 gtheory (m/s2) % Diff Agree? 2.0 Yes 0.2 1.8 1.9 Yes Yes Yes Conclusions and Discussion When a steel ball is dropped and falls through the air, both the gravitational and air drag forces will act on the ball. For free fall motion with constant gravitational force, the acceleration of the ball should be the same value for all initial heights, and equal to 9.81 m/s2. If air resistance had a significant effect, the acceleration should decrease with increasing initial height, and be less than 9.81 m/s2. The results support the conclusion that free-fall motion describes the motion of the falling steel ball over a range of heights up to 13.5 m. This conclusion is supported by both graphical and numerical analysis. For free fall motion with constant gravitational force, the value of g near Earth's surface should be 9.81 ± 0.01 m/s2 ( source: Lab Manual, p. 302 ). Tables R1 and R2 in the Results section provide a summary of the experimentally determined values of g for different heights. Table R2 shows that the experimentally-determined acceleration of the steel ball is consistent with 9.81 m/s2 for all four heights investigated – each experimental value agrees with the accepted value of 9.81 ± 0.01 m/s2 within their uncertainties, with percentage differences between 0.2% and 2%. In addition, Graph 1: Calculated values of g for different fall distances, gives results consistent with a value for g which is independent of the initial height. This is seen from the fact that it is possible to draw a horizontal line through the data + error bars; a horizontal line on a graph indicates that the "dependent variable" is independent of the "independent variable." It can also be noted that, if air drag was significant, its effect would be to reduce the apparent value of g with increasing height. On the graph Calculated values of g for different fall distances it can be seen that a straight line of negative slope could also fit the results, which would be consistent with a conclusion that air resistance plays a part in the motion. However, the data points themselves do not appear to have any obvious negative slope trend, so a conclusion of significant air resistance effects is not very well supported by the data. (See answer to Question below.) In summary, the results of this activity supports the assumption that a 500-g steel ball, falling from a height of 13.5 m or less, can be considered as having free fall motion. 6 Question: 1) How would the presence of air resistance show up in the data? If air resistance were present, the net force acting on the falling ball would be decreased (Fg down, but FR up, opposite the direction of travel), so the acceleration of the ball would be reduced from its free fall value. Because the effects of air resistance increase with speed, the effects should also increase with initial height (the greater the fall distance, the longer the fall time, and the longer the time for the ball to accelerate). Therefore, the calculated value of g should decrease with increasing initial height. A possible line that would support the presence of air resistance is shown and labeled in Graph 1. 7 PHYS 164 LAB #24 Interference of Light Waves Instrumental uncertainty 0,2 cm dX = dD = DATA & ANALYSIS Source Wavelength ltrue = 546,1 nm Part I. DIFFRACTION GRATING GRATING 1 100 l i nes/ mm ORDER m 1 2 3 GRATING 2 300 l i nes/ mm ORDER m 1 2 3 GRATING 3 600 l i nes/ mm ORDER m 1 2 3 N= d = 1/N = D= XL (cm) 2,5 5 7,6 1000 1,00E-03 44,4 XR (cm) 3,0 5,7 7,9 lines/cm cm cm Xavg (cm) 2,8 5,4 7,8 Grating constant Screen distance ± tan q (cm) 0,2 0,06 0,2 0,12 0,2 0,17 N= d = 1/N = D= XL (cm) 4,7 9,3 14,4 3000 lines/cm cm 3,333E-04 cm 26,3 XR Xavg (cm) (cm) 4,5 4,6 8,7 9,0 12,3 13,4 Grating constant Screen distance ± tan q (cm) 0,2 0,17 0,2 0,34 0,2 0,51 N= d = 1/N = D= XL (cm) 3,9 8,3 13,3 6000 lines/cm cm 1,667E-04 cm 9,8 XR Xavg (cm) (cm) 3,7 3,8 9,4 8,9 14,2 13,8 Grating constant Screen distance ± tan q (cm) 0,2 0,39 0,2 0,90 0,2 1,40 ± 0,00 0,01 0,01 ± 0,01 0,01 0,01 ± 0,02 0,03 0,03 q (rad) 0,06 0,12 0,17 q (rad) 0,17 0,33 0,47 q (rad) 0,37 0,73 0,95 6. Compare appearance of mercury spectrum w/ monochromatic source (use 600 lines/mm grating) Color diffracted most? Orange Color diffracted least? Part II. THIN-FILM INTERFERENCE 8. Make a sketch on a sheet of paper of the fringes 9. Describe effect on fringes by depressing one edge 10. Measure fringes over cm  sin q ± 0,00 0,00 0,01 0,06 0,12 0,17 0,00 0,00 0,01  sin q ± 0,01 0,01 0,01 0,17 0,32 0,45 0,01 0,01 0,01  sin q ± 0,02 0,02 0,01 0,36 0,67 0,81 0,02 0,01 0,01 Purple lexp (nm) 618,2 598,2 573,2 ± (nm) 47,6 24,7 16,9 % Diff lexp (nm) 574,3 539,6 502,9 ± (nm) 26,3 12,6 7,5 % Diff lexp (nm) 602,5 558,5 452,4 ± (nm) 32,9 10,1 3,8 % Diff 13,2 9,5 5,0 5,2 1,2 7,9 10,3 2,3 17,2 Agree w/ Uncert ? No No No 570,6 573,5 556,3 Agree w/ Uncert ? No Yes No 548,0 552,2 510,4 Agree w/ Uncert ? No No No 569,6 548,4 448,6 67 #24 Interference of Light Waves Objective The objective of this experiment is to observe various manifestations of the interference of light waves. Introduction and Theory Because light has a wave nature it can exhibit the same phenomena that, say, water waves do when they interact with sharp boundaries or with other waves. The effects of interaction include reflection, refraction, dif- fraction, and interference. The effects of diffraction and interference will be the subject of this laboratory. I. The Diffraction Grating The diffraction grating consists of a very large number of fine, equal- ly spaced parallel slits. There are two types of diffraction gratings: the reflecting type and the transmitting type. The lines of the reflection grat- ing are ruled on a polished metal surface: the incident light is reflected from the unruled portions. The lines of the transmission grating are rule on glass: the unruled portions of the glass act as slits. Gratings have typi- cally between 100 to 800 lines per millimeter. A diffraction grating pro- vides the simplest and most accurate method for measuring wavelengths of light. The principles of diffraction and interference are applied to the meas- urement of wavelengths with a diffraction grating. Let the vertical broken line in Fig. 24.1 represent a magnified portion of a diffraction grating. Let a beam of parallel monochromatic light, originally from a single source and having passed through a slit, impinge upon the grating from the left. By Huygens' principle, the light spreads out in every direction from the apertures of the grating, each of which acts as a separate new source of light. The envelope of the secondary wavelets determines the position of the advancing wave. In Fig. 24.1 we see the instantaneous positions of several successive wavelets after they have advanced beyond the grating. Lines drawn tangent to these wavelets connect points which are in phase: hence they represent the new wave fronts. One of these wave fronts is tan- gent to wavelets which have all advanced the same distance from the slits, and the wave front formed is thus parallel to the original wave front. A converging lens placed in the path of these rays would form the central image. Another wave front is tangent to wavelets whose distances from adjacent slits differ by one wavelength. This wave front advances in the direction 1 and forms the first-order image. The next wave front is tangent to wavelets whose distances from adjacent slits differ by two wavelengths. This wave front advances in the direction 2 and forms the second-order image. Images of higher orders will be found at correspondingly greater angles. 69 If the light is polychromatic, however, there will be as many images of the source, the diffracting angle for each wavelength being given by the above slit in each order as there are different wavelengths in the light from the spectrum, and so on. Spectra produced in this manner will be discussed in detail in Experiment #24. 11 ng 13 12 C P 1 @ t Fig. 24.3. Interference of two beams incident on the left and right boundaries of a thin film II. Thin-Film Interference Interference is the combining by superposition of two or more waves that meet at one point in space. Thin-film interference can be understood as the combining by superposition of rays reflected from opposite sides of the thin film. Examples of thin films are soap films, lens coatings, or a thin wedge of air formed between two glass plates as shown in Fig. 24.3. Ray rı is reflected from the left surface of the film and ray r2 is re- flected from the right surface. If t is the thickness of the air wedge at the point where ray rı strikes it, then ray r2 will travel a distance 2t further than ray ri. Therefore, if the rays were in phase before they reached the wedge, when they combine they will no longer be in phase. Three effects must be taken into account in determining the net result of this type of superposition. First, not only does a ray undergo a partial reflection when encountering another medium (glass-to-air or air-to-glass), but it may also experience a 180° phase shift (which corresponds to a shift of 1/2 in the wavetrain). A 180° phase shift occurs whenever the index of refraction of the first medium is less than that of the second (e.g., air-to- glass). Such a change of phase does not occur, however, when the index of refraction of the first medium is greater than that of the second (e.g., 70 glass-to-air). Because of this effect, the two rays will travel essentially the same distance (assuming t
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