SIMPLE HARMONIC MOTION OBJECTIVE To calculate the spring constant ‘k’ of a spring by using Hooke’s Law and N2L and compare the results. EQUIPMENT 1. 2. 3. 4. 5. 6. 2-support rods and clamp spring masses and hanger stopwatch 2-m stick Triple-Beam Balance THEORY I. Using Hooke’s Law Consider a spring suspended vertically in its equilibrium position. Suppose you add a mass ‘m’ to the end of the spring that displaces the spring an amount ‘x’ from equilibrium. Fs m x mg x m Applying N2L gives: ∑F x = mg − kx = 0 mg = kx 1. The graph of mg vs. x will give a straight line with the slope of the line equal to the spring constant ‘k’. 2. We will take this value of ‘k’ to be the expected value. 1 II. Using N2L Suppose you add a mass ‘m’ to the end of a suspended vertical spring. The mass displaces the spring an amount ‘x1’ from equilibrium. In this position the mass is in equilibrium. Consider displacing the spring an amount ‘x’ from the new equilibrium position. x1 kx1 m k(x+x1) m m x x x mg mg m Applying N2L for the first displacement gives: ∑ Fx = mg − kx1 = 0 mg = kx1 Applying N2L for the second displacement gives: ∑ Fx = mg − k ( x + x1 ) ∑F ∑F ∑F x = mg − kx − kx1 x = kx1 − kx − kx1 x = − kx Net Force on Mass ∑F = − kx = m x d 2x dt 2 d 2x  k  +   x = 0 Simple Harmonic Motion Equation dt 2  m  2 1. Confirm that the solution to this equation is given by: x(t ) = A cos(ωt + φ ) Solution to SHM Equation Where, x(t) = amplitude of oscillation (rad) A = maximum amplitude of oscillations from equilibrium (rad) k (angular frequency in units of rad/s) It is a measure of how fast ω= m the oscillations occur. t = time (s) φ = phase angle (rad) (determined by initial conditions) 2. The cosine and sin function repeat every period T. Thus: θ (t ) = θ (t + T ) θ m cos(ωt + φ ) = θ m cos[ω (t + T ) + φ )] θ m cos(ωt + φ ) = θ m cos[(ωt + φ ) + ωT )] The sine and cosine repeat when their phase changes by 2π . Thus, ωT = 2π T= 2π ω = 2π m = 2π k k m 1 T 2 =   4π 2 m k 3. 4. The graph of T2 vs. 4π2m will give a straight line with the slope equal to 1/k. We will take this ‘k’ value to be the experimental result and compare to the expected value. PROCEDURE 1. Attach spring to horizontal rod and measure equilibrium position. 2. Attach 100g to end of spring and measure displacement ‘x’ from equilibrium. 3. Displace the mass slightly from equilibrium and release. 4. Measure the time for 10 oscillations and calculate the period. Repeat for a total of 3 runs. 5. Repeat steps (1) – (4) for the masses listed in the table below and fill in the rest of the data. 6. Make a graph of mg vs. x using EXCEL and from the equation of the best curvefit determine the expected value of k. 2 vs. 4π 2 m using EXCEL and from the equation of the best 7. Make a graph of Tave curve-fit determine the experimental value of k. 8. Compare both values of k. 3 DATA TABLE m(gram) mg x t1 T1 t2 T2 t3 T3 Tave 2 Tave 4π2 m 100 150 200 250 300 4 PHYSICS SAMPLE LAB WRITE-UP Title - Newton’s 2nd Law Objective In this experiment we will attempt to confirm the validity of Newton’s 2nd Law by analyzing the motion of two objects (glider and hanging mass) on a horizontal air-track. First, we will calculate the theoretical acceleration by applying Newton’s 2nd Law (Fnet = MA), neglecting friction, to the glider and hanging mass. Next, we will calculate the experimental acceleration of the glider by applying the kinematic equations of motion as it moves between two markers (photogates) on the track. We will then compare the experimental acceleration to the theoretical acceleration. Theory a) Acceleration using Newton’s 2nd Law Apparatus Setup V1 Photogates V2 glider M1 d +X Airtrack M2 +Y hanging mass Free-Body Diagram N T T M1 M1g M2 M2g Apply Newton’s 2nd Law to mass M1 and M2. Mass ‘M1’ ΣFx = T = M1a Mass ‘M2’ 1 ΣFY = M2g - T = M2a Adding both equations gives: M2g = M1a + M2a atheo = M2g/(M1 + M2) b) Acceleration using Kinematic Equations Using the kinematic equation V22 = V12 + 2a ( x − x0 ) we will calculate the experimental acceleration of the glider as it moves between the two photogates. We will take the origin of our coordinate system at the first photogate. d = distance between photogates V1 = (s/t1) velocity of the glider through photogate 1 V2 = (s /t2) velocity of the glider through photogate 2 s = diameter of small flag on glider t1 = time for small flag to go trough photogate 1 t2 = time for small flag to go trough photogate 2 a exp = V22 − V12 2d Apparatus Refer to theory section for apparatus setup One air track(#21), blower(#2), blower hose and power supply One digital photogate(#2C) and one accessory photogate(#2A) One glider(#1B) One flat accessory box(#22A) String Electronic pan balance(#1) Vernier Calypers (#12c) Procedure 1. 2. 3. 4. 5. Measure the mass of the glider and hanging mass. Setup the air track and blower as indicated by instructor. Measure the distance between photogates. Measure the diameter of the small flag on glider with vernier calipers. Release glider 10 cm away from photogate 1 and record times trough both photogates. 6. Repeat step (5) four more times. 2 Data M1= 4750 g M2=50.00 g g = 9.80 m/s2 d = 60.65 cm s = 1.01 cm Run # t1 t2 1 2 3 4 5 0.039 0.043 0.044 0.041 0.038 0.023 0.024 0.023 0.023 0.032 V1 (cm/s) 25.5 23.0 22.5 24.5 26.0 V2 (cm/s) d (cm) 43.0 41.5 42.5 42.5 43.5 60.65 60.65 60.65 60.65 60.65 aexp (cm/s2) 9.91 9.86 10.7 9.97 10.1 Calculations Theoretical Acceleration: atheo = M2g/(M1 + M2) = 50.00 g*980 cm/s2/(4750g + 50.00 g) atheo = 10.2 cm/s2 Experimental Acceleration: a exp = V22 − V12 = (43.5 cm/s)2 - (26.0 cm/s)2 /(2*60.65 cm) (sample calculation Run #5) 2d aexp = (9.91 +9.86+10.7+9.97+10.1)/5 = 10.1 cm/s2 (average experimental acceleration) % error = exp− theo % error = � theo 10.1−10.2 10.2 × 100 � X 100 = 0.98 % 3 Conclusion 1. The theoretical acceleration using Newton’s 2nd Law was 10.21 cm/s2 and the average experiment acceleration using the kinematic equations was 10.10 cm/s2. The percent error between experiment and theory was only 1%. Although the percent error was small, there were still systematic and random errors present. 2. Based on the relative small % error of 0.98% we can conclude that the objective of confirming Newton’s 2nd Law was accomplished. 3. In measuring the velocity of the gliders through the photogates we used the average velocity instead of the instantaneous velocity. This resulted in the average velocity always being smaller than the instantaneous velocity. This will V 2 − V12 then cause a exp = 2 to be consistently smaller than atheo which resulted in a 2d systematic error. A second systematic error was that in applying Newton’s 2nd Law to derive atheo of the glider we neglected the frictional force. The resulting equation should have been atheo = (M2g – fk)/(M1 + M2). Neglecting friction on the atheo equation should result in atheo being consistently larger than aexp. The data shows this to be true with the exception of one data point. 4. In addition to the random errors involved due to the uncertainty of the measuring devices, other random errors involved in the experiment include: a) Not releasing the glider from same initial point every run. b) Trying to balance the air track. c) Having the hanging mass M2 swinging when releasing M1 from rest. All these random errors contributed to the uncertainty in the final results for the accelerations. These random errors also contributed to the 0.98% error in the final results. 4 Lab Report Format Physics 2A, 4A-D 1. TITLE Place the title of the lab experiment at the top of the first page. 2. OBJECTIVE State the objective of the experiment clearly. The objective of the experiment is what you’re trying to prove or accomplish. 3. THEORY Explain relevant concepts and provide any appropriate definitions. Pertinent equations should be derived in a clear, logical manner. Any relevant background information may also be included in this section. 4. APPARATUS Record a list of the equipment being used. Write down the serial number, model, and make of the equipment. You will need this information for reference in case you need to repeat the experiment or collect additional data. Describe the equipment being used and draw a diagram or picture of the equipment. 5. PROCEDURE This section includes your plan for performing the experiment. The experimental plan should be written in a step-by-step, orderly fashioned method. It describes in detail your procedure for performing the lab such that you or anyone else could re-create the experiment exactly as it was performed. The experiments in the lab manual/handouts already have a step-by-step written procedure. You may cut and paste the written procedure from your lab manual into your lab notebook. Keep in mind that you may need to repeat an experimental procedure during the lab final. (The lab handouts have the TITLE, OBJECTIVE, THEORY, APPARATUS, and PROCEDURE sections already written out, so you may cut and paste these sections into your lab notebook.) 6. DATA Your data should be well organized and easy to read. Label each set of data with the appropriate quantity being measured along with the trial/run number. Use ‘table’ format for easier reading. Your data should have the correct number of significant figures and appropriate units. If your data is represented by graphical methods, make sure your graph has been appropriately labeled with the correct axis, units, and scale. Any work that is printed out from the computer must be attached securely (taped, glued, stapled ....)to your lab notebook. DO NOT FOLD PRINTOUTS AND SLIP INTO NOTEBOOK! Any work that is done on the computer should not be saved on the hard drive! Once finished with your work, delete it and empty Recycle Bin. 7. CALCULATIONS Write down equations being used to do calculations. The calculations should be clear and readable. Show and label calculations in complete detail for any quantity. If you are repeating a calculation several times, you only need to show one sample calculation. The calculation for % error between experiment and theory should be included here. 8. CONCLUSION & RESULTS Include a discussion of the results and their significance. Address the experimental objective and state whether it was accomplished. Comment on the % error between theory and experiment. Identify at least two sources of experimental error (systematic or random) to account for the percentage error involved in the experiment. Explain how these errors effected the outcome of the experiment? Was the experimental result greater than or less than the theoretical value? Was this what you had anticipated? Explain why or why not? You may also discuss methods to eliminate or minimize these experimental errors. LAB REPORT GRADING RUBRIC TITLE: b Section Description of Section Possible Points 1 PARNERS NAMES 1. Partners names at the upper right-hand corner of the first page of the lab report. 1. Includes handout containing these components stapled to lab report or each component is explicitly written. THEORY 1. The theory/principle/law associated with the lab is clearly stated. 2. Explained how the theory will be used to accomplish the objective of the lab. 3. Derivation/proof of equations are done clearly and logically. 1. All data is labeled, organized, and easy to read. 2. Used table-format whenever appropriate 3. Data measurements taken correctly. 4. Used appropriate significant figures and units. 5. Graphs have correct axis labeled, units, and scale. 2 CALCULATIONS 1. Calculations are labeled with equations shown. 2. Calculations are clear and legible. 3. Calculations are done correctly 4. Calculations have correct number of significant figures and units. 5. % error calculation shown. 5 CONCLUSION 1. Includes a summary of the results of the experiment and the % error involved. 2. Addresses the experimental objective and EXPLAINS if it was accomplished or not based on experimental results and % error involved. 3. Provides and explains one systematic error involved in the experiment and explains how it affected the outcome of the experiment and the % error involved. 4. Provides and explains one random error involved in the experiment and explains how it affected the outcome of the experiment and the % error involved. 5 TITLE OBJECTIVE THEORY EQUIPMENT PROCEDURE DATA Write the conclusion in outline-number form to obtain credit! SCORE 2 5 20 Your Score
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