Exam 3: Free and Damped Harmonic Oscillation
Exam 3: Free and Damped Harmonic Oscillation In this exercise you’ll use the KET Virtual Physics Lab “Harmonic” to find the natural oscillation frequency and the 1/e decay time of a harmonic oscillator. The apparatus is described in the Simple Harmonic Motion User’s Guide. It consists of an ideal (i.e. massless) vertical spring from which you can hang different weights. The oscillation can be damped or undamped. Experiment 1: Measure the elastic constant of the spring from its free (undamped) motion. The spring exerts a force on the suspended mass, , where [N/m] is the elastic constant of the spring. The natural oscillation period is , and hence the elastic constant can be estimated from the mass and the period of the motion. Procedure: Measure the period of oscillation for the 100, 200 and 300 [g] masses. Use the 25% speed setting. To estimate the period, import the data into Grapher and find the time between two successive zero-crossings of the displacement. It’s easy to see the approximate crossing times in the graph. Use these rough estimates to find more exact values in the in the data table. Since it’s unlikely that there will be a data point where is exactly zero, you’ll have to find the two times where the displacement brackets zero and interpolate linearly to estimate the time of the zero-crossing. Algorithm: Denote the values of displacement immediately above and below zero (at times ) by . From the line through these two points, estimate the zero-crossing time (i.e. ) as . Since the line passes through the points the slope () and intercept () must satisfy the two equations . Solve these for 1. Find 2 adjacent zero-crossings where the slope is negative and 2 where the slope is positive and fill in the data tables below (2 for each mass) 100 g: Downward zero-crossings (i.e. with negative slope) Crossing # t+ y+ t- y- t0 (down) 1 2 100 g: Upward zero-crossings (i.e. with positive slope ) Crossing # t+ y+ t- y- t0 (up) 1 2 200 g: Downward zero-crossings (i.e. with negative slope) Crossing # t+ y+ t- y- t0 (down) 1 2 200 g: Upward zero-crossings (i.e. with positive slope) Crossing # t+ y+ t- y- t0 (up) 1 2 300 g: Downward zero-crossings (i.e. with negative slope) Crossing # t+ y+ t- y- t0 (down) 1 2 300 g: Upward zero-crossings (i.e. with positive slope) Crossing # t+ y+ t- y- t0 (up) 1 2 · Append a screenshot of your Grapher plot for the free oscillation of the 200 g mass to the end of this report. 2. For each mass calculate the period from the successive down- and up-crossings as and . Average these to find the period , and compute the spring constant for each mass (report the spring constant in [N/m]). Enter your results in the table below Mass [g] [s] [s] [s] [N/m] 100 200 300 Average your 3 results to estimate the spring constant and enter it in the box: [N/m] Repeat the procedure in the first experiment to find the “mystery mass” (denoted by the question mark) by measuring the period of the free (undamped) oscillation and using your measured value of . [s] [g] 3. Record the values in the box below: Experiment 3: In this experiment you’ll find the 1/e decay time, , of a damped oscillator (this is the time required for the motion to decay by 1/e – i.e. ). The displacement of a linear, damped, unforced oscillator is described by where . Note that a phase angle has been included to allow the solution to match the initial displacement . Procedure: Use the same masses (100, 200 and 300 g) as in Experiment 1. Be sure to check the box on the right in the VPL that selects “Damp Spring”, and select “25% speed”. Import your data into Grapher and find the displacements and times of the first 3 maxima beyond the initial displacement. Denote the maximum displacement and the time (for a given mass) by for where . The maxima of occur when is an integer multiple of . Comment: Since , the first point on your graph will be equal to only if . Since you can’t easily ensure that condition, to be on the safe side use the first 3 maxima of the displacement beyond the initial value at . · Append a screenshot of your Grapher plot for the 200 g mass to the end of this report. Taking the natural logarithm of the equation above for and evaluating it at the times gives where is a constant. This is a linear relation between and , and so defines a line with slope . To find the decay time for each mass, fit a line using least squares to the points vs (for ). Then . There are lots of fitting programs on-line. A good one (it also will directly fit an exponential model, so you don’t have to find ) is: https://www.wolframalpha.com/widgets/view.jsp?id=ce8e5d2e9be09346c2d6da0a921983fd 4. Enter your results for the decay time for each mass in the box below Mass [g] 100 200 300 Decay time [s] Experiment 3: The previous experiments all used dynamics to infer various parameters of the spring/mass system. However, gravity acts on the mass, and for the system to achieve static equilibrium the spring must stretch by (note that this not the final length). 5. Measure the static equilibrium stretch for masses 100, 200 and 300 g, and estimate for each case. Record your values in the table below: Elastic constant estimated from static equilibrium Mass [g] 100 200 300 [cm] [N/m] Estimate the spring constant by averaging your 3 results, and enter it into the box: [N/m] 6. Although you need the gravitational acceleration ( 9.8 m/s2 ) to calculate the elastic constant from static equilibrium, explain why doesn’t appear anywhere in the procedure used above to estimate dynamically from the decay of the oscillatory motion. Your explanation should start by writing the equation of motion for the undamped oscillator, but with the inclusion of the gravitational force. 7. Why did we suggest that you to use the “25% Speed” setting? To answer this, re-run the free oscillator with a 200 g mass at the “100% Speed” setting. Inspect the graph and data table to see what changed (you don’t have to do a detailed analysis). What is the effect of the speed setting on your ability to measure the oscillation period with high precision? [%] 8. Denoting the radian frequency of the undamped oscillator by ( ), the oscillation frequency of the damped oscillator can be written as . Use your measured values of and to show that in this case it is justified to take . Estimate the fractional change in frequency caused by damping and enter it into the box below. m ln()/ nn ytB t =-+ B ln() n y n t 1/ t - ln() n y n t 1/ slope t =- t k / ymg k D= k k y D k g 0 (1/)100 ww -´= 0 w 22 4/ Tm pk = / m k º 222 00 /1() wwwt - =- 0 2/ T wp º t 0 ww ; y t ± y ± yatb =+ 0 t 0 ()0 yt = 0 / tba =- (,) ty ±± a b yatb ±± =+ 0 ()/() tytytyy +--++- =-- 00 (2)(1) DownDownDown Ttt =- 00 (2)(1) UpUpUp Ttt =- T Down T Up T k k k = ? m T = ? m = t ()/()1/1/2.7180.3679 ytyte t +=»» ()exp(/)cos() ytAtt twj =-+ 2222 0 / m wktwt -- =-=- j (0) yt = (,) nn ty 1,2,3 n = Fy k = () nn yyt º () yt n t wj + 2 p (0)cos() ytA j == A 0 j = 0 t = () yt n t Microsoft Word - 07Springs.docx 7 –Simple Harmonic Motion User’s Guide: 06/10/2014 The spring. A massless spring is the focal point of this lab. By default there is no damping, but dampening can be turned on using the check box in the lower right corner of the screen. The masses. Masses from 100g to 300g plus one unknown (?g) mass are found toward the bottom of the screen. Click and drag one to the bottom of the spring and it will attach itself. If a mass is already on the hook, it will replace it. Once on the spring, continue to drag the mass to extend or compress the spring, and then release the mouse to set the spring in motion. The mass and spring system will automatically release once it reaches its maximum position. To remove a mass from the spring, just click and drag it away from the spring and drop it anywhere else on the screen. The stopwatch. Located just left of the masses, the stopwatch is an invaluable tool for this lab. By default, the “Auto-Stopwatch” setting is on, which starts