Take the log of both sides and then expand according to rules for logs y y o e

Take the log of both sides and then expand according

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Take the log of both sides and then expand according to rules for logs. y = y o e kt log y = log (y o e kt ) = log y o + log (e kt ) 9
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At this point we observe a significant difference. If we are using natural logs (base e) rather than common logs (base 10) then we can make a substantial additional simplification. Recall that the symbol "ln" is used to indicate natural logs in circumstances when it is important to know the base of the log calculation. Then: n y = n y o + n (e kt ) = n y o + kt So! If the relationship between y and t is exponential, then a plot of n y vs. t (NOT vs. n t) will give a straight line whose slope provides the value of k and whose y- intercept (at t = 0) provides y o . (A) Now add a 3 rd column (neatly) to the data table in section 3(A). The new column should contain the values of n V for each of the 10 values of V. I. Graph your data for n V vs. t on the sheet of Cartesian paper following this page. n V is dependent, t is independent. Sketch the best straight line. II. Find Vo and the slope k (k = - = -1/ T = -1/ T e ) from your straight line. Show your calculation for K below. Circle y o on your graph. V o = ________ _____ K = ________ _____ = ________ _____ T e = T = ________ _____ value units III. Compare your values of V o and from part II above to the values of V o and from parts 3(C) and (G). 3(C) and (G): V o = ________ = ________ part 2 above: V o = ________ = ________ Comment: 10
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PH262 Simple Harmonic Motion and the Pendulum Lab # ____ Name ______________________________________________ Date _________ Lab Partner(s) Name ___________________________________________________________ III. EQUIPMENT LIST Tall Lab Stand C-Clamp 15” Spring Pendulum Clamp Cardboard Dampener Assorted Hooked Masses Spring Scale (in Newtons) Cotton String (≈ 1.0m long) IV. BACKGROUND INFORMATION III. EXPERIMENTAL PROCEDURE 1. Simple Harmonic Motion - Mass/Spring System Find a support base and rod. Use a "C" clamp to fasten the stand to a lab table. Attach a pendulum clamp near the top of the rod. Attach the spring to the pendulum clamp to the innermost support point. Hang a 1 kg mass from the spring. The point where the weight is hanging at rest is the static equilibrium point, y = 0. Pull the weight down about 10 -15 cm from equilibrium and release it. Measure Δy before you release the weight. Δy = y 0 = ________ _____ value units (A) Measure the period, T - let the spring/mass oscillate about 10 complete cycles, find the total t using a stop watch and calculate T . Repeat this measurement at least once for consistency. Use T to find f (frequency) and ω (angular frequency). T = ________ ___ f = ________ ___ Note: 2 or 3 ω = ________ ___ significant figures (B) Measure K (spring constant) for your spring. Don't spend a lot of time on this; it is easy to accurately determine K with no more than 3 values of F vs. Δy or F vs. Δx. (You can use F g = mg or a spring scale.) K = ________ ___ m = ________ ___ 13
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(C) Calculate ω C = (k/m) 1/2 and compare to your experimental value in part (A). Use the space below to compare and comment on the values of ω A and ω C .
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