System Dynamics, 2nd Edition by William Palm III

By William Palm III

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Taking the logarithm of both sides gives log y = log (bemx ) = log b + mx log e This has the form Y = B + Mx if we let Y = log y, B = log b, and M = m log e. Thus if we plot Y versus x on rectilinear scales, we will obtain a straight line whose slope is M and whose intercept is B. This is the same as plotting log y versus x on rectilinear scales, so we will obtain a straight line whose slope is m log e and whose intercept is log b. This is equivalent to plotting y on a log axis and x on a rectilinear axis.

The capacitor is initially charged to 100 V. When the power supply is detached, the capacitor voltage decays with time as shown in the following data table. Find a functional description of the capacitor voltage v as a function of time t. 22 Introduction Water (of volume 425 ml) in a glass measuring cup was allowed to cool after being heated to 207◦ F. The ambient air temperature was 70◦ F. The measured water temperature at various times is given in the following table. Time (sec) ◦ Temperature ( F) 0 300 600 900 1200 1500 207 182 167 155 143 135 Time (sec) ◦ Temperature ( F) 1800 2100 2400 2700 3000 128 123 118 114 109 Obtain a functional description of the relative water temperature ( T = T − 70) versus time.

8 Plot the data and obtain a function that describes the data. Estimate when the population will be double its 1990 size. 18 The half-life of a radioactive substance is the time it takes to decay by half. The half-life of carbon-14, which is used for dating previously living things, is 5500 years. When an organism dies, it stops accumulating carbon-14. The carbon-14 present at the time of death decays with time. Let C(t)/C(0) be the fraction of carbon-14 remaining at time t. In radioactive carbon dating, it is usually assumed that the remaining fraction decays exponentially according to the formula C(t) = e−bt C(0) a.

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