Page 266 - Excel for Scientists and Engineers: Numerical Methods

CHAPTER 10 ORDINARY DIFFERENTIAL EOUATIONS. PART I 243

8. Pendulum Motion 11. The motion of a simple pendulum as a function of
time is described by the following second-order differential equation:
d2B g
-+-e=o
dt2 L
where the terms in the equation are as defined in the preceding problem.
Generate a table of angle of displacement as a function of time from t = 0 to t
= 2 seconds, with B= 10' and dB/d = 0 at t = 0 .

9. Liquid Flow. A cylindrical tank of diameter D is filled with water to a
height h. Water is allowed to flow out of the tank through a hole of diameter
din the bottom of the tank. The differential equation describing the height of
water in the tank as a function of time is

where g is the acceleration due to gravity. Produce a plot of height of water
in the tank as a function of time for D = 10 ft, d = 6 in and ho = 30 ft.
2
-
kt/2)
Compare your results with the analytical solution h = (6 , where
k = (d / D2)& .
10. Chemical Kinetics I. Calculate concentrations as a function of time for the
second-order reaction
k
A+B-+C
for which 4A]/dt = -d[B]/dt = d[C]/dt = k[A][B]. Use [A], = 0.02000,
[B]o = 0.02000, k = 0.050 s-'. Calculate concentrations over the time range
from 0 to 500 seconds.

1 1. Chemical Kinetics 11. Use the Runge custom function to calculate [A], [B]
and [C] for the coupled reaction scheme
kl k3
A=B=C
k2 k4
using [Ale = 0.1, [BIo = 0, [C], = 0 mol L-I, kl = 1 s-I, k2 = 1 s-I, k3 = 0.1 s-'
and k4 = 0.01 s-', over the range 0-100 s.

12. Chemical Kinetics 111. Repeat #8, using [A], = 0, [B]o = 0.1, [C], = 0 mol
L-'