TEST BANK FOR System Dynamics 3rd Edition By William J. Palm III

1.1 W = mg = 3(32.2) = 96.6 lb.
1.2 m = W/g = 100/9.81 = 10.19 kg. W = 100(0.2248) = 22.48 lb. m = 10.19(0.06852) =
0.698 slug.
1.3 d = (50 + 5/12)(0.3048) = 15.37 m.
1.4 d = 3(100)(0.3048) = 91.44 m
1.5 d = 100(3.281) = 328.1 ft
1.6 d = 50(3600)/5280 = 34.0909 mph
1.7 v = 100(0.6214) = 62.14 mph
1.8 n = 1/[60(1.341× 10−3)] = 12.43, or approximately 12 bulbs.
1.9 5(70 − 32)/9 = 21.1 C
1.10 9(30)/5+ 32 = 86 F
1.11 ! = 3000(2)/60 = 314.16 rad/sec. Period P = 2/! = 60/3000 = 1/50 sec.
1.12 ! = 5 rad/sec. Period P = 2/! = 2/5 = 1.257 sec. Frequency f = 1/P = 5/2 =
0.796 Hz.
1.13 Speed = 40(5280)/3600 = 58.6667 ft/sec. Frequency = 58.6667/30 = 1.9556 times
per second.
1.14 x = 0.005 sin 6t, x˙ = 0.005(6) cos 6t = 0.03 cos 6t. Velocity amplitude is 0.03 m/s.
¨x = −6(0.03) sin 6t = −0.18 sin 6t. Acceleration amplitude is 0.18 m/s2. Displacement,
velocity and acceleration all have the same frequency.
1.15 Physical considerations require the model to pass through the origin, so we seek a
model of the form f = kx. A plot of the data shows that a good line drawn by eye is given
by f = 0.2x. So we estimate k to be 0.2 lb/in.
c 2013 McGraw-Hill. This work is only for non-profit use by instructors in courses for which
the textbook has been adopted. Any other use without publisher’s consent is unlawful.
1.16 The script file is
x = [0:0.01:1];
subplot(2,2,1)
plot(x,sin(x),x,x),xlabel(0x (radians)0),ylabel(0x and sin(x)0),...
gtext(0x0),gtext(0sin(x)0)
subplot(2,2,2)
plot(x,sin(x)-x),xlabel(0x (radians)0),ylabel(0Error: sin(x) - x0)
subplot(2,2,3)
plot(x,100*(sin(x)-x)./sin(x)),xlabel(0x (radians)0),...
ylabel(0Percent Error0),grid
The plots are shown in the figure.
0 0.5 1
0
0.2
0.4
0.6
0.8
1
x (radians)
x and sin(x)
x
sin(x)
0 0.5 1
−0.2
−0.15
−0.1
−0.05
0
x (radians)
Error: sin(x) − x
0 0.5 1
−20
−15
−10
−5
0
x (radians)
Percent Error
Figure : for Problem 1.16.
From the third plot we can see that the approximation sin x x is accurate to within
5% if |x| 0.5 radians.
c 2013 McGraw-Hill. This work is only for non-profit use by instructors in courses for which
the textbook has been adopted. Any other use without publisher’s consent is unlawful.
1.17 For near /4,
f() sin

4
+ cos

4 −

4
For near 3/4,
f() sin
3
4
+ cos
3
4 −
3
4
1.18 For near /3,
f() cos

3 − sin

3 −

3
For near 2/3,
f() cos
2
3 − sin
2
3 −
2
3
1.19 For h near 25,
f(h)
p25 +
1
2p25
(h − 25) = 5 +
1
10
(h − 25)
1.20 For r near 5,
f(r) 52 + 2(5)(r − 5) = 25 + 10(r − 5)
For r near 10,
f(r) 102 + 2(10)(r − 10) = 100 + 20(r − 10)
1.21 For h near 16,
f(h)
p16 +
1
2p16
(h − 16) = 4 +
1
8
(h − 16)
f(h) 0 if h > −16.
c 2013 McGraw-Hill. This work is only for non-profit use by instructors in courses for which
the textbook has been adopted. Any other use without publisher’s consent is unlawful.
1.22 Construct a straight line the passes through the two endpoints at p = 0 and p = 900.
At p = 0, f(0) = 0. At p = 900, f(900) = 0.002p900 = 0.06. This straight line is
f(p) =
0.06
900
p =
1
15, 000
p
1.23 (a) The data is described approximately by the linear function y = 54x − 1360. The
precise values given by the least squares method (Appendix C) are y = 53.5x− 1354.5.
(b) Only the loglog plot of the data gives something close to a straight line, so the data is
best described by a power function y = bxm where the approximate values are m = −0.98
and b = 3600. The precise values given by the least squares method (Appendix C) are
y = 3582.1x−0.9764.
(c) Both the loglog and semilog plot (with the y axis logarithmic) give something close
to a straight line, but the semilog plot gives the straightest line, so the data is best described
by a exponential function y = b(10)mx where the approximate values are m = −0.007 and
b = 2.1 × 105. The precise values given by the least squares method (Appendix C) are
y = 2.0622× 105(10)−0.0067x.
c 2013 McGraw-Hill. This work is only for non-profit use by instructors in courses for which
the textbook has been adopted. Any other use without publisher’s consent is unlawful.