630 Chapter 18 Mathematics in Engineering
18.17. Given matrices: [A] ,
[B] , and {C} , perform
the following operations.
a. [A] [B] ?
b. [A] [B] ?
c. 3[A] ?
d. [A][B] ?
e. [A]{C} ?
f. [A]
2
?
g. Show that [I][A] [A][I] [A]18.18. Given the following matrices: [A]
and [B] , calculate the determi-
nant of [A] and [B] by direct expansion. Which
matrix is singular?
18.19. Solve the following set of equations using the Gaussian
method.18.20.Solve the following set of equations using the Gaussian
method.18.21.Solve the following set of equations using the Gaussian
method.18.22.As we explained in Chapter 13, an object having a mass
mand moving with a speedVhas a kinetic energy,
which is equal to£
111
251
315
§•
x 1
x 2
x 3
¶•
6
15
14
¶
x 1 x 2 10
2 x 1 3 x 2 5
4 xy 1
x 3 y 14
£
210 0
420 0
12 418
§
£
210 0
16 6 14
12 418
§
1
2
4
£ ¶
12 1
53 3
45 7
§
£
421
70 7
1 53
§
Plot the kinetic energy of a car with a mass of 1500 kg
as the function of its speed. Vary the speed from zero to
35 m/s (126 km/h). Determine the rate of change of
kinetic energy of the car as function of speed and plot
it. What does this rate of change represent?
18.23.In Chapter 13, we explained that when a spring is
stretched or compressed from its unstretched position,
elastic energy is stored in the spring and that energy will
be released when the spring is allowed to return to its
unstretched position. The elastic energy stored in a
spring when stretched or compressed is determined fromObtain expressions for the elastic energy of a linear
spring described byFkxand a hard spring whose
behavior is described byFkx
2
.
18.24.For Example 1 in Table 18.8, verify that the given
solution satisfies the governing differential equation and
the boundary conditions.
18.25.For Example 3 in Table 18.8, verify that the given
solution satisfies the governing differential equation and
the initial conditions.
18.26.We presented Newton’s Law of Gravitation in Chapter 10.
We also explained the acceleration due to gravity. Create
a graph that shows the acceleration due to gravity as
a function of distance from the earth’s surface.
Change the distance from sea level to an altitude of
5000 m.
18.27.For Problem 18.26, plot the weight of a person with a
mass of 80 kg as a function of distance from the earth’s
surface.
18.28.An engineer is considering storing some radioactive
material in a container she is creating. As a part of
her design, she needs to evaluate the ratio of volume
to surface area of two storage containers. Create
curves that show the ratio of volume to surface area
of a sphere and a square container. Create another
graph that shows the difference in the ratios. Vary the
radius or the side dimension of a square container
from 50 cm to 4 m.
18.29.As we mentioned in Chapter 10, engineers used to use
pendulums to measure the value of gat a location. TheElastic Energy
x0F dx
Kinetic Energy
1
2
mV
2
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