EGR
1101: Unit
9
Lecture #1
Applications of Derivatives:
Electric
Circuits
(Section 8.4 of Rattan/Klingbeil text)
Review: Some Derivative Rules
where
a
,
c
,
n
, and
are
constants.
0
)
(
c
dt
d
)
sin(
))
(cos(
)
cos(
))
(sin(
t
t
dt
d
t
t
dt
d
1
)
(
n
n
nt
t
dt
d
at
at
ae
e
dt
d
)
(
Two
New Derivative Rules
(
Product rule
)
dt
dg
dg
df
dt
df
dt
df
t
g
dt
dg
t
f
t
g
t
f
dt
d
)
(
)
(
))
(
)
(
(
•
If
f(g)
is a function of
g
and
g(t)
is a function of
t
,
(Chain rule)
Today’s Examples
1.
Voltage, current, & power
2.
Current & voltage in an inductor
3.
Current & voltage in an inductor
(graphical and working backwards)
4.
Current & voltage in a capacitor
5.
Current & voltage in a capacitor
(graphical and working backwards)
Review: Some
Electrical Quantities
Quantity
Symbol
Unit
Symbol for
Unit
Voltage
V
or
v(t)
volt
V
Current
I
or
i(t)
ampere
A
Charge
Q
or
q(t)
coulomb
C
Energy
W
or
w
(
t
)
joule
J
Power
P
or
p(t)
watt
W
Time
t
second
s
Review: More
Electrical Quantities
Quantity
Symbol
Unit
Symbol for
Unit
Resistance
R
ohm
Inductance
L
henry
H
Capacitance
C
farad
F
Voltage

versus

Current Relations
For resistors,
R
t
i
t
v
)
(
)
(
dt
di
L
t
v
)
(
dt
dv
C
t
i
)
(
•
For inductors,
•
For capacitors,
EGR
1101: Unit
9
Lecture
#
2
Applications of Derivatives:
Beams
(Section 8.5
of Rattan/Klingbeil text)
Some Beam Terminology
Types of beams
Simply supported
Cantilever
Types of load on a beam
Concentrated
Distributed
More Beam Terminology
In addition to the type of beam and load,
a beam’s behavior also depends on its
geometry and the material it is made of.
Its geometry is summarized in a quantity
called the
second moment of area
(
I
)
.
Its material is summarized in a quantity
called the
modulus of elasticity
(
E
).
The product of these two (
E
I
) is called
the
flexural rigidity
.
Some Beam Quantities
Quantity
Symbol
Unit
Modulus of
elasticity
E
lb/in
2
or
N/m
2
Second moment of
area
I
in
4
or m
4
Flexural rigidity
EI
lb
in
2
or
N
m
2
Deflection
y(x)
in or m
Slope
(x)
radians
Excellent Online Resource
University of Wisconsin’s online lessons
on Strength of Materials:
http://www3.uwstout.edu/faculty/
scotta/upload/Foley

StaticsStrengths.pdf
See especially Topic 4 (Beams) and
Topic 8.2 (Stress on Incline Planes).
Review
Given a function
f
(x), the function’s local
maxima
occur at values of
x
where
and
Its local
minima
occur at values of
x
where
and
0
dx
df
0
2
2
dx
f
d
0
dx
df
0
2
2
dx
f
d
Today’s Examples
1.
Deflection of a cantilever beam with
end load
2.
Deflection of a simply supported beam
with central load
3.
Deflection of a simply supported beam
with distributed load
4.
Maximum stress under axial loading
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