Homework assignment 1
*
Due date: Wednesday February 8, 2008.
1. (# 1.2.9) Consider a thin onedimensional rod whose lateral surface area is
not insulated
.
(a) Assuming exact conservation of energy, and assuming that the amount of heat energy
flowing out laterally at
x
per lateral unit area and per unit of time is
w
(
x,t
), derive the
PDE describing the temperature
u
(
x,t
). (b) Assuming that
w
(
x,t
) is proportional to the
difference of the inside temperature
u
(
x,t
) and the outside temperature
γ
(
x,t
) with positive
proportionality factor
h
(
x
), show that the PDE for
u
(
x,t
) becomes:
c
(
x
)
ρ
(
x
)
∂u
∂t
(
x,t
) =
∂
∂x
parenleftbigg
K
0
(
x
)
∂u
∂x
(
x,t
)
parenrightbigg
+
Q
(
x,t
)

P
A
h
(
x
)(
u
(
x,t
)

γ
(
x,t
))
,
where
P
is the lateral perimeter. (d) Specialize the previous PDE to the case of a rod with
constant thermal properties, without internal heat sources, constant zero outside temperature
and constant circular cross section.
2. (# 4.2.1) This problem shows that in case the external force on a string is due to gravity
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Summer '06
 DeLeenheer
 Thermodynamics, Energy, Heat, heat energy

Click to edit the document details