??????? ?? (Derivation of Euler Equation) ??????? ??? ?? ?? Newton? ?2??? ??? Euler ??????? ????.

1
2-(1,2,3,4,5)

• ?????
• 20081450
• ???

2
2-1 Prove that the all the pressure acting on the
one point in the water are all the same for all
the direction.(??? ??? ???? ??? ?? ??? ??? ??? ??
????.)
????? ??? ?? ???? ? ???? ??? ????. ? ???? ?? ?? ?
?? ???? ?? ??? 0?? ??? ? ? ?? ???? ???? ?? ? ??
?? ?????. ?????? ?? ?? ???? ??? ?? ??? ? ?? ???
?? ???. ??? ??? ???? ?? ?? ?? ?? ?? ??? ? ???
????? ???? ??? ?. ? ??? ???? ??? ??? ?? ?? ???
????? ?? ????.
??? ??? ???? ??? ?????? ? ? (x,y)?? ?? ?? ??
????? ????? ??? ??? ?? ?? ??? ???? ??.
3
?? 1 ?? ?? ??? ?? ????.
? ?? ???? ???? ??, ?? ?????? ???? ?????. x? y ???
?? ???? ?? ??? ?? ????.
4
???, px, py, ps

? ?? ? ?? ???? ??????, ? ??? ???, ?
??, ax, ay
? ???? x, y ???? ? ??? ????? ???? ? (x, y)?
?? ??? ???? ????? ??? 0?? ???? ??? ??? ???? ???
? ???? ???? ??? ?? ??? ??. ? ?? ?? ??? ?? ???
?????? ??? ? ??. dy?dx? ?? ?? ? ???? ???? ?? ???
???.
? ??? ??? ?????? ?? ?? ???? ??? ? ?? ????
??? ?? ???? ??? ??? ?? ??. ? ??? 2??? ??? ???
????, 3?? ????? 1?? ??? ???? ?? ?? ???? ?????
?????? ???? 3??? ???? ??? ??? ? ??. ??? ???? ? ??
???? ??? ????? ??? ????? ???? ?? ????? ? ?? ????
????? ?? ???? ???? ?? ??. ? ?? ? ?? ???? ??? ? ??
???? ??? ?? ?? ????? ???? ????.
5
????? 0? ????, ? ??? ?? ????? ??? ?? ??? ????
????? ???? ???? ? ???? ??? ?? ???? ??.
6
2-2??????? ?? (Derivation of Euler Equation)
??????? ??? ?? ?? Newton? ?2??? ??? Euler
??????? ????.

• ? ??
• ?? ? ???

?????? ??? ???? ?? ???(surface force)? ???(body
force)?? ????(?? 1). ?????? ???? ????? ??? ?, ??
???? y??? ??? ???? y???? ?
????. ?? (x, y, z)??? ??? p? ? ? y?? ???? ??? ??
??? ?? ???? ?? ?????
?? ?? ?? ???? ??
7
?? 1 ?????? ???? ??.
??. ??? dy/2? ?????? y?? ???? ???? ????. y????
???? ?? ??? ??? ??.
x? z??? ????, ? ??? ?????? ???? ????
8
???? ??1 ? ??? ???? ? ?? dF? ??? ??.
? ??? ???, ??? ??? 0?? ?????
(1)
? ?? ????. ? ?? ? ??? ????? ???? ???? ????, ????
???? ? ??? 0? ??? ??.
???? ?? ??(gradient)?? ?? ???? del??? ???.
8.2???? ? ? ?? ???
(2)
??? p? ???? -?p? ??? ?? ????? ???? ???? ??? f??.
(3)
?? ????? ?? ?????? ??? ?? ??.
(4)
9
???? ?? ??? ???, ?? ??? ?? ??? ????? 0? ??? ???
??? Newton ??? ? 2??? ??? ?? ??? ????.
(5)
??? a? ????? ?????. (f - jr)? ??? ??????? ??? ?
??? ???? ?????. ? (5)? 2.9??? ????? ???? ? ???
3?? 8???? Euler???? ???? ? ????. ?(4)? ???? ???
???? ??? ?? ??.
(6)
????? ????? ???? ????? ()? ??? Pascal? ??? ???
???. ?, ???? ?? ???? ??? ?? ??? ?? ? ???? ??? ???
?? ????. p? y?? ?????
(7)
??. ? ??? ?????? ???, ?????? ??? ???? ???, ????
????? ???? ??? ??? ???? ???? ??. ??, ????????? r?
????? ? (7)? ???? ????? c? ?? ??? ?? ?? ??.
10
?????? ?? ?????(?????) ??(??)? ?? ?? ??? ?? ??.
(8)
??? ? ???????? ???? ?? ?? ?
??????? ???? ??? ??? ??? ????. ? ???
????(hydrostatic pressure)?? ???. ? (8)? ???
????? ???? ??? ? ????? ????? ??? ?????? ??????
??? ?? ??.
11
2 - 3 Derive the pressure equation of the
compressible fluid for the constant temperature
and varying temperature.   ??? ??? ??? ????
?????? ????? ????.
12
2-4 Derive the moments of inertia of simple areas
about centroidal axis for the figures such as
follows

Figure A.2 Moments of inertia of simple areas
about centroidal axes
For rectangle (Figure A.2)
13
For triangle
(A.9)
14
2-5 The triangular gate CDE (Fig. 2.12) is hinged
along CD and is opened by a normal force P
applied at E. It holds oil, relative density
0.80, above it and is open to the atmosphere on
its lower side Neglecting the weight of the gate,
find (a) the magnitude of force exerted on the
gate by integration and by Eq. (2.5.2) (b) the
location of pressure center (c) the force P
needed to open the gate.?? ??(gate) (?? 2.12)?
? ?? ??(hinge)? ???? ?? ? ???? ??? ? ??? ????. ??
????? ?? O.8? ??? ??? ?? ?? ??? ?? ?? ???? ??.
??? ??? ???? ??? ????. (a) ??? ???? ?? ??? ????,
??? ? (2.5.2)? ???? ???. (b)????? ??, (c)??? ???
??? ?? ?? .
15
??2.12 ?? ??
(a) ?? 2.12? ???? ????
16
x? y? ?? ????? ????, y 4?? x O, y 6.5??
x3???
a, b? ??? ??, a, b? ?? ? 1?? ???? x? ?? y? ???
??? ? ??.
?? ???? y 6.5, x 3 y 9, x 0 ???? x
6/5(9 - y) , ???
? ?? ???? ? ?? ????, ??? ??? ???.
? (2.5.2)???
17
(b) ??? ???? ??? ??? ??? ??. ?
(2.5.8)? ?? ??
???? x?? ??? ???? ??? ????? , ???
??. ? (2.5.11)???
?, ????? ???? ???? ?? ???? O.32 ft ??? ????. (c)
??? ?? ??? ???? ????, CD? ?? ?????? ???