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Centre of Mass
Centre of mass definition.
∑
m
i
r
i
= M r
G
(1)
M
 total mass
r
G
 centre of mass position
From diagram,
r
i
= r
G
+
ρ
i
therefore
∑
m
i
r
i
=
∑
m
i
(r
G
+
i
)
=
∑
m
i
r
G
+
∑
m
i
i
=
M r
G
+
∑
m
i
i
Since (1),
therefore
∑
m
i
i
= 0
(2)
From (1), we can deduce
∑
m
i
=
M
and
∑
m
&
r
i
&
r
G
i
&&
rM
r
iG
=
From (2), we can deduce
∑
m
i
&
i
= 0
and
∑
m
i
i
= 0
1
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View Full Document Triple Scalar Product
Consider 3 vector
A
,
B
,
C
,
.
A
,
B
on xy plane
C
on yz plane
A
x
B
= 
A
 
B
 sin
α
k

A
 
B
 sin
is the area of the parallelogram abcd.
Therefore
A
x
B
•
C
= 
A
 
B
 sin
k
•
C
=
(

A
 
B
 sin
) (
C
 cos
β
)
By symmetry, the volume can be written as :
A
x
B
•
C = B
x
C
•
A = C
x
A
•
B
Since dot product is commutative, the vol. is
C
•
A
x
B = A
•
B
x
C = B
•
C
x
A
i.e. Position of dot and cross operator is interchangeable
therefore
A
x
B
•
C = A
•
B
x
C
=
AAA
BBB
CCC
xy
z
z
z
The sign of the result is unaltered provided that the cyclic order is preserved. Reverse order
will introduce a change of sign.
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This note was uploaded on 05/10/2009 for the course MXXM 2XX9 taught by Professor Gxxy during the Spring '09 term at City University of Hong Kong.
 Spring '09
 GXXY

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