Problem 1.43 [Difficulty: 2]
Given: Pet food can
Hmmto
Dmmto
mgto
=±
=±
=±
102 1 20 1
73 1 20 1
397 1 20 1
()
()
()
Find: Magnitude and estimated uncertainty of pet food density.
Solution: Density is
ρππρρ
=∀== =
mm
RH
m
DH or m
2DH
2
4(,,)
From uncertainty analysis:
2
1
222
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
±= HDm u
H
H
u
D
D
u
m
m
u
ρ
ρ
ρ
ρ
ρ
ρ
ρ
Evaluating:
m
m
m
DH DH u
D
D
Dm
DH
m
DH u
H
H
Hm
DH
m
DH u
m
D
H
ρ
ρ
ρπ ρπ
ρ
ρ
ρπρπ
ρ
ρ
ρπρπ
∂
∂=== =
±=±
∂
∂=− =− =− =
±=±
∂
∂=− =− =− =
±=±
41 1
411
397 0 252%
24214 21
73 137%
14114 11
102 0 980%
22
32
22 2
m;.
() () ; .
() () ; .
Substituting:
()( )( )
[]
%92.2
980.0137.12252.01 2
1
222
±=
×−+×−+×±=
ρ
ρ
u
u
∀= = × × × = ×
=∀=××=
−
−
ππ
ρ
44
73 102 4 27 10
930
22 4
D H mm mm m
10 mm m
m397 g
4.27 10 m
kg
1000 g kg m
23
93 3
43 3
() .
Thus:
ρ
=±930 27 2 20 1.()kg m to
3