Problem 1.52 [Difficulty: 4]
Given: American golf ball, m = 1.62 ± 0.01 oz, D = 1.68 in.
Find: Precision to which D must be measured to estimate density within uncertainty of ± 1percent.
Solution: Apply uncertainty concepts
Definition: Density, 3
3D
m4
36
R
π
ρπ
∀
≡∀= =
Computing equation:
1
2
1
2
1
Rx
1
R
uu
Rx
x
⎡
⎤
⎛⎞
∂
⎢
⎥
=± +
⎜⎟
∂
⎢
⎥
⎝⎠
⎣
⎦
L
From the definition,
3/6 3
6m
m
DD
(m, D)
ππ
ρρ
===
Thus mm1
ρ
ρ
∂
∂=and DD3
ρ
ρ
∂
∂=, so
1
2
22
mD
222
mD
u[(1u)(3u)]
uu 9u
ρ
ρ
=± +
=+
Solving,
1
2
22
1
Dm
3
u[uu]
ρ
=± −
From the data given,
1
2
m
22
D
u 0.0100
0.01oz
u 0.00617
1.62 oz
1
u [(0.0100) (0.00617) ] 0.00262 or 0.262%
3
ρ
=±
±
==±
=± − =± ±
Since D
DD
u
δ
=± , then
Dx
D D u 1.68 in. 0.00262 0.00441 in.
δ
=± =± =±
The ball diameter must be measured to a precision of ± 0.00441 in.( ± 0.112 mm) or better to estimate density
within ± 1percent. A micrometer or caliper could be used.