Page 138 - Mechanical Engineers' Handbook (Volume 2)

3 Error Analysis 127

Suppose that
V F(a, b, c, d, e, , x, y, z) (29)

where a, b, c, x, y, z represent quantities which must be individually measured to determine
V. Then
V
F
n n
and
C
F
E e n (30)
n
The sum of the squares of the error contributions is given by
e 2
F
2
E
n e n (31)
are independent and
Now, as in the discussion of internal errors, assume that errors e n
symmetrical. This justifies taking the sum of the cross products as zero:
nm
F
F
n m ee 0 n m (32)
Hence
) 2
F
(C E 2 e 2 n
n
or
e 1 / 2
2
F
2
E
n e n (33)
. It is much less than the worst case:
This is the most probable value of e E

C [ C C C C ] (34)
e a b c z
As an application, the determination of g, the local acceleration of gravity, by use of a
simple pendulum will be considered:
2
4 L
g (35)
T 2
where L length of pendulum
T period of pendulum
If an experiment is performed to determine g, the length L and the period T would be
measured. To determine how the accuracy of g will be influenced by errors in measuring L
and T write
2
g 4 2 g 8 L
and (36)
L T 2 T T 3
The error in g is the variation in g written as follows: