Page 140 - Mechanical Engineers' Handbook (Volume 2)

References 129

L
5.5 10 4
L
Suppose T is determined with the pendulum swinging in a vacuum with an arc of 5 using
a stop watch that has an inherent accuracy of one part in 10,000. (If the arc is greater than
5 , a nonisochronous swing error enters the picture.) This means that the error in the watch
reading will be no more than 10 4 s. However, errors are introduced in the period determi-
nation by human error in starting and stopping the watch as the pendulum passes a selected
point in the arc. This error can be minimized by selecting the highest point in the arc because
the pendulum has zero velocity at that point and timing a large number of swings so as to
spread the error out over that number of swings. Human reaction time may vary from as
low as 0.2 s to as high as 0.7 s. A value of 0.5 s will be assumed. Thus the estimated
maximum error in starting and stopping the watch will be 1 s ( 0.5 s at the start and 0.5
s at the stop). A total of 100 swings will be timed. Thus the estimated maximum error in
the period will be 1/100 s. If the period is determined to be 1.92 s, the estimated maximum
error will be 0.01/1.92 0.005. Compared to this, the error in the period due to the inherent
inaccuracy of the watch is negligible. The nominal value of g calculated from the measured
2
values of L and T is 982.03 cm/s . The most probable error [Eq. (29)] is
421 / 2
2
[4(0.005) (5.5 10 ) ] 0.01 (43)
The uncertainty in the value of g is then 9.82 cm/s , or in other words the value of g will
2
2
be somewhere between 972.21 and 991.85 cm/s .
Often it is necessary for the engineer to determine in advance how accurately the mea-
surements must be made in order to achieve a given accuracy in the final calculated result.
For example, in the pendulum problem it may be noted that the contribution of the error in
T to the most probable error is more than 300 times the contribution of the error in the
length measurement. This suggests, of course, that the uncertainty in the value of g could
be greatly reduced if the error in T could be reduced. Two possibilities for doing this might
be (1) find a way to do the timing that does not involve human reaction time or (2) if that
is not possible, increase the number of cycles timed. If the latter alternative is selected and
other factors remain the same, the error in T timed over 200 swings is 1/200 or 0.005,
second. As a percentage the error is 0.005/1.92 0.0026. The most probable error in g
then becomes
32
421 / 2
e [4 (2.6 10 ) (5.5 10 ) ] 0.005 (44)
g
This is approximately half of the most probable error in the result obtained by timing just
100 swings. With this new value of e the uncertainty in the value of g becomes 4.91 cm/
g
2
2
s and g then can be said to be somewhere between 977.12 and 986.94 cm/s . The procedure
for reducing this uncertainty still further is now self-evident.
Clearly the value of this type of error analysis depends upon the skill and objectivity
of the engineer in estimating the errors in the individual measurements. Such skills are
acquired only by practice and careful attention to all the details of the measurements.
REFERENCES
1. W. A. Wildhack, ‘‘NBS Source of American Standards,’’ ISA Journal 8(2) (February 1961).
2. P. Giacomo ‘‘News from the IBPM,’’ Metrologia 20(1) (April, 1984).
3. NIST-F1 Cesium Fountain Atomic Clock.