ELIMINATION OF ARM-LENGTH ERRORS IN WEIGHING V. R. DAMERELL Western Reserve University, Cleveland, Ohio
IN THE quantitative laboratory it occasionally becomes necessary or convenient for a student to use two balances during a determination. This frequently happens when there are more students than balances. As a convenience, it seemed profitable to have a simple method for findingand listing the conversion factors for changing apparent masses obtained on any balance to what they should be on any other balance in the laboratory. At first glance this seemed to involve a lot of work, but the writer devised a method that gave the necessary relationships between twenty-one balances in a few hours. The object of this paper is to describe the method, so that it may be available to anyone wishing to do likewise. If m is the trne mass of any object, and w and w' are its apparent masses when it is weighed on any two balances I and 11, and if r and 1are the arm len,@hs of the beam of balance I, and r' and 2' the arm lengths of the beam of balance 11, then: E
m l = wr ml' = w'r'
and
w/w' = nlr'/rnl'r w = w'k.
=
k
The constant k will then be the conversion factor to use in order to convert the weight of any object weighed on balance I1 to what it should weigh on balance I. Also the constant l / k will bring about the opposite conversion. The writer determined these constants for the laboratory balances as follows. A ten-gram weight was weighed with another tengram weight on each of the balances in the laboratory. The apparent masses of the 6 r s t ten-gram weight (always on the left-hand pan) varied from 10.0003 grams to 10.0017 grams. The values for k for each possible pair of balances were rapidly obtained by adding to or subtracting from 1.00000 a tenth of the difference between the two apparent masses of the ten-gram weight obtained on the two balances. Using the two extreme
values as an example of the calculations of the constants, we have: k '- -- 10'0003 10.0017
= 0.99986 = 1.00000 - 0.00014
k,=-= 100017 10.0003
1.00014 = 1.00000
+ 0.00014
The constant used would depend upon which way the conversion was to be made. It is interesting to note that the failure to use the correction in this case would cause an m o r of 2.8 milligrams in the weight of a twenty-gram crucible. Factors for 420 conversions were calculated for 21 balances in this manner in a short time. These were listed in. at'able similar to the abbreviated table given below. In employing such a table, it is of course necessary that the same set of weights be used on both balances, or if two sets are used, that they be calibrated in terms of the same standard. TABLE
To convert the weight of an object weighed on any balance in the vertical6ow to what it should weigh on any balance in the horizontal row, multiply by the factor indicated. Thus, if a crucible weighed 16.4678 grams on balance 4, and its weight w a s desired on balance 6, the factory or 1.00010 would be used. and weight would be 16.4694 gram? the B
~
I
l 2 3 4 5 G
o k
7
m
I
j r h
i
2~
3~
s o n
l p
u l
%
m
9
-
d = 0.99989 e 0.99990 f = 0.99991 g = 0.99992 h 0.99993 i = 0.99994
-
jk = 0.99996 0.99085 1 = 0.99997 m n. 0.99998 n = 0.99999 o = 1.ooooo
4 ~
a % n
~5 v r
k g
f
G
7
u
a
m ~
g
d c
o s
i
I
Y
n o
9 = O.OWOI g = 1.00002 r r. l.WO03 s = 1.00004 1 = 1.00005 u = 1.00006 u = 100oo7 ul = 1.00008 z = 1.00009 y = 1.00010 a n. 1.00011
U
j k
l
