Crayton M. Crawford Mississippi Stote University Stote College, Mississippi
I
I
The Rest Point of the Balance
The problem of finding the distance P of the rest point of a balance from some arbitrarily selected origin has found various solutions, including the use of an odd1 or even2 number of readings and a single deflection method.3 Building on the treatment of S t a ~ ythis , ~ paper offers a new approach having the advantages of simplicity and exactness. Let the i-th turnaround point of the indicator he Ri units from the origin, hence (R, - P) units from the rest point. We postulate constant damping, meaning that
where r is a negative number because successive swings are in opposite directions. By the first equality, (Rz - P ) %= (R1 - P)(Ra - P). Solving for P,
Provided the mathematical model correctly describes the physical situation, equation (2) permits the rest point to be calculated from any three successive turnaround points. No approximations are involved.
After P has been found from equation ( Z ) , the valne of r follows from equation (1). Since r is presumably independent of P, equation (1) can then he used to find other rest points from only two readings;
A special case is that of an undamped balance, for which r = - 1. If this valne of r is substituted into eqnation (3), one obtains; P
Co.,
SIzn
'ldR,
+ R*)
(4)
which is the basis for the "rest sum"2 method. Evidently it becomes less valid the more highly damped the balance. Extensive data have not been compiled, hut a few for purposes of illustration are shown in the table. It would appear that r is not dependent on load. The apparent dependence of r on amplitude of swing (thus contradicting the assumption of equation (1)) is believed to result from a loss of significant figures when short swings are used. Comparison o f Methods (Ainsworth Balance)
WIRTSCHAFTER, J. D., J. CHEM.EDUC., 33, 216 (1956). a OLSON, A. R., KOCH, C. W., AND PIMENTEL, G. C., 'Tntroductory Quantitative Chemistry," W. H. Freeman & Francisco, 1956, p. 15. ~NUDSO G. N E.,, J. CHEM.EDUC., 41, 449 (1964). STACY, I., J. CHEM.EDUC., 32, 90 (1955).
=
Load 0 50 d p s n 50 =/pan
(transposed]
RI
RI
Ra
( e q n 2)
P
r
P' ( e m . 4)
P - P'
-4.9 -1.3 -2.7
8.4 4.8 5.2
-4.3 -1.0 -2.5
1.89 1.83 1.30
-0.980 -0967 -0.975
1.75 1.75 1.25
0.14 0.08 0.05
-2.7
4.8
-2.0
1.07
-0.975
1.05
0.02
Volume 42, Number 12, December 1965
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669
