JOURNAL OF CHEMICAL EDUCATION
FINDING THE REST POINT OF AN UNDAMPED ANALYTICAL BALANCE' IRVING F. STACY2 Columbia University, New York, N. Y.
IN
two usual derivations, together with the more exad one, are presented below. One proof3 depends upon the assumption that the amplitude of each swing is less than the preceding one by a constant amount k. According to Swift, if the first turning point of the pointer is b divisions away from the actual rest point (the position of the pointer after a very long time), then the next turning point is b- k divisions away, the third b-2k, and so on, as shown in Figure 1. The average of the first two readings on the left is b-k divisions to the left of the rest point and the first reading on the right is the same number of divisions 1 Presented before the Division of Chemical Education at to the right. For a total of five readings, the average of t.ha%fit,h Meetine of the American Chemioal Society, New .-.. the three on the left is b-2k divisions to the left of the York, September, i954. 2 Present address: RCA Victor Division, Radio Corporation rest point while the two readings on the right give an of America, Harrison, New Jersey. average which is b-2k divisions t o the right of the rest point. In each case, the point midway between the I REST POINT I left and right averages is the rest point. Although the result is the desired one, and is valid with negligible error for a good analytical balance, the assumption of a constant decrement implies that after a finite number of swings the pointer will suddenly come to rest. The other proof sometimes referred to4 assumes damped harmonic motion of the beam, i. e . . the MOST textbooks on quantitative analysis, instructions for finding the rest point of an analytical balance are presented without explanation. The student is told to allow the beam of the balance to swing freely and to take three or five consecutive readings of the extremes of the oscillations. The point midway between the left- and right-hand averages is taken as the rest point. Of the two derivations of the method to which reference is sometimes made, neither is completely satisfactory. A mathematitically exact method for finding the rest point from three readings is possible. The
~
AMPLITUDE Fig-
1
SWIFT,E. H., "Introductary Quantitative Analysis," Prentice-Hall, Inc., New York, 1950, p. 18. ~FALES, H. A,, AND F. KENNY,"Inorganic Quantitative Andyeis," Appleton-Century-Crofts, Ino., New Yark, 1939, pp. 678-80.
FEBRUARY, 1955
91
are small for good analytical balances, but starting with the same assumption of damped oscillations, a method of calculating the rest point without computing error is possible. If the constant ratio of amplitudes
is r (the reciprocal of the K used by Fales and Kenny and less than unity), then the number of scale divisions between consecutive readings of the pointer will also decrease by the same factor, e. g.,
amplitudes of the vibrations of the balance pointer decrease by a constant factor, so that
It can he seen from Figure 3 that if the initial reading on the left is c (negative if the first turning point is to the left of the balance scale zero), and the first swing takes the pointer d scale divisions to the right (S, S2in the notation of Fales and Kenny), the second reading will be c d, the next c d - dr, then c d - dr dr2, and so on. The true rest point is c plus the sum of the infinite geometric series:
+
(see Figure 2). According to the method of proof and the nomenclature used by Fales and Kenny, this constant is characteristic of the balance, and is equal to eaT12.where e is the hase of the natural loearithms. a is related to the coefficient of friction, a& T is the period of vibration in seconds. Usually, for a good having the common ratio -r, or analytical balance, a is very small, so that the quantity aT/2 is also small, say 0.1, whence the constant earl2 is slightly greater than unity (eo' = 1.10). Expanding the constant gives exactly. For example, the three readings 5.1, 15.7, 5.4 give the rest point aT @TI2 s 1 + + + ...
+
+
+
+
(9
Only the first two terms
are considered. It
follows that
The average of the two amplitudes on the left is
If the squared term (small compared with unity) is neglected, then this average is the same as Sz, the amplitude on the right. Thus, to a first order of approximation, the rest point lies halfway between the average of the two readings on the left and the one on the right. (A similar result holds for five readings if terms in aT/2 of degree higher than the first are neglected.) Expanding eaT12and considering only the first two terms of the expansion, as Fales and Kenny do, implies the introduction of an error. However, the introduction of the exponential is completely unnecessary. For a good balance the constant K is slightly greater than unity, say (1 x) exactly. The rest of the proof would be as before, the only error being due to the omission of squared and higher terms in z. The two proofs presented above involve errors which
Of course, if the balance scale zero were shifted, the values of d and r would be unaffected but the reading c would he different and the new calculated rest point would he shifted by the same number of scale divisions as the scale zero. Ordinarily, finding an arithmetic mean is simpler than calculating the formula given above, and the difference between the results of the two methods of computing rest points is insignificant for values of r close to unity (less than 0.05 divisions for initial readings 30 divisions apart a t r = 0.90). However, it is interesting to note that there is a mathematically exact method for finding the rest point of an analytical balance from only three readings. REST POINT
+
1
AMPLITUDE Figur. 3
