FINDING THE REST POINT OF A BALANCE A Mathematical Justi$cation for a n Odd Number of Swings JONATHAN D. WIRTSCHAFTER1 Reed College, Portland, Oregon
Tm
determination of the rest point of the undamped If the pointer follows the equation for damped haranalytical balance is one of the critical steps in gravi- monic motion, the ratio between the successive dismetric analysis. Chemistry students are taught that placements is a constant. the rest point (also called the zero point or the center of swings) is calculated by reading an odd number of displacements, averaging the displacements to the right and the left separately, and then averaging the two averages. This calculation is taught as a rule and the Expanding the constant, student is never certain that some other method could not be used. All textbooks give instructions for the procedure. Fales and Kenny offer a proof.2 Stacya writing in Since the terms after the second are small compared THIS JOURNAL has offered another proof as nell as reto the error in the readings, they are dropped. viewing that of Fales and Kenny. Yet neither of these answer the student's question: "What is wrong with an even number of readings?" While the chemistry course cannot go into the mathematical physics Because of the proof, it can give the student an opportunity to work out the algebra of the rest point calculation. This can readily be done if the terms of the exponentials of the Fales and Kenny are presented in chart . proof form. Sp can be written in terms of S6. Similarly all other five the pointer. The terms can be written as powers of Ss. These terms are readings represent the maximum displacement from expanded and then approximated in Table 1. The rest the true rest point during each swing. The distances pointcan be calculated in terms of the data from the rest point are SI . . . SL. there. It is apparent that since the rest point is the unknown, i t can be found only if two points equidistant 1 The author will receive his B.A. from Reed College, June, 1956. on either side of it are knoum. The student must find FALES, H. A,, AND F.KENNY,"Inorganic Quantitative Analyway to a displacement to the right of the sis," Appleton-Century-Crafts, Inc., New York, 1925, pp. 84-7. rest point approximately equal to a displacement to the a STACY, I. F., J. CHEM.EDUC.,32,90 (1955). left. Assume that swings 1. 3, and 5 are to the right and *pprozimnSuing that swings 2 and 4 &e to the left. If the rest point is I z n Dzsplaeernent as functwn of Ss no. oalculated from an even number of swings, use of the aT values from the table ~ i e l d s : 1 S , = S, rl + 2aT + 6 (?) + S,(l + ZaT)
+ aT + (a$)2]
3
S = 8, [ l
4
&=S5[1+0.5aT] S6 = S S
5
(7)A
But
S O
+ aT)
S6(1
+ 0.5aT) This inequality shows that an even number of swings
S5
&(l
+ 1.5oT) # Ss(l + a T )
cannot establish the rest point. In contrast to this,
VOLUME 33, NO. 5, MAY, 1956
217
an equality will result when an odd number of swings are used for the calculation as follows: +
+
" - "(l
+
2aT)
- =+ " "
2
2 "
+
+
=
Sdl ia T )
&(I ia T )
The use of an odd number of swings as the correct
basis of rest point determination thus can be demonstrated aleebraicallv. The loeic of this will overcome the objections of tge student who "intuitively knows" that the average of an even number of swings should give the correct value. This method of proof allowsthe teacher to give a convincing demonstation of an important operation instead of merely prescribing another rule for the student t o follow in the laboratory.
