IV: The Sensitivity Change of Analytical Balances KAROL J. MYSELS University of Southern California, Los Angeles, California
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bending of the beam of an analytical balance is generally2 brought into discussions of sensitivity of analytical balances as an important and occasionally as the main factor. I am indebted to Mr. Milton Gray (now of Thackabery Tool Company) for calling my attention to the lack of significa.nceof this factor and to the fact that it is the relative position of the knife edges tha,t determines the variation of sensitivity with load. We may mea,surethe sensitivities of a balance by t,he I Suggestions of material suitable for this column are eagerly sought and will be acknowledged. They should be sent with as many details as possible to K. J. M. at the above address. Since the purpose of this column is to prevent the spread and continuation of errors and not the evduation of individual texts, the source of the emon discussed will not be cited. The error must occur in s t least two independent standard books to be presented.
millimeters deflection d of the pointer per milligram of excess weight on one pan. This may be readily calculated as follows: As shown schematically in the figure, a balance beam weighing B grams, loaded by weights W and W w, reaches equilibrium after deflecting by an angle a under the influence of the moments of three forces with respect to its fulcrum E. These are (a) the deflecting force w with a moment wl cos a where I is the length of the balance arm; (b) the restoring force B of the weight of the beam acting at the center of gravity G of the rigid part of the oscillating system with a moment Bg sin a, where g is the distance of this center of gravity from the fulcrum; and (c) the restoring or deflecting force 2W of the two equal parts of the load applied at the midpoint M of the two extreme knife edges, its moment 2W6 sin a, is restoring if M is below the fulcrum (by 6), d o fleeting if i t is above, and zero if M coincides with the fulcrum; i. e., if the three knife edges are exactly in one plane. Thus:
+
wl eos n = (Rg
+ 2W6) sin ar
Introducing the length of the pointer P, we have tgcr = d / P and after rearranging:
If the beam is rigid the two expressions in brackets are constant for a given balance and the inverse of the sensitivity is a linear function of the load. It increases or decreases depending on the sign of 6. By plotting 11s versus W, a straight line is obtained whose intercept a t zero load on the knife edges (i. e., nfter correrring for rhr weight of the pans and urirrups) eivrs o when rnnltivlied 111' PI ' B nnd whose I o n r vi\.es b whe; multiplied dy P L / ~ On the other hand, if the beam is not rigid both g and 6 will increase with W. Hence the above plot of l / s versus W will curve upward. This curvature will be due mainly to 6 which would obviously be affected much more than g. It is thus possible to determine very accurately any bending of the beam as well as the deviation of the knife edges from a plane and the position of the center of gravity, by determining the sensitivity a t three widely differing loads. &
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OCTOBER, 1955
519
Mr. G. S. Clayson, President of Wm. Ainsworth full load. To place any bending amounting to a and Sons, kindly told me that he once had a standard small fraction of a micron in proper perspective it beam adjusted so carefully that it showed the same should be compared with the other dimensions. In a sensitivity of five divisions per two mg. a t no load and balance which has a zero load sensitivity of 2.5, g at the full load of 200 g. When this beam was over- amounts to about 150 microns. If this sensitivity loaded by 400 per cent to one kg., the sensitivity in- decreases by 20 per cent a t 200 g. load, 6 is about 4 creased to 5L/8divisions per two mg. Assuming reason- microns. Hence the bending of the beam, while it able constants these results are compatible uithin must, of course, occur, can only be a very secondary experimental error with a perfectly rigid beam whose factor. knife edges are one micron (four millionths of an inch) I t may be worth noting, incidentally, that if the out of plane, or alternately that the maximum possible sensitivity of a balance is independent of load, its bending of the beam was one micron under a load of period will be increasing rapidly with load, because one kg., i. e., 0.2 microns at full normal load. constant sensitivity means that the restoring torque I have recently tested balances of two other makes remains the same while the moment of inertia rises and could not detect any bending up to 250 per cent of manyfold.
