. Physical laboratory experiments : mechanics, optics and heat. of the Balance.—Reduced to its elements, an equalarm balance consists of a rigid beam supported horizontally atits centre on a knife edge and loaded at its ends. The variousconditions upon which the sensitiveness of a balance dependswill be readily seen from the following demonstration: Let I = length of arms (assumed equal). w = weight of beam acting downward through centre of = distance of centre of gravity, CG, below point ofsupport of the beam. With a load of P grams in each pan, the balance will evi-dently be in equ
. Physical laboratory experiments : mechanics, optics and heat. of the Balance.—Reduced to its elements, an equalarm balance consists of a rigid beam supported horizontally atits centre on a knife edge and loaded at its ends. The variousconditions upon which the sensitiveness of a balance dependswill be readily seen from the following demonstration: Let I = length of arms (assumed equal). w = weight of beam acting downward through centre of = distance of centre of gravity, CG, below point ofsupport of the beam. With a load of P grams in each pan, the balance will evi-dently be in equilibrium. If a small additional weight p beadded to the right-hand pan, the balance will swing throughan angle a and take up a new position of equilibrium as inFig. 12. The condition for equilibrium will evidently befound by taking moments of the three effective forces,P, P -\- p, and w about an axis passing through the knifeedge, , PI cos a -\- wV sin a = (P + p) Z cos a, , sin a Ip f^s or tana ^= ^ -^,. (1) cos a Wl 30 PHYSICAL LABORATORY EXPERIMENTS. Fig. 12. Now the sensitiveness of a balance is measured by theangle through which the beam will rotate when a definiteweight, usually one milligram, is added to either pan. Asthe angle practically coincides with its tangent for verysmall angles, the expression deduced for tan a may be takenas a measure of the sensitiveness of the balance whenp = 1 milligram. From this it follows that the sensitiveness of a balance is di-rectly proportional to the length of the balance arms; in-versely proportional to the weight of the beam; and inverselyproportional to the distance of the centre of gravity belowthe point of support. It is also evident that the observedscale deflection is proportional to the length of the pointer. Unfortunately the above conditions for maximum sensi-tiveness conflict with each other in the mechanical construc-tion of a balance. Thus long arms are incompatible withminimum weight. The length of the arms is al
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