JOURNAL OF
CHEMICAL EDUCATION
DETERMINATION OF THE CENTER OF GRAVITY OF THE BEAM OF A CHEMICAL BALANCE HARVEY V. MOYER Ohio State University, Columbus, Ohio
A LABORATORY exercise to determine the center of gravity of the beam and pointer of an analytical balance is an excellent method of emphasizing some of the fundamental principles of the theory of the balance. The necessary measurements are easy to make and the derivation of the equation for calculating the distance from the middle knife edge to the center of gravity is quite simple if the sensitivity is measured horizontally along the pointer scale instead of along the arc described by the pointer. Procedure.-Determine the sensitivity of the balance in the usual way by observing the differencebetween the zero point of the empty balance and the rest point with an excess load of one mg. on one side. Remove the pans and measure the distance between the two rest points with a small celluloid metric rule placed horizontally along the pointer scale tangent to the arc described by the pointer. Remove the beam and weigh it without the stirrups on a rough balance. Lay the beam and pointer on a sheet of paper and mark the position of the knife edges and the tip of the pointer. Use the marks to measure the distance in millimeters between the knife edges and the distance from the middle knife edge to the tip of the pointer. Calculations-The equation for calculating the location of the center of gravity is derived as a part of the exercise. It is possible to simplify this derivation so that all trigonometric functions are eliminated.
excess weight (usually one mg.) in the right pan, G is the center of gravity of the unloaded beam and pointer, CH is the length of the pointer, and PH is the horizontal displacement of the rest point caused by the excess weight, w. It is assumed that the zero point of the empty balance is located a t the mid-point of the pointer scale. If the beam is in stable equilibrium in this position, the moments tending to turn the beam on its axis are equal, hence, wXAD= WXGE
(1)
The right triangles, GCE, PCH, and CAD are similar because they have equal angles, therefore, the ratios of their corresponding sides are equal.
and,
Substituting the value of AD in equation (1) and solving for GE gives,
Substituting the above value of GE in equation (2) gives the desired form of the equation for expressing the location of the center of gravity of the beam and pointer. GC
=
w X CA X t H
WXPH
The following measurements were made on an Ainsworth Student Type balance: w CA CH W PH
=
0.0010g.
= 76mm. = 242 mm. =
-
35.2g. 3.5mm.
If the above values are substituted in equation (5), GC = 0.15 mm. Students are surprised to find that the center of gravity is so small a distance below the central knife edge. After making the measurements and calculating the location of the center of gravity it is a simple matter to solve the equation for the sensitivity, P H , thus Tns STABLE POSITION OF T ~ BEAM, E A' A , AND emphasizing that the sensitivity of a balance is directly PomrER, p, WITH AN EXCESS LOAD,W, I N THE proportional to the lengh of the beam and to the RIGHTPANOP THE BALANCE length of the pointer and that it is inversely proportional In the accompanying drawing, CA is one arm of the to the weight of the beam and to the distance of the beam, W is the weight of the beam and pointer, w is the center of gravity below the middle knife edge.
