Page:Elementary Text-book of Physics (Anthony, 1897).djvu/94

 beam from the position of equilibrium. If the balance be accurately made and perfectly adjusted, and equal weights placed in the scale-pans, the pointer will remain at rest, or will oscillate through distances regularly diminishing on each side of the zero of the scale.

If the weight of a body is to be determined, it is placed in one scale-pan, and known weights are placed in the other until the balance is in equilibrium, or nearly so. The final determination of the exact weight of the body is then made by one of three methods: we may continue to add very small weights until equilibrium is established; or we may observe the deviation of the pointer from the zero of the scale, and, by a table prepared empirically, determine the excess of one weight over the other; or we may place a known weight at such a point on a graduated bar attached to the beam that equilibrium is established, and find what its value is, in terms of weight placed in the scale-pan, by the relation between the length of the arm of the beam and the distance of the weight from the middle point of the beam.

If the balance be not accurately constructed, we can, nevertheless, obtain an accurate value of the weight desired. The method employed is known as Borda's method of double weighing. The body to be weighed is placed in one scale-pan, and balanced with fine shot or sand placed in the other. It is then replaced by known weights till equilibrium is again established. It is manifest that the replacing weights represent the weight of the body.

If the error of the balance consist in the unequal length of the arms of the beam, the true weight of a body may be obtained by weighing it first in one scale-pan and then in the other. The geometrical mean of the two values is the true weight; for let $$l_{1}$$, and $$l_{2}$$ represent the lengths of the two arms of the balance, $$P$$ the true weight, and $$P_{1}$$ and $$P_{2}$$ the values of the weights placed in the pans at the extremities of the arms of lengths $$l_{1}$$ and $$l_{2}$$, which balance it. Then $$Pl_{2} = P_{1}l_{1}$$ and $$Pl_{1} = P_{2}l_{2}$$; from which $$P = \sqrt{P_{1}P_{2}}.$$

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