True Weight from False Balance/Unequal Arms

Theorem

Let $B$ be a body whose weight is $W$.

Let $B$ be weighed in a false balance with unequal arms.

Let the readings of the weight of $B$ be $a$ and $b$ when placed in opposite pans.

Then:

$W = \sqrt {a b}$

Proof

We have that the false balance has arms of different lengths.

Let the lengths of the arms of the false balance be $x$ and $y$.

Without loss of generality, placing $B$ in the pan at the end of $x$ gives:

$W x = a y$

and placing $B$ in the pan at the end of $y$ gives:

$W y = b x$

A particular theorem is missing. In particular: We need to invoke the physics of couples to justify the above statements, but we haven't done the work yet to cover it. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding the theorem. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{TheoremWanted}} from the code.

Then:

\(\ds W x\)

\(=\)

\(\ds a y\)

\(\ds W y\)

\(=\)

\(\ds a x\)

\(\ds \leadsto \ \ \)

\(\ds \dfrac x y\)

\(=\)

\(\ds \dfrac W a\)

\(\ds \)

\(=\)

\(\ds \dfrac b W\)

\(\ds \leadsto \ \ \)

\(\ds W^2\)

\(=\)

\(\ds a b\)

multiplying both sides by $W a$

\(\ds \leadsto \ \ \)

\(\ds W\)

\(=\)

\(\ds \sqrt {a b}\)

$\blacksquare$