True Weight from False Balance/Imbalanced Pans

Theorem

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

Let $B$ be weighed in a false balance with imbalanced pans.

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

Then:

$W = \dfrac {a + b} 2$

Proof

We have that the false balance has pans such that one weighs more than the other.

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

Let one of the pans weigh $m$ more than the other.

Placing $B$ in the lighter pan gives:

$W x = \paren {a + m} x$

and placing $B$ in the heavier pan gives:

$\paren {W + m} x = b x$

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:

\(\ds W\)

\(=\)

\(\ds a + m\)

\(\ds W + m\)

\(=\)

\(\ds b\)

\(\ds \leadsto \ \ \)

\(\ds 2 W\)

\(=\)

\(\ds a + b\)

\(\ds \leadsto \ \ \)

\(\ds W\)

\(=\)

\(\ds \dfrac {a + b} 2\)

$\blacksquare$