Answer
The equation $m=\sqrt {m_1m_2}$ is proved with the use of the concept of net torque about the fulcrum equal to 0 each for both given situations where two equations will be derived and after simplification, the result is the equation to be proved.
Work Step by Step
Let the whole length of the pan balance be $d$ and the length of the left end to fulcrum $O$ be $x$.
The length of from the fulcrum to the right end will become $d-x$.
For the first situation:
$\tau_{net,O}=0$
$m_1(d-x)-mx=0$
$m_1d-m_1x-mx=0$
$m_1d-(m_1+m)x=0$
$d=\frac{(m_1+m)x}{m_1}$
For the second situation:
$\tau_{net,O}=0$
$m(d-x)-m_2x=0$
$md-mx-m_2x=0$
$md-(m+m_2)x=0$
$d=\frac{(m_1+m_2)x}{m}$
Equate the two equations:
$\frac{(m_1+m)x}{m_1}=\frac{(m+m_2)x}{m}$
Cancel $x$:
$\frac{m_1+m}{m_1}=\frac{m+m_2}{m}$
$m(m_1+m)=m_1(m+m_2)$
$mm_1+m^2=mm_1+m_1m_2$
Cancel $mm_1$:
$m^2=m_1m_2$
Take the square root of both sides:
$m=\sqrt{m_1m_2}$
Therefore, the equation $m=\sqrt{m_1m_2}$ is proved.
