Q:

Taylor's "Classical mechanics" - inertial balance

I am currently reading John Taylor's "Classical Mechanics" book and I am wondering how does the inertial balance work (as explained in figure 1.2, page 10).
The autor says about an inertial balance and two masses attached to the opposite ends of a rigid rod - the masses are equal if and only if a force applied at the rod's midpoint causes them to accelerate at the same rate, so that the rod does not rotate. I searched on the web hoping to see how an inertial balance works but my attempt to make the connection between the inertial balance described, for example, here and the inertial balance in Taylor's book failed. Can someone explain to me how the inertial balance works in Taylor's book?
Here is the redirection to Taylor's book (page 10)

A:

Both links you provide to 'inertial balances' refer to quite different devices. 
The first one, the so called 'wig wag' is rather like a pendulum, where oscillation frequency is inversely proportional to inertia, i.e. mass.
The second type relies on unequal inertial forces causing torque around a point of symmetry.

When $F$ acts it causes acceleration in the horizontal direction acc. $F=(m_1+m_2)a$. This acceleration causes inertial reaction forces on $m_1$ and $m_2$ respectively of $F_1=m_1a$ and $F_2=m_2a$.
These cause torques respective to $0$, the net torque being $T=F_1L-F_2L=m_1aL-m_2aL=(m_1-m_2)aL$.
As long as $m_1=m_2$ $\Rightarrow$ $T=0$ but for $m_1 \neq m_2$ $\Rightarrow$ $T \neq 0$ and this net torque causes rotation of the weights around $0$.

