Taylor's "Classical mechanics" - inertial balance

Question

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)

Answer 1

You have a rod. It is accelerating. Either it passes through the other objects like a ghost or it pushes them until they accelerate with an equal acceleration.

To accelerate each of them it must exert a force on them (Newton 2nd) and so by Newton 3rd the rod feels an equal an opposite force on each end.

If the rod is free to rotate about the exact center then it will rotate unless the torques are equal. The torques are equal if the forces are equal (its the mid point) the forces on the two ends are themselves equal and opposite to the forces on the masses. So they are equal if the forces on the masses ate equal.

The forces on the masses are such as to produce the same acceleration (the acceleration of the device) so the accelerations are the same. If the accelerations are he same the forces can only be the same if the masses are the same.

This is actually what you see if you are in a frame freely falling due to gravity. In that frame a balance scale is accelerating upwards and the masses fail to torque the balance scale because the inertial masses are equal.

In the frame stationary to the earth we think the scale is not accelerating upwards so we say there is an inertial force (inertial force just means a force that is proportional to inertial mass so that everything feels the same acceleration, the acceleration of the frame when no other forces act). The inertial force is called gravity. If you pretend gravity isn't an inertial force you have to invent passive gravitational mass and then note it exactly equals inertial mass.

But any force proportional to inertial mass is an inertial force and really comes from using a non inertial frame. Gravity included.

Answer 2

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 $0$.