Mechanics of an inertial balance

[Quantum55151]

In the following diagram (from Taylor's Classical Mechanics), an inertial balance is shown.

Intuitively, I totally understand that unequal masses would cause unequal accelerations and therefore rotational motion of the rod. However, how does one prove this mathematically?

The first thing that came to mind was to look at the situation from the point of view of angular momentum. The system is in rotational equilibrium if and only if the sum of the angular momenta of the two masses is zero, where L = r x p. But since the masses are subject to unequal accelerations, at any instant in time after the force is applied, p₁ is not equal to p₂, and so the sum of L cannot possibly zero. However, this explanation seems quite hand-wavy to me, so I'm wondering if anyone could suggest a more rigorous treatment of the problem.

 

[Orodruin]

Just look at the CoM frame of the system (ie, an accelerating frame accelerating with the CoM). This factors out the translational part of the motion and you are left with the inertial forces mg acting on each mass. If the masses are different there is a net torque about the mid point (meaning a net torque everywhere as the total force is zero in the CoM frame by definition).

 

[Quantum55151]

Just look at the CoM frame of the system (ie, an accelerating frame accelerating with the CoM). This factors out the translational part of the motion and you are left with the inertial forces mg acting on each mass. If the masses are different there is a net torque about the mid point (meaning a net torque everywhere as the total force is zero in the CoM frame by definition).

Is there a simpler way of looking at it that does not involve CoM? I'm just not too familiar with the concept.

Edit: should I maybe just wait until CoM is formally introduced in the book before tackling the problem?

 

[Mister T]

Is there a simpler way of looking at it that does not involve CoM?

The force on each object is the same, therefore they will have identical acceleration iff they have identical masses.

 

[Quantum55151]

The force on each object is the same, therefore they will have identical acceleration iff they have identical masses.

What is this assumption based on? Sure, there is a force applied at the center of the rod, but why are we allowed to say that the same force acts on both masses?

 

[Nugatory]

What is this assumption based on? Sure, there is a force applied at the center of the rod, but why are we allowed to say that the same force acts on both masses?

Third law says that the force on the rod from each mass is opposite and equal to the force on each mass from the rod. The rod is not rotating so the forces of the rod are equal; therefore the forces on the masses must also be equal.

Whether this is less hand-wavey than your original solution is an open question. It is certainly less elegant than @Orodruin ’s.

 

[Mister T]

What is this assumption based on?

The fact that the rod doesn't rotate.