Mechanics of an inertial balance

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Discussion Overview

The discussion revolves around the mechanics of an inertial balance, specifically focusing on the mathematical proof of how unequal masses lead to unequal accelerations and rotational motion of the rod. Participants explore various approaches to understanding the dynamics involved, including the concept of angular momentum and the center of mass (CoM) frame.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests using angular momentum to analyze the system, noting that the sum of angular momenta cannot be zero due to unequal accelerations, but finds this explanation lacking rigor.
  • Another participant proposes examining the system from the CoM frame, indicating that this perspective reveals inertial forces acting on each mass and leads to a net torque if the masses are unequal.
  • A later reply questions the necessity of using the CoM frame and expresses uncertainty about the concept, suggesting a potential wait for formal introduction in the relevant textbook.
  • Another participant asserts that the same force acts on both masses, leading to identical acceleration only if the masses are equal, prompting questions about the basis for this assumption.
  • One participant references Newton's third law to argue that forces on the rod and masses must be equal if the rod is not rotating, while another participant questions the elegance of this reasoning compared to earlier suggestions.
  • There is a reiteration of the assumption that the rod's lack of rotation implies equal forces on the masses.

Areas of Agreement / Disagreement

Participants express differing views on the best approach to analyze the problem, with no consensus reached on the most rigorous method or the validity of assumptions regarding forces acting on the masses.

Contextual Notes

Participants highlight the dependence on the center of mass concept and the implications of rotational motion, but do not resolve the mathematical steps or assumptions involved in their reasoning.

Quantum55151
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In the following diagram (from Taylor's Classical Mechanics), an inertial balance is shown.

1719947221301.png


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.
 
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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).
 
Orodruin said:
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?
 
Last edited:
Quantum55151 said:
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.
 
Mister T said:
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?
 
Quantum55151 said:
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.
 
Quantum55151 said:
What is this assumption based on?
The fact that the rod doesn't rotate.
 

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