How to Determine the Axis of Rotation

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Discussion Overview

The discussion revolves around determining the appropriate axis of rotation for calculating the rotational kinetic energy of a system consisting of a massless bar with two different masses attached to springs. Participants explore the implications of choosing different axes, such as the center of the bar or the center of mass, and how this choice affects the calculation of kinetic energy.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant suggests that when determining the kinetic energy of the system, it may not be necessary to focus on an axis of rotation, proposing to simply add the kinetic energies of the two masses.
  • Another participant notes that the choice of axis influences the distribution of kinetic energy into "rotational" and "linear" components, but the total kinetic energy remains unchanged.
  • A further contribution emphasizes the importance of selecting a convenient axis, particularly one for which the moment of inertia is known, to facilitate the calculation of rotational kinetic energy.
  • Another participant introduces a more complex approach involving the Lagrangian method, suggesting the use of independent variables for position and angle to derive the kinetic and potential energies.

Areas of Agreement / Disagreement

Participants express differing views on the necessity and implications of choosing an axis of rotation, indicating that multiple competing perspectives remain without a clear consensus on the best approach.

Contextual Notes

Some assumptions about the system's motion and the definitions of kinetic and potential energy are not fully detailed, and the discussion does not resolve the mathematical steps involved in the Lagrangian approach.

Diracobama2181
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Let's say I have a massless bar of length ##l## with two different masses, ##m_1## and ##m_2##. Suppose an identical spring is attached to each individual mass, with the other end being attached to the ceiling. How would I go about determining the rotational kinetic energy of the system. Do I choose the axis about the center or the center of mass?
 
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Diracobama2181 said:
Summary: When determining the kinetic energy of a rotating system, which point should I use as an axis to determine rotation kinetic energy.

Let's say I have a massless bar of length ##l## with two different masses, ##m_1## and ##m_2##. Suppose an identical spring is attached to each individual mass, with the other end being attached to the ceiling. How would I go about determining the rotational kinetic energy of the system. Do I choose the axis about the center or the center of mass?
If your goal is to determine the kinetic energy of the system, there is no need to bother with an axis of rotation. You have two masses. Add up their kinetic energies.
 
Diracobama2181 said:
When determining the kinetic energy of a rotating system, which point should I use as an axis to determine rotation kinetic energy.
The chosen axis merely determines how much of the kinetic energy is considered "rotational", and how much "linear". Their sum (total KE) won't change.

You should choose an axis which is most convenient. For example, the one for which you know the moment of inertia, which is needed for the rotational KE calculation:
https://en.wikipedia.org/wiki/Rotational_energy
 
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Diracobama2181 said:
Summary: When determining the kinetic energy of a rotating system, which point should I use as an axis to determine rotation kinetic energy.

Let's say I have a massless bar of length ##l## with two different masses, ##m_1## and ##m_2##. Suppose an identical spring is attached to each individual mass, with the other end being attached to the ceiling. How would I go about determining the rotational kinetic energy of the system. Do I choose the axis about the center or the center of mass?

We can assume a flat motion system. Then we have 3 independent variables in the context of an inertial coordinate system : (x,y) for the position of the center of masses and φ for the bar's angle. From this information you can calculate the potential energy due to gravity an due to the strings and complete the Lagrangian as L=KineticEnergy-PotentialEnergy; where KineticEnergy is the sum of the kinetic energy of the center of masses plus rotational energy relative to the center of masses = 1/2 I (dφ/dt)^2 and I= moment of inertia relative to the center of masses.
 

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