Why Does Distance From Axis Affect Moment of Inertia?

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Discussion Overview

The discussion revolves around the concept of moment of inertia, particularly how the distance from the axis of rotation affects it. Participants explore the relationship between mass, distance, and resistance to changes in angular velocity in the context of rotational dynamics.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about the definition of moment of inertia and its analogy to mass in linear dynamics, suggesting that resistance to rotation should be inversely proportional to distance from the axis of rotation.
  • Another participant corrects the first by stating that moment of inertia is proportional to mass and emphasizes that the relationship is given by I = mr², not I = m/r.
  • A participant clarifies that while the torque applied with a force is proportional to the distance from the axis (r), the rotational inertia itself remains constant as the mass distribution does not change.
  • There is a reiteration that the moment of inertia for a point mass is proportional to r², indicating a need for deeper understanding of this relationship.

Areas of Agreement / Disagreement

Participants do not reach consensus on the initial confusion regarding the relationship between moment of inertia and distance from the axis. Multiple viewpoints are presented, with some participants correcting earlier claims and others seeking clarification.

Contextual Notes

Participants express uncertainty about the implications of applying force at different distances and how this relates to torque versus moment of inertia. The discussion highlights the need for clarity on the definitions and relationships involved in rotational dynamics.

ninja319
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this might sound like a strange and general question but I am a bit confused so please help me, i was wondering what exactly moment of inertia is

- we know just like mass offers resistance in linear dynamics, we need something that offers a similar kind of resistance in rotational dynamics
- i imagined a flat, somewhat heavy, metal plate (lets assume completely cylindrical and a relatively small thickness compared to the radius. more like a flat metallic pizza) floating in empty space. it has a chunk (a slice) cut away from it (so it looks like a metallic pizza with one slice missing).
- now i have to find what is causing the resistance when I am trying to apply force and rotate it.
- i know of the plate causes rotation . so 'I' is proportional to 'm'.
- now a similar amount of force is applied on a point closer to the axis of rotation and a point further away from it. the force applied further away (from observation) shows less resistance. that is it is harder to change the angular velocity when you apply the force closer to a point on the axis of rotation. so further the distance we move from the axis of rotation (more the r) lesser the resistance becomes. therefore, 'I' is inversely proportional to r.
- but that is clearly not the case as I = mr^2 not I = m/r
- I am sure i went wrong in the reasoning. can someone please tell me what it is?
 
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sorry, correction

"i know of the plate causes rotation . so 'I' is proportional to 'm'."

- i know that more the mass of the plate, more the resistance. so 'I' is proportional to 'm'
 
ninja319 said:
- now a similar amount of force is applied on a point closer to the axis of rotation and a point further away from it. the force applied further away (from observation) shows less resistance. that is it is harder to change the angular velocity when you apply the force closer to a point on the axis of rotation. so further the distance we move from the axis of rotation (more the r) lesser the resistance becomes. therefore, 'I' is inversely proportional to r.
- but that is clearly not the case as I = mr^2 not I = m/r
But the amount of angular acceleration is proportional to how far from the axis the force is applied. Since you are not changing the mass distribution of your object, its rotational inertia remains the same. But the torque you are applying with a given force is proportional to r.

In your example you are changing the torque, not the rotational inertia. (Analogous to changing the force, but not the mass, and getting a different linear acceleration.)
 
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