Why Does Distance From Axis Affect Moment of Inertia?

AI Thread Summary
Moment of inertia represents an object's resistance to changes in its rotational motion, similar to how mass functions in linear dynamics. The discussion highlights that while mass influences resistance, the distance from the axis of rotation affects torque rather than the moment of inertia itself. Specifically, applying force further from the axis results in greater torque, leading to more angular acceleration, but does not alter the object's rotational inertia, which is defined as I = mr^2. The confusion arises from equating resistance with distance, whereas moment of inertia is inherently linked to the square of the distance from the axis. Understanding this distinction clarifies why I is proportional to r^2, not just r.
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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