Moment of inertia of a metal plate about three points

[Krushnaraj Pandya]

Homework Statement

Moment of inertia of a metal plate (center at origin) which was a square before 1/4th of it was cut off (4th quadrant), about three points a)- top leftmost corner in 2nd quadrant b)- origin and c) (where the right-lowest corner used to be in 4th quadrant) ranked in decreasing order

Homework Equations

I=Mr^2

The Attempt at a Solution

using my intuition, I figured the order is c<a<b, since more points are farther away from c, then a and then b- but according to parallel axis theorem moment of inertia at b and c should be same...i'm confused how to do this mathematically

 

[Nathanael]

but according to parallel axis theorem moment of inertia at b and c should be same...

Can you explain this?

I have a feeling you meant to say “a and c should be the same”? If so, your mistake is that the parallel axis theorem must use the center of mass. The CoM is not at the origin.

 

[Gene Naden]

Start by determining the moment of inertia of the plate before the square was cut off. Determine it for each of the three points. Then determine the moment of inertia of the plate that was cut off. Determine that for each of the three points. Now subtract. The moment of inertia of the plate after the square was cut off is the moment of the plate before the square was cut off, minus the moment of the square that was cut off.

 

[Krushnaraj Pandya]

Can you explain this?

I have a feeling you meant to say “a and c should be the same”? If so, your mistake is that the parallel axis theorem must use the center of mass. The CoM is not at the origin.

yes, that's what I meant. Thanks for pointing out my mistake

 

[Krushnaraj Pandya]

Start by determining the moment of inertia of the plate before the square was cut off. Determine it for each of the three points. Then determine the moment of inertia of the plate that was cut off. Determine that for each of the three points. Now subtract. The moment of inertia of the plate after the square was cut off is the moment of the plate before the square was cut off, minus the moment of the square that was cut off.

Excellent! Thanks a lot

 

[Nathanael]

Moment of inertia is additive, so what Gene says is true. But, if we’re only asked to rank the three points in descending order, then we don’t need to do all that work.

Your intuition (and answer) is correct. The parallel axis theorem can be used to restate the problem as, “order a,b,c by their distance to the center of mass,” which is simpler.

 

[Krushnaraj Pandya]

Moment of inertia is additive, so what Gene says is true. But, if we’re only asked to rank the three points in descending order, then we don’t need to do all that work.

Your intuition (and answer) is correct. The parallel axis theorem can be used to restate the problem as, “order a,b,c by their distance to the center of mass,” which is simpler.

Certainly, I understand it much better now. Thank you.