# Distance between 2 axis in Parallel Axis Theorem

• Sunbodi
In summary: I am glad I could help. In summary, the moment of inertia for a perpendicular axis through the center of a uniform, thin, rectangular metal sheet with sides a and b is (1/12)M(a2 + b2). If the axis is through a corner, the moment of inertia is given by I = 1/3 M(a^2 + b^2). The distance from the center of mass to a diagonal can be found using Pythagoras' theorem, where the distance to one axis is a/2 and the distance to the other axis is b/2.
Sunbodi

## Homework Statement

The moment of inertia for a perpendicular axis through the center of a uniform, thin, rectangular metal sheet with sides a and b is (1/12)M(a2 + b2). What is the moment of inertia if the axis is through a corner?

The answer is given as this was a powerpoint lecture and it is: I = 1/3 M(a^2 + b^2)
I'm not looking for how to solve the whole thing however as the moment of Inertia is given in the context of the problem. I'm trying to understand how D was found to be: (a^2/2^2 + b^2 /2^2).

## Homework Equations

I = I (cm) + Md^2

## The Attempt at a Solution

D is meant to be the distance between the the new axis and the center of mass, if I personally were to solve this I'd state that the center of mass is half the distance between the axis that contains a and the axis that contains b. If that's the case, the distance from the center of mass to a diagonal would be a^2 + b^2 because you're increasing both a and b values by the same distance it originally was from the center of mass.

For clarification: You're trying to understand how the distance between two different axis on a sheet depends on the lengths of the sheet?

I'm asking cause I see questions, but then answers to those questions in later sentences.

RomegaPRogRess said:
For clarification: You're trying to understand how the distance between two different axis on a sheet depends on the lengths of the sheet?

I'm asking cause I see questions, but then answers to those questions in later sentences.
I trying to understand why D = (a^2/2^2 + b^2 /2^2). To me it seems as if it should simply be D = (a^2 + b^2)

Sunbodi said:
the center of mass is half the distance between the axis that contains a and the axis that contains b
a and b are the dimensions of the plate. In what sense does an axis contain them?
Sunbodi said:
the distance from the center of mass to a diagonal
The distance from the centre to a diagonal is zero. You want the distance to a corner.
It's a/2 parallel to one axis, then b/2 parallel to the other. What does Pythagoras have to say on the matter?

Sunbodi
haruspex said:
a and b are the dimensions of the plate. In what sense does an axis contain them?

The distance from the centre to a diagonal is zero. You want the distance to a corner.
It's a/2 parallel to one axis, then b/2 parallel to the other. What does Pythagoras have to say on the matter?

Thank you so much! The second part of your comment really helped me. I've noticed how you're consistently on this forums helping out people whether it comes to high level physics or high school stuff and it's really well appreciated.

Sunbodi said:
Thank you so much! The second part of your comment really helped me. I've noticed how you're consistently on this forums helping out people whether it comes to high level physics or high school stuff and it's really well appreciated.
You are welcome.

## What is the Parallel Axis Theorem?

The Parallel Axis Theorem is a principle in physics that relates to the moment of inertia of a rigid body about a certain axis to its moment of inertia about a parallel axis that is a specific distance away.

## What is the formula for calculating the distance between two axes in the Parallel Axis Theorem?

The formula for calculating the distance between two axes in the Parallel Axis Theorem is: d = h2 * m, where d is the distance between the two axes, h is the distance between the original axis and the parallel axis, and m is the mass of the object.

## Does the distance between two axes affect the moment of inertia in the Parallel Axis Theorem?

Yes, the distance between two axes has a direct impact on the moment of inertia in the Parallel Axis Theorem. The further apart the two axes are, the higher the moment of inertia will be.

## Can the Parallel Axis Theorem be applied to all types of rigid bodies?

Yes, the Parallel Axis Theorem can be applied to all types of rigid bodies, as long as the body is symmetric around its axis of rotation and has a constant mass distribution.

## What are some real-life applications of the Parallel Axis Theorem?

The Parallel Axis Theorem has various applications in fields such as engineering, physics, and mechanics. It is commonly used in designing and analyzing the stability and motion of objects, such as cars, airplanes, and satellites. It is also used in calculating the moment of inertia of rotating objects, such as wheels and gears.

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