Parallel Axis Theorem

In summary, the conversation involves a person seeking help with a prove problem involving a rectangular sheet with sides a and b and uniform density. The problem consists of two parts: in part (a), the person needs to show that the moment of inertia of the sheet about an axis L that is perpendicular to the sheet and passes through its center is equal to 1/12 times the total mass M multiplied by the sum of the squares of the sides a and b. In part (b), the person needs to use the Parallel Axis Theorem to show that the moment of inertia about an axis L' that is perpendicular to the sheet and passes through one of its corners is equal to 1/3 times the total mass M multiplied by the sum
  • #1
mayaitagaki
8
0
Hi everyone,
I've got stuck on this prove problem:cry:
Please help me!

Let S be a rectangular sheet with sides a and b and uniform density, and total
mass M.

(a) Show that the moment of inertia of S about an axis L that is perpendicular to S,
meeting S through its center, is

I =1/12*M(a^2 + b^2)

(b) Use the Parallel Axis Theorem in combination with part (a) to show that the moment of inertia of S about an axis L' that is perpendicular to S, meeting S through one of its corner, is

I =1/3*M(a^2 + b^2)

Please see the attachment.
Theorem and an example!

Thank you,
Maya
 

Attachments

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  • Parallel Axis Theorem (ex).JPG
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  • #2
You need to show some work first so we can see where you're getting stuck.
 
  • #3
Ok, sorry about that! :shy:

I think I kind of got the part a. Then, I don't know how to go from there...

Thank you,
 

Attachments

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  • #4
In the parallel-axis theorem, h is the distance from the center of mass of the object to the new axis. In your case, it would be the distance from the center of the slab to the corner. What is that in terms of a and b?
 

1. What is the Parallel Axis Theorem?

The Parallel Axis Theorem is a principle in physics that relates to the moment of inertia of an object. It states that the moment of inertia of an object about any axis parallel to its center of mass is equal to the moment of inertia about its center of mass plus the product of the object's mass and the square of the distance between the two axes.

2. Why is the Parallel Axis Theorem important?

The Parallel Axis Theorem is important because it allows us to easily calculate the moment of inertia of an object about an axis that is not its center of mass. This is useful in various fields such as physics, engineering, and astronomy where the moment of inertia is an important factor in understanding an object's motion and behavior.

3. How is the Parallel Axis Theorem derived?

The Parallel Axis Theorem can be derived using the parallel axis theorem for an infinitesimal element of mass, which states that the moment of inertia of an infinitesimal element of mass about an axis is equal to the product of the element's mass, its distance from the axis squared, and a geometrical factor. By integrating this equation over the entire object, we can derive the Parallel Axis Theorem.

4. Can the Parallel Axis Theorem be applied to any object?

Yes, the Parallel Axis Theorem can be applied to any object as long as its mass distribution is known. It is a general principle that applies to all types of objects, regardless of their shape, size, or composition.

5. How is the Parallel Axis Theorem used in real-life applications?

The Parallel Axis Theorem is used in various real-life applications, such as in the design of vehicles, machinery, and structures. It is also used in sports, such as in the design of sports equipment like golf clubs and tennis rackets. Additionally, it is used in astronomy to calculate the moment of inertia of celestial bodies, and in medicine to analyze the movement of the human body.

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