Homework Help: What is the moment of inertia of the plate about z axis?

1. Sep 19, 2010

zorro

1. The problem statement, all variables and given/known data

What is the moment of inertia of the plate about z axis?

2. Relevant equations

3. The attempt at a solution

Consider the isosceles triangle to be a part of a square of side l/root(2)
Its mass will be 2M
We know that its moment of inertia about the centre perpendicular to the plane is 2M(l^2/2)/6=Ml^2/6
Applying parallel axes theorem, moment of inertia about z axis will be Ml^2/6 + 2M(l^2/4)
which is 2/3 (Ml^2)
Moment of inertia of the triangle will be half of it i.e. Ml^2/3
Please explain if there is any other easier approach.
So the moment of inertia of the triangle will be half of it i.e. Ml^2/12

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2. Sep 19, 2010

rl.bhat

What is the moment of inertia of the plate about z axis?

What is the shape of the plate and the point of suspension?

3. Sep 19, 2010

zorro

I am sorry I did not post the full question. Here it is.

The figure shows an isosceles triangular plate of mass M and base L. The angle at the apex is 90°. The apex lies at the origin and the base is parallel to X-axis

4. Sep 19, 2010

ehild

The moment of inertia of the triangle is not half that of the square. See the picture: the points of the upper triangle are farther than those of the lower triangle.

It is easy to get the moment of inertia by integrating (x^2+y^2)dm for the triangle.

ehild

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5. Sep 19, 2010

zorro

The base of the triangle is fixed ( it is of length l). Since the apex angle is 90 degrees and it is isosceles, You get a square when it is rotated about l. So there is no chance of upper triangle being bigger or smaller.

6. Sep 19, 2010

ehild

How is the moment of inertia defined for an extended body?

ehild

7. Sep 20, 2010

zorro

here the extended body is a square. So its moment of inertia is same as that of a square.

8. Sep 20, 2010

ehild

You did not understand me. Forget for the moment that this triangle is half of a square. It is just a set of elementary masses, organized in a certain shape. How would you get the moment of inertia of a set of point masses?

ehild

9. Sep 22, 2010

zorro

By using the equation ∫x^2 dm

10. Sep 22, 2010

ehild

What do you mean on x? If x is the horizontal axis, a rod along the y axis has zero moment of inertia?

ehild