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

In summary: The moment of inertia is the product of the mass and the square of the distance from the center of the mass to the center of the rotation.
  • #1
zorro
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Homework Statement



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


Homework Equations





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
but the answer is Ml^2/6
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
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
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
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
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
How is the moment of inertia defined for an extended body?

ehild
 
  • #7
here the extended body is a square. So its moment of inertia is same as that of a square.
 
  • #8
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
By using the equation ∫x^2 dm
 
  • #10
What do you mean on x? If x is the horizontal axis, a rod along the y-axis has zero moment of inertia?



ehild
 

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

What is the definition of moment of inertia?

The moment of inertia is a physical property of a rigid body that describes its resistance to rotational motion about a particular axis. It is also known as rotational inertia.

How is moment of inertia calculated?

The moment of inertia of a plate about the z axis can be calculated using the formula I = MR², where M is the mass of the plate and R is the distance between the axis of rotation and the center of mass of the plate.

What factors affect the moment of inertia of a plate?

The moment of inertia of a plate is affected by its mass and shape. The further the mass is distributed from the axis of rotation, the higher the moment of inertia will be. In addition, the shape of the plate also plays a role in determining the moment of inertia.

How does moment of inertia affect the rotational motion of a plate?

The higher the moment of inertia of a plate, the more difficult it is to change the rotational motion of the plate. This means that a plate with a higher moment of inertia will require more torque to rotate compared to a plate with a lower moment of inertia.

Can the moment of inertia of a plate change?

Yes, the moment of inertia of a plate can change if the mass or shape of the plate is altered. For example, if more mass is added to the plate, the moment of inertia will increase. Similarly, changing the shape of the plate can also affect its moment of inertia.

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