Question

moment of inertia of uniform Square Plate of mass M and side a about an Axis passing through centre of mass of the square and in the Plane of the square as shown in the figure ​

Answer 1

Answer:

The moment of inertia of uniform Square Plate of mass M and side a about an Axis passing through center of mass of the square and in the Plane of the square was given below:

Explanation:

According to perpendicular axis theorem:

$dI_{x} = dmy^{2}$       where $I_{y}$ = ∫ dmx²

$I_{x}$ = ∫ dmy²

$dI_{z}$ = dm$(\sqrt{x^{2}+y^{2} } )^{2}$

$I_{z} =$ ∫ dmx² + ∫ dmy²

Where,

$I_{z} = I_{x} + I_{y}$ ( for x and y plane)

$I_{x} = I_{y} + I_{z}$ ( for y and z plane)

$I_{y} = I_{x} + I_{z}$ ( for x and z plane)

• The perpendicular axis theorem is applicable for planar body.
• The sum of moment of inertia of a planar body along the perpendicular axis lying its plane is equal to the moment of Inertia of that body about the line passing through the intersection point of the two perpendicular axis and perpendicular to its plane.

I₃ = I₁ + I₂

I + I = I₃

2 I = $\frac{ma^{2} }{6}$

$I = \frac{m a^{2} }{12}$

The moment of inertia of uniform square plate of mass M and side a about an axis passing through center of mass of the square and in the plane of the square is $I = \frac{ma^{2} }{12}$

 

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Answer 2

Answer: The moment of inertia of uniform Square Plate of mass m and side a about an Axis passing through center of mass of the square and in the plane of square is $\frac{ma^2}{12}$

Moment of Inertia: The moment of inertia can be calculated by adding the "sum of the product of mass" for each particle to the "square of its distance from the axis of the rotation." This gives the formula for the moment of inertia.

Explanation:

• From perpendicular axis theorem:

• $dI_x = dmy^2$ where $I_y = \int dmx^2$

• $I_y = \int dmy^2$

• $dI_z = dm[(x^2+ y^2)^\frac{1}{2}]^2\\I_z = \int dmx^2 + \int dmy^2$

The theorem of the perpendicular axis can be applied to bodies that are planar.

The sum of the moments of inertia of a planar body along an axis perpendicular to its plane is equal to the moments of inertia of that body around a line that passes through the point where two perpendicular axes join and is perpendicular to its plane.

I₃ = I₁ + I₂

I + I = I₃

2 I = $\frac{ma^2}{6}$

I = $\frac{ma^2}{12}$

The moment of inertia of uniform Square Plate of mass m and side a about an Axis passing through center of mass of the square and in the plane of square is $\frac{ma^2}{12}$

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