State Perpendicular Axis Theorum & Paralell Axis Theorum Respectively?

Theorem of Perpendicular Axis

The moment of inertia of planar body about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent with perpendicular axis and lying in the plane of the body.

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Theorem of Parallel axes

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The moment of inertia of a body about any axis is equal to the sum of the moments of inertia of the body about a parallel axis passing through its centre of mass and the product of its mass and the square of distance between the two parallel axes.

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Statement:
The moment of inertia about Z-axis can be represented as:


Where
Icmis the moment of inertia of an object about its centre of mass
m is the mass of an object
r is the perpendicular distance between the two axes.

Proof
Assume that the perpendicular distance between the axes lies along the x-axis and the centre of mass lies at the origin. The moment of inertia relative to z-axis that passes through the centre of mass, is represented as 


Moment of inertia relative to the new axis with its perpendicular distance r along the x-axis, is represented as:


We get,


The first term is Icm,the second term is mr2and the final term is zero as the origin lies at the centre of mass. Finally, 
 this the required parallel axis theorem                                                 perpendicular axis theorem (or plane figure theorem) can be used to determine the moment of inertia of a rigid object that lies entirely within a plane, about an axis perpendicular to the plane, given the moments of inertia of the object about twoperpendicular axes lying within the plane. The axes must all pass through a single point in the plane.

Define perpendicular axes x,y,, and z, (which meet at origin O,) so that the body lies in the xy, plane, and the z, axis is perpendicular to the plane of the body. Let IxIy and Iz be moments of inertia about axis x, y, z respectively, the perpendicular axis theorem states that[1]

I_z = I_x + I_y,

This rule can be applied with the parallel axis theorem and the stretch rule to find moments of inertia for a variety of shapes.

If a planar object (or prism, by the stretch rule) has rotational symmetry such that I_x, and I_y, are equal, then the perpendicular axes theorem provides the useful relationship:

I_z = 2I_x = 2I_y,

[edit]Derivation

Working in Cartesian co-ordinates, the moment of inertia of the planar body about the z, axis is given by[2]:

I_{z} = int left(x^2 + y^2right), dm = int x^2,dm + int y^2,dm = I_{y} + I_{x}

On the plane, z=0,, so these two terms are the moments of inertia about the x, and y, axes respectively, giving the perpendicular axis theorem. The converse of this theorem is also derived similarly.

Note that int x^2,dm  = I_{y} ne I_{x}  because in int r^2,dm  , r measures the distance from the axis of rotation, so for a y-axis rotation, deviation distance from the axis of rotation of a point is equal to its x co-ordinate.

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