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Question
Answer in brief:
State the conditions under which the theorems of parallel axes and perpendicular axes are applicable. State the respective mathematical expressions.
Solution
The theorem of the parallel axis is applicable to any object of any shape. The moment of inertia (IO) of an object about any axis is the sum of its moment of inertia (IC) about an axis parallel to the given axis, and passing through the centre of mass and the product of the mass of the object and the square of the distance between the two axes (M.h2 ). Therefore, we can write,
`I_O = I_(CM) + M.h^2`
This is the mathematical form of the theorem of parallel axes.
The theorem of perpendicular axes relates the moment of inertias of a laminar object about three mutually perpendicular and concurrent axes, two of them in the plane of the object and the third perpendicular to the object. If Ix , Iy and Iz are the respective moments of inertia of the body about x, y and z axes. The moment of inertia (Iz) of a laminar object about an axis (z) perpendicular to its plane is the sum of its moments of inertia about two mutually perpendicular axes (x and y) in its plane, all the three axes being concurrent. Therefore, we can write,
`I_z = I_x + I_y`
This is the mathematical form of the theorem of perpendicular axes.
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