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## State and prove parallel axis theorem

Elements of Civil Engineering and engineering mechanicsAsked On2017-06-14 06:38:18 by:Aparna-Dasgupta

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Download question setAnswersIn physics, the parallel axis theorem, also known as Huygensâ€“Steiner theorem, or just as Steiner's theorem, after Christiaan Huygens and Jakob Steiner, can be used to determine the mass moment of inertia or the second moment of area of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's center of gravity and the perpendicular distance between the axes.

Derivation:

We may assume, without loss of generality, that in a Cartesian coordinate system the perpendicular distance between the axes lies along the x-axis and that the centre of mass lies at the origin. The moment of inertia relative to the z-axis isThe moment of inertia relative to the axis zâ€², which is a perpendicular distance d along the x-axis from the centre of mass, is

Expanding the brackets yields

The first term is Icm and the second term becomes md2. The integral in the final term is the x-coordinate of the centre of mass, which is zero by construction. So, the equation becomes:

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