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Elements-of-Civil-Engineering--Engineering-Mechanics---CV13-OR-CV23-->View question

State and prove parallel axis theorem

Elements of Civil Engineering and engineering mechanics

Asked On2017-06-14 06:38:18 by:Aparna-Dasgupta

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In physics, the parallel axis theorem, also known as Huygens–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.


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 is
I_{\mathrm {cm} }=\int (x^{2}+y^{2})\,dm.
The moment of inertia relative to the axis z′, which is a perpendicular distance d along the x-axis from the centre of mass, is
{\displaystyle I=\int \left[(x+d)^{2}+y^{2}\right]\,dm}
Expanding the brackets yields
{\displaystyle I=\int (x^{2}+y^{2})\,dm+d^{2}\int dm+2d\int x\,dm.}
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:
I=I_{\mathrm {cm} }+md^{2}.

Answerd on:2017-06-14 Answerd By:tarun101

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