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प्रश्न
State and prove the theorem of the parallel axis about the moment of inertia.
State and prove the theorem of parallel axes.
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उत्तर
A body's moment of inertia along an axis is equal to the product of two things: Its moment of inertia about a parallel axis through its centre of mass and the product of the body's mass and the square of the distance between the two axes. This is known as the parallel axis theorem.
Proof: Let ICM represent a body of mass M moment of inertia (MI) about an axis passing through its centre of mass C and let I stand for that body's MI about a parallel axis passing through any point O. Let h represent the separation of the two axis.
Think about the body's minuscule mass element dm at point P. It is perpendicular to the rotation axis through point C and to the parallel axis through point O, with a corresponding perpendicular distance of OP. CP2 dm is the MI of the element about the axis through C. As a result, `I_(CM) = int CP^2 dm` is the body's MI about the axis through the CM. In a similar vein, `I = int OP^2 dm` is the body's MI about the parallel axis through O.
Draw PQ perpendicular to OC produced, as shown in the figure. Then, from the figure,
`I = int OP^2 dm`
= `int (OQ^2 + PQ^2) dm`
= `int [(OC + CQ)^2 + PQ^2] dm`
= `int (OC^2 + 2OC.CQ + CQ^2 + PQ^2) dm`
= `int (OC^2 + 2OC.CQ + CP^2)dm` ...(∵ CQ2 + PQ2 = CP2)
= `int OC^2 dm + int 2OC.CQ dm + int CP^2 dm`
= `OC^2 int dm + 2OC int CQ dm + int CP^2 dm`
Since OC = h is constant and `int dm = M` is the mass of the body,
`I = Mh^2 + 2h int CQ dm + I_(CM)`
The integral `int CQ dm` now yields mass M times a coordinate of the CM with respect to the origin C, based on the concept of the centre of mass. This position and the integral are both zero because C is the CM in and of itself.
∴ I = ICM + Mh2
This proves the theorem of the parallel axis.
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संबंधित प्रश्न
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