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Demonstrate that in the case of a thin plate of arbitrary shape there is the following relationship between the moments of inertia: I1 + I2= I3, where subindices 1, 2, and 3 define three mutually perpendicular axes passing through one point, with axes 1 and 2 lying in the plane of the plate. Using this relationship, find the moment of inertia of a thin uniform round disc of radius R and mass m relative to the axis coinciding with one of its diameters.

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(a) Let us consider a lamina of an arbitrary shape and indicate by 1,2 and 3, three axes coinciding with x, y and z-axes and the plane of lamina as x - y plane.

(b) Let us take the plane of the disc as x - y plane and origin to the centre of the disc (Fig.) From the symmetry Ix = Iy . Let us consider a ring element of radius r and thickness dr, then the moment of inertia of the ring element about the y-axis. 

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