Q. 104.0( 16 Votes )

# A. Find the moment of inertia of a sphere about a tangent to the sphere, given the moment of inertia of the sphere about any of its diameters to be, where M is the mass of the sphere and R is the radius of the sphere.

B. Given the moment of inertia of a disc of mass M and radius R about any of its diameters to be, find its moment of inertia about an axis normal to the disc and passing through a point on its edge.

Answer :

(A)The moment of inertia (M.I) of sphere about its diameter = kg m^{4}

According to the parallel axis theorem, the moment of inertia of a body about an axis is equal to the sum of the moment of inertia of the about a parallel axis passing through its center of mass and product of square of distance between axis of its mass.

The M.I about tangent of the sphere

kg m^{4}

(B)The moment of inertia of a disc about its diameter = kg m^{4}

According to the perpendicular axis theorem, the moment of inertia of a planar body (lamina) about an axis perpendicular to its plane is equal to the sum of its moments of inertia about its planar axis in which the body lies.

The M.I of the disc about its center:-

The situation can be shown as,

Applying the parallel axis theorem,

The moment of inertia about an axis normal to the disc and passing through a point on its edg

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A. Prove the theorem of perpendicular axes.

(Hint: Square of the distance of a point (x, y) in the x–y plane from an axis through the origin and perpendicular to the plane is x^{2} + y^{2}).

B. Prove the theorem of parallel axes.

(Hint: If the centre of mass is chosen to be the origin ).

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