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Derive an expression for the moment of inertia of a thin spherical shell about a diameter. |
Answer» Solution :Consider a solid sphere. Suppose a smaller concentric sphere is removed from the solid sphere we get a hollow or a thick spherical shell (See Fig. 7.2.58). Let Mbe the mass, R the external radius and r the internal radius of the hollow sphere. Volume of the shell `=4/3 pi(R^(3)-r^(3))`  Mass PER unit volume of the shell `=rho` `rho = M/((4/3)pi(R^(3)-r^(3)))` `=(3M)/(4PI(R^(3)-r^(3))` M.I. of the solid sphere of radius R about a diameter `=2/3 XX "mass" xx ("radius")^(2)` `=2/5 xx 4/3pi R^(3) rho R^(2)` `=8/15 pi R^(5) rho` M.I. of the solid sphere of radius r about a diameter `=8/15 pi r^(5) rho` M.I. of the hollow sphere about the diameter = `L` `=8/15 piR^(5) - 8/15 pi r^(5) rho = 8/15 pi rho (R^(5)-r^(5))` `l=2/5 M (R^(5)-r^(5))/(R^(3)-r^(3))` |
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