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Asked by mufeedatvp2000 | 18 Apr, 2020, 02:22: PM
Let us find the centre of mass of semi circular ring of radius R .
From the given figure above, if we consider origin of X-Y coordinate system is at center of semi-circular ring,
then centre of mass is in y-axis by symmetry, i.e. x-coordinate Cx of centre of mass is zero.

y-coordinate Cy of centre of mass is given by

........................(1)
where M is mass of semi-circular ring, dm is mass of a small element of length dl .

Let radial line connecting the element dl makes angle θ with x-axis and dθ be angle subtended  by element dl at centre.

ρ is mass per unit length of wire.  Mass M of wire is obtained as M = ρ ( π R )

Using the substitution  ( ρ/M ) = 1/( π R ) ,  equation (1) for y-coordinate of centre of mass becomes

Cy = 2R/π

Moment of inertia I about centre of mass is given by

we get from above integration,
Answered by Thiyagarajan K | 19 Apr, 2020, 07:54: AM

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