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Q.

A circular arc (AB) of thin wire frame of radius $R = 2 π$ cm and mass M makes an angle of $90 ∘$ at the origin. Find the y coordinate of the center of mass taking O as the origin in cm .

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Detailed Solution

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$\begin{matrix} dm = \frac{m}{\frac{π R}{2}} (dl) \\ = \frac{2 M}{π R} (Rd θ) \\ = \frac{2 Md θ}{π} \end{matrix}$

$Y_{cm} = = \frac{\int ydm}{\int dm}$

$= \frac{1}{M} \int Rcosθ \frac{2 M}{π} dθ$

$= \frac{2 R}{π} \int_{- π / 4}^{+ π / 4} cosθdθ$

$= \frac{2 \sqrt{2} R}{π} = 4 cm$

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A circular arc (AB) of thin wire frame of radius R= 2 π cm and mass M makes an angle of 90 ∘ at the origin. Find the y coordinate of the center of mass taking O as the origin in cm .