Find the centre of mass of a uniform plate having semicircular inner and outer boundaries of radii R-1 and R-2 -figure 9 - E5- -None of these-2-3 -R-2-2 - R-1-2-2R-1 R-2-R-1 - R-2 above the centre-4-R-2-2 - R-1-2-R-1 R-2-R-1 - R-2 above the centre-4-3 -R-2-2 - R-1-2-R-1 R-2-R-1 - R-2 above the centre

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Standard IX

Physics

A Half Disc

Find the cent...

Question

Find the centre of mass of a uniform plate having semicircular inner and outer boundaries of radii $R_{1}$ and $R_{2}$ (figure 9 - E5)

= \frac{4}{π} \frac{(R_{2}^{2} + R_{1}^{2} + R_{1} R_{2})}{R_{1} + R_{2}}

above the centre

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= \frac{4}{3 π} \frac{(R_{2}^{2} + R_{1}^{2} + R_{1} R_{2})}{R_{1} + R_{2}}

above the centre

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= \frac{2}{3 π} \frac{(R_{2}^{2} + R_{1}^{2} + 2 R_{1} R_{2})}{R_{1} + R_{2}}

above the centre

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None of these

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Solution

The correct option is B $= \frac{4}{3 π} \frac{(R_{2}^{2} + R_{1}^{2} + R_{1} R_{2})}{R_{1} + R_{2}}$ above the centre
The centre of mass of the plate will be on the symmetrical axis.
$\Rightarrow ¯ Y c m = \frac{(\frac{π R_{2}^{2}}{2}) (\frac{4 R_{2}}{3 π}) - (\frac{π R_{1}^{2}}{2}) (\frac{4 R_{1}}{3 π})}{\frac{π R_{2}^{2}}{2} - \frac{π R_{1}^{2}}{2}}$
$= \frac{(\frac{2}{3}) R_{2}^{3} - (\frac{2}{3}) R - 1^{3}}{\frac{π}{2} 2 (r_{2}^{2} - R_{1}^{2})} = \frac{2}{3 π} \frac{(R_{2} - R_{1}) (R_{2}^{2} + R_{1}^{2} + R_{1} R_{2})}{(R_{2} - R_{1}) (R_{2} + R_{1})}$
$= \frac{2}{3 π} \frac{(R_{2}^{2} + R_{1}^{2} + R_{1} R_{2})}{R_{1} + R_{2}}$ above the centre.

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Find the centre of mass of a uniform plate having semicircular inner and outer boundaries of radii $R_{1}$ and $R_{2}$ (figure 9 - E5)

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Find the centre of mass of a uniform plate having semicircular inner and outer boundaries of radii $R_{1}$ and $R_{2}$ (figure 9 - E5)