MCQ
A uniform square plate $S$ (side $c$) and a uniform rectangular plate $R$ (sides $b$, $a$) ($a > b$) have identical areas and masses figure. Which of the following is correct

$(a)\ I_{xR}/I_{xS} <1;\  (b)\ I_{yR}/I_{yS}  >1; \  (c)\ I_{zR}/I_{zS} >1$

  • A
    $a$,$b$
  • B
    $b$,$c$
  • $a$,$b$,$c$
  • D
    None

Answer

Correct option: C.
$a$,$b$,$c$
c
$a b=c^{2}$           $a>c>b$

$\mathrm{I}_{\mathrm{XR}}=\frac{\mathrm{Mb}^{2}}{12}, \mathrm{I}_{\mathrm{YR}}=\frac{\mathrm{Ma}^{2}}{12}, \mathrm{I}_{\mathrm{ZR}}=\frac{\mathrm{M}\left(\mathrm{a}^{2}+\mathrm{b}^{2}\right)}{12}$

$\mathrm{I}_{\mathrm{XS}}=\frac{\mathrm{Mc}^{2}}{12}, \mathrm{I}_{\mathrm{YS}}=\frac{\mathrm{Mc}^{2}}{12}, \mathrm{I}_{\mathrm{zs}}=\frac{\mathrm{Mc}^{2}}{6}$

$\frac{\mathrm{I}_{\mathrm{ZR}}}{\mathrm{I}_{\mathrm{ZS}}}=\frac{1}{2}\left(\left(\frac{\mathrm{a}}{\mathrm{c}}\right)^{2}+\left(\frac{\mathrm{b}}{\mathrm{c}}\right)^{2}\right)$

$=\frac{1}{2}\left(\left(\frac{\mathrm{a}}{\mathrm{c}}\right)^{2}+\left(\frac{\mathrm{c}}{\mathrm{a}}\right)^{2}\right)$

$\frac{I_{\mathbb{Z R}}}{I_{Z S}}>1$

$x+\frac{1}{x}>2$ if $x>0$

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