Question

If $27$ small spheres each of $M.I.$ $I$ about their diametrical axis are combined to form a big sphere,then the $M.I$ of the big sphere about its diameter is:

• A. $243I$
• B. $81I$
• C. $27I$
• D. $9I$

Answer 1

The correct option is A $243I$

REF.Image

Solution

Let the mass and radius of spheres be

M and R resp. Then,

After joining total volume and mass will

be remain unconserved. So,

$M^1=27M$

$\frac{4\pi }{3}(R^{\prime }{)}^3=(27\left)\right[\frac{4\pi }{3}{R}^3]$

$\Rightarrow R^{\prime }(27{)}^{1/3}R\Rightarrow R^{\prime }=3R$

MI of each small spheres $=\frac{2}{5}M{R}^2=I$

$\Rightarrow M{R}^2=\frac{5}{2}I$

MI of bigger sphere = $\frac{2}{5}{M}^{\prime }{R}^{\prime 2}=\frac{2}{5}\left(27M\right)(3R{)}^2$

$=\left(\frac{2}{5}\right)(27\times 9)M{R}^2=243I$

Option - A is correct.