Question

A sphere of styrofoam has radius $R$. A cavity of radius $R/2$ having center at a distance $R/2$ directly above the center of the sphere is hollowed out and filled with a solid material of density five times the density of styrofoam (Figure). Where is the center of mass of the new sphere?

• A. $\frac{R}{3}$
• B. $\frac{R}{4}$
• C. $\frac{R}{6}$
• D. $\frac{3R}{16}$

Answer 1

Answer: (C)

$\frac{R}{6}$

Case I

When cavity of radius $R/2$ removed, center of mass of remaining sphere.

Mass of cavity removed,

Let $f\to$ initial density of sphere.

$R\to$ Radius of sphere

$M\to$ mass of full sphere.

Mass of cavity removed $m^{\prime }$

$m^{\prime }=\rho v^{\prime }$

$m^{\prime }=f\left[\frac{4}{3}\pi {\left(\frac{R}{2}\right)}^3\right]$

$m^{\prime }=\frac{1}{8}(f.\frac{4}{3}\pi R^3)$

$m^{\prime }=\frac{M}{8}$ ___(1)

Mass of remaining part $=M-\frac{M}{8}=\frac{7M}{8}$ ___(2)

Since before removing, center of mass is at $(0,0)$

$y-$ component of center of mass,

$y_{cm}=\frac{\left(\frac{M}{8}\right)\left(\frac{R}{2}\right)+\left(\frac{7M}{8}\right)\left(y\right)}{\frac{M}{8}+\frac{7M}{8}}=0$

Centre of mass of remaining solid,

$y=-\frac{R}{14}$ ___(3)

Case II

It is filled with material of density $5\rho$

new mass, $m"=\left(5f\right)\left[\frac{4}{3}\pi {\left(\frac{R}{2}\right)}^3\right]$

$=\frac{5}{8}(f.\frac{4}{3}\pi R^3)$

$m"=\frac{5M}{8}$ ___(4)

y- component pf center of mass,

$y_{cm}=\frac{\frac{5M}{8}\left(\frac{R}{2}\right)+\left(\frac{7M}{8}\right)\left(\frac{-R}{14}\right)}{\frac{5M}{8}+\frac{7M}{8}}$

$y_{cm}=\frac{R}{6}$