Question

From a sphere of radius 1m ,a sphere of radius 0.5m is removed from the edge. The shift in centre of mass is ?

Answer 1

we know, centre of mass of sphere is located at centre of sphere. Let mass of sphere of radius 1m is M and it is located in Cartesian system such that centre of it lies on origin.
so, centre of mass of sphere is (0,0)
now a sphere of radius 0.5m is removed from it, centre of mass of small sphere is located at (0.5, 0) as shown in figure.

then, mass of small sphere , m = M/{4/3π1³} × {4/3π(0.5)³} = M × (0.5)³ = M/8
now, use basic formula to find centre of mass,

$x=\frac{m_1x_1-m_2x_2}{m_2-m_2}$

$x=\frac{M\times0-\frac{M}{8}\times0.5}{M-\frac{M}{8}}$

$x=\frac{-\frac{M}{16}}{\frac{7m}{8}}$

$x=-\frac{1}{14}$

similarly, we can find value of y using $y=\frac{m_1y_1-m_2y_2}{m_1-m_2}$

e.g., y = 0

so, centre of mass of new shape is (-1/14, 0)

Answer 2

Sphere of radius R =1m and

sphere of radius removed from the edge r =0.5m

X shift= r^3÷ R^2+r^2+Rr

=(0.5)^3 ÷ 1^2+(0.5)^2+1*0.5

= 0.125 ÷1+0.25+0.5

=0.125÷1.75

=125/1000*100/175

=5/70. =1/14