Question

Four ifour identical spheres each of mass m and radius 10 centimetre each are placed on a horizontal surface touching one another so that their centres are located at the corners of square of a side 20 centimetre then the distance of their centre of mass from centre of any of the sphere is ___cm

Answer 1

$\Large\boxed{\sf{\quad10\sqrt2\ cm\quad}}$

Let a coordinate system be introduced with the four spheres as shown in the figure.

The centers of each spheres are located at points (0, 0), (20, 0), (20, 20) and (0, 20). The point (x, y) is assumed as the center of mass.

The x coordinate of the center of mass is,

$\longrightarrow\sf{x=\dfrac{m(0)+m(20)+m(20)+m(0)}{m+m+m+m}}$

$\longrightarrow\sf{x=\dfrac{0+20m+20m+0}{4m}}$

$\longrightarrow\sf{x=10}$

And y coordinate of the center of mass is,

$\longrightarrow\sf{y=\dfrac{m(0)+m(0)+m(20)+m(20)}{m+m+m+m}}$

$\longrightarrow\sf{y=\dfrac{0+0+20m+20m}{4m}}$

$\longrightarrow\sf{y=10}$

So the center of mass is located at the point (10, 10). Hence its distance from center of any of the sphere is,

$\longrightarrow\sf{d=\sqrt{x^2+y^2}}$

$\longrightarrow\sf{d=\sqrt{10^2+10^2}}$

$\longrightarrow\sf{\underline{\underline{d=10\sqrt2\ cm}}}$