Three identical metal balls,each of radius $r$,are placed touching each other on a horizontal surface such that an equilateral triangle is formed when the centres of the three balls are joined. The centre of mass of the system is located at:

The integral $\int \vec{r} dm = 0$ for a homogeneous body is defined for which point?

Two particles of masses $m_1$ and $m_2$ are separated by a distance $d$. The shift in the centre of mass when the two particles are interchanged is:

The centre of mass of three particles of masses $1 \ kg, 2 \ kg$ and $3 \ kg$ is at $(2, 2, 2)$. The position of the fourth mass of $4 \ kg$ to be placed in the system so that the new centre of mass is at $(0, 0, 0)$ is:

Mention the position of the centre of mass of particles of equal mass.

Look at the figure drawn with ink of uniform linear density. $A$ mass $m$ of ink is used to draw each of the two inner circles and each of the two line segments. $A$ mass $6m$ of ink is used to draw the outer circle. The coordinates of the centers of the different parts are: outer circle $(0, 0)$,left inner circle $(-a, a)$,right inner circle $(a, a)$,and the horizontal line segment $(0, -a)$. Find the $y$-coordinate of the center of mass of the ink in the figure.