Consider the following statements regarding the Centre of Mass $(CM)$ of objects of radius $R$ from their geometric centre:
$[1]$ $CM$ of a uniform semicircular disc is at $2R/\pi$.
$[2]$ $CM$ of a uniform semicircular ring is at $4R/3\pi$.
$[3]$ $CM$ of a solid hemisphere is at $4R/3\pi$.
$[4]$ $CM$ of a hemispherical shell is at $R/2$.
Which of these statements are correct?

Obtain the general expression for the centre of mass of a system of $n$ distributed particles in three dimensions.

Infinite rods of uniform mass density and lengths $L, L/2, L/4, \dots$ are placed one upon another up to infinity as shown in the figure. Find the $x-$ coordinate of the centre of mass.

An object comprises a uniform ring of radius $R$ and its uniform chord $AB$ (not necessarily made of the same material) as shown. Which of the following can not be the centre of mass of the object?

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.

Three point masses $m_1 = 1.6 \, kg$,$m_2 = 2.0 \, kg$,and $m_3 = 2.4 \, kg$ are placed at the corners of a thin massless rectangular sheet $(1.2 \, m \times 1.0 \, m)$ as shown. The center of mass will be located at the point ........... $m$.