Masses of $8 \ kg, 2 \ kg, 4 \ kg,$ and $2 \ kg$ are placed at the corners $A, B, C,$ and $D$ respectively of a square $ABCD$ with a diagonal of $80 \ cm$. The distance of the center of mass from $A$ is ....... $cm$.

$A$ rod of length $5L$ is bent at a right angle,keeping one side length as $2L$. Find the position of the center of mass of the system. (Consider $L = 10 \ cm$)

$A$ thin rod of length $L$ is placed along the $X$-axis such that one of its ends is at $x = 0$ and the other is at $x = L$. Its linear mass density (mass/length) varies with $x$ as $k(x/L)^n$,where $n$ is any positive number or zero. If the graph of the center of mass $x_{cm}$ versus $n$ is plotted for the rod,which of the following graphs is correct?

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 ........... $m$.

Three masses of $2\,kg$,$4\,kg$,and $4\,kg$ are placed at the three points $(1, 0, 0)$,$(1, 1, 0)$,and $(0, 1, 0)$ respectively. The position vector of its center of mass is:

For a body,the center of volume is defined as $\frac{\int \vec{r} \, dV}{\int dV}$ over the complete body,where $dV$ is a small volume element of the body and $\vec{r}$ is the position vector of that small volume from the origin. Which of the following statements is correct?