Consider a system of two particles of masses $m_1$ and $m_2$. If mass $m_1$ is pushed towards the center of mass of the system by a distance $d$,by what distance should the mass $m_2$ be displaced so that the center of mass of the system remains unchanged?

$A$ 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?

The centre of mass of a triangle shown in the figure has coordinates

Find the centre of mass of a uniform $L$-shaped lamina (a thin flat plate) with dimensions as shown. The mass of the lamina is $3 \; kg$.

Two particles of mass $5\, kg$ and $10\, kg$ respectively are attached to the two ends of a rigid rod of length $1\, m$ with negligible mass. The centre of mass of the system from the $5\, kg$ particle is nearly at a distance of $..........\, cm$.

The centre of mass is a point: