$A$ uniform thin rod $AB$ of length $L$ has linear mass density $\mu(x) = a + \frac{bx}{L}$,where $x$ is measured from $A$. If the $CM$ of the rod lies at a distance of $\frac{7}{12}L$ from $A$,then $a$ and $b$ are related as

$A$ uniform rod of length $L$ is placed along the $x$-axis with one end at the origin. If the linear mass density of the rod varies as $\lambda(x) = ax$,where $a$ is a constant,find the position of the centre of mass of the rod.

$A$ metre stick of mass $400 \ g$ is pivoted at one end and displaced through an angle $60^o$. The increase in its potential energy is ..... $J$.

What is the torque acting on the center of mass of an object due to the gravitational force?

Which of the points is the likely position of the centre of mass of the system shown in the figure?

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.