$A$ 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

Four identical spheres each of mass $m$ are placed at the corners of a square of side $2 \, m$. Taking the point of intersection of the diagonals as the origin,the coordinates of the centre of mass are

On a horizontal frictionless frozen lake,a girl of mass $36 \,kg$ and a box of mass $9 \,kg$ are connected to each other by means of a rope. Initially,they are $20 \,m$ apart. The girl exerts a horizontal force on the box,pulling it towards her. How far has the girl travelled when she meets the box?

Four particles of masses $m_1 = 2m$,$m_2 = 4m$,$m_3 = m$,and $m_4$ are placed at the four corners of a square of side $a$. Let the corners be $(0,0)$,$(a,0)$,$(a,a)$,and $(0,a)$ respectively. What should be the value of $m_4$ so that the centre of mass of the system is at the centre of the square,i.e.,at $(\frac{a}{2}, \frac{a}{2})$?

Write the difference between the centre of gravity and the centre of mass of a body.

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.