A piece of wood of mass $0.03\, kg$ is dropped from the top of a $100\, m$ height building. At the same time, a bullet of mass $0.02\, kg$ is fired vertically upward, with a velocity $100\, ms^{- 1}$, from the ground. The bullet gets embedded in the wood. Then the maximum height to which the combined system reaches above the top of the building before falling below is ........ $m$. $(g = 10\, ms^{-2})$

Find the centre of mass of a uniform :

$(a)$ half-disc,$(b)$ quarter-disc. 

The centre of mass of system of particles does not depend on

The centre of mass of a solid hemisphere of radius $8\, cm$ is $X \,cm$ from the centre of the flat surface. Then value of $x$ is$......$

$A$ man weighing $80\, kg$ is standing at the centre of a flat boat and he is $20\, m$ from the shore. He walks $8\, m$ on the boat towards the shore and then halts. The boat weight $200\, kg$. ........ $m$ far is he from the shore at the end of this time.

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