The distance of centre of mass from end $A$ of a one dimensional rod $( AB )$ having mass density $\rho=\rho_{0}\left(1-\frac{ x ^{2}}{ L ^{2}}\right)\,kg / m$ and length $L$ (in meter) is $\frac{3 L }{\alpha} m$. The value of $\alpha$ is $\ldots \ldots \ldots . .$ (where $x$ is the distance form end $A$ )

The centre of gravity of a body on the earth coincides with its centre of mass for a small object whereas for an extended object it may not. What is the qualitative meaning of small and extended in this regard ? For which of the following two coincides ? A building, a pond, a lake, a mountain ? 

Three point masses $m_1, m_2$ and $m_3$ are placed at the corners of a thin massless rectangular sheet ($1.2 \,m \times$ $1.0 \,m$ ) as shown. Centre of mass will be located at the point ........... $m$

Figure shows a composite system of two uniform rods of lengths as indicated. Then the coordinates of the centre of mass of the system of rods are ...........

A circular plate of diameter ' $a$ ' is kept in contact with a square plate of side $a$ as shown. The density of the material and the thickness are same everywhere. The centre of mass of composite system will be ...........

A uniform circular disc of radius $a$ is taken. A circular portion of radius $b$ has been removed from it as shown in the figure. If the centre of hole is at a distance $c$ from the centre of the disc, the distance $x_2$ of the centre of mass of the remaining part from the initial centre of mass $O$ is given by