If the linear mass density of a rod of length $L$ varies as $\lambda = kx^3$,determine the position of its centre of mass (where $x$ is the distance from one of its ends and $k$ is a constant):

Four identical particles each of mass $m$ are kept at the four corners of a square of side $a$. If one of the particles is removed,the shift in the position of the centre of mass is

In the figure shown,$ABC$ is a uniform wire. If the center of mass of the wire lies vertically below point $A$,then $\frac{BC}{AB}$ is close to:

Two identical uniform rectangular blocks (with longest side $L$) and a solid sphere of radius $R$ are to be balanced at the edge of a heavy table such that the centre of the sphere remains at the maximum possible horizontal distance from the vertical edge of the table without toppling as indicated in the figure. If the mass of each block is $M$ and of the sphere is $M/2$,then the maximum distance $x$ that can be achieved is

The position vectors of two particles of masses $1 \ kg$ and $3 \ kg$ are $\hat{i} + 2\hat{j} + \hat{k}$ and $-3\hat{i} - 2\hat{j} + \hat{k}$ respectively. The position vector of the center of mass of this system is:

The centre of mass of two thin uniform rods of the same length $L$ but made of different materials and kept as shown in the figure can be,if the meeting point is the origin of coordinates: