The variation of density of a solid cylindrical rod of cross-sectional area $\alpha$ and length $L$ is given by $\rho = \rho_0 \frac{x^2}{L^2}$,where $x$ is the distance from one end of the rod. The position of its centre of mass from that end $(x=0)$ is:

Two bodies of mass $1\,kg$ and $3\,kg$ have position vectors $\hat{i}+2\hat{j}+\hat{k}$ and $-3\hat{i}-2\hat{j}+\hat{k}$ respectively. The magnitude of the position vector of the centre of mass of this system will be equal to the magnitude of which of the following vectors?

Three identical spheres each of mass $M$ are placed at the corners of a right-angled triangle with mutually perpendicular sides equal to $3\,m$ each. Taking the point of intersection of mutually perpendicular sides as the origin,the magnitude of the position vector of the centre of mass of the system will be $\sqrt{x}\,m$. The value of $x$ is

Four masses are arranged along a circle of radius $1 \ m$ as shown in the figure. The center of mass of this system of masses is at

$A$ 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.

Two uniform plates of the same thickness and area but of different materials,one shaped like an isosceles triangle and the other shaped like a rectangle,are joined together to form a composite body as shown in the figure. If the centre of mass of the composite body is located at the mid-point of their common side,then the ratio between the masses of the triangle to that of the rectangle is: