Mass is distributed uniformly over a thin rectangular plate and positions of two vertices are given by $(1, 3)$ and $(2, -4)$. What is the position of $3^{rd}$ vertex if centre of mass of the plate lies at the origin ?

A rigid body can be hinged about any point on the $x$ -axis. When it is hinged such that the hinge is at $x$, the moment of inertia is given by $I = 2x^2 - 12x + 27$ The $x$ -coordinate of centre of mass is

The distance between the vertex and the centre of mass of a uniform solid planar circular segment of angular size $\theta$ and radius $R$ is given by

Three point particles of masses $1.0\; \mathrm{kg} .1 .5 \;\mathrm{kg}$ and $2.5\; kg$ are placed at three comers of a right angle triangle of sides $4.0\; \mathrm{cm}, 3.0 \;\mathrm{cm}$ and $5.0\; \mathrm{cm}$ as shown in the figure. The center of mass of the system is at a point

The linear mass density $(\lambda)$ of a rod of length $L$ kept along $x$-axis varies as $\lambda=\alpha+\beta x$; where $\alpha$ and $\beta$ are positive constants. The centre of mass of the rod is at ..........

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 point of intersection of mutually perpendicular sides as origin, the magnitude of position vector of centre of mass of the system will be $\sqrt{x} m$. The value of $x$ is