A thin uniform wire is bent to form the two equal sides $AB$ and $AC$ of triangle $ABC$, where $AB = AC = 5\,cm.$ The third side $BC$, of length $6\,cm,$ is made from  uniform wire of twice the density of the first. The distance of centre of mass from $A$ is

A circular disc of radius $R$ is removed from a bigger uniform circular disc of radius $2R$ such that the circumferences of the discs coincide. The centre of mass of the new  disc is $\alpha R$ from the centre of the bigger disc. The value of $\alpha$ is

The disc of mass $M$ with uniform surface mass density $\sigma$ is shown in the figure. The centre of mass of the quarter disc (the shaded area) is at the position $\frac{x}{3} \frac{a}{\pi}, \frac{x}{3} \frac{a}{\pi}$ where $x$ is ....... .

(Round off to the Nearest Integer) $[ a$ is an area as shown in the figure $]$

From a uniform disk of radius $R$, a circular hole of radius $R/2$ is cut out. The centre of the hole is at $R/2$ from the centre of the original disc. Locate the centre of gravity of the resulting flat body.

On a horizontal frictionless frozen lake, a girl $36 \,kg$ and a box $9 \,kg$ are connected to each other by means of a rope. Initially, they are $20 \,m$ apart. The girl exerts a horizontal force on the box, pulling it towards her. How far has the girl travelled when she meets the box?

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