A piece of paper Fig. (a) is in form of a square. Now an arrow sign is made by cutting and rejoining a quarter part of the square of side '$L$ ' as shown. If $O$ is taken to be origin $(0,0),$ the centre of mass of Fig. (b) will be at

Two planks $A$ and $B$ are sliding in the same direction on smooth horizontal surface such that $A$ is faster than $B.$ Packets of equal weight are transferred from $A$ to $B.$ Due tò this: 

An object comprises of a uniform ring of radius $R$ and its uniform chord $A B$ (not necessarily made of the same material) as shown. Which of the following can not be the centre of mass of the object?

A machinist starts with three identical square plates but cuts one corner from first, two corners from the second and three corners from the third. Rank the three according to the $x$ -coordinate of their centre of mass, from smallest to largest.

An inverted$T$ -shaped object is placed on a smooth horizontal floor as shown in figure.

A force $F$ is applied on the system as shown in figure. The value of $x$ so that the system performs pure translational motion is

A circular plate of uniform thickness has a diameter of $28 \mathrm{cm}.$ A circular portion of diameter $21 \mathrm{cm}$ is removed from the plate as shown. $O$ is the centre of mass of complete plate. The position of centre of mass of remaining portion will shift towards left from ' $O$ ' by

Figure shows a thin uniform $rod50 \mathrm{cm}$ long and has a mass of $100 \mathrm{g}.$ A hollow metal ball is filled with air and has a diameter $10 \mathrm{cm}$ and total mass $50 \mathrm{g}$ is fixed to one end of the rod. At what point along its length will the ball and rod balance horizontally?

Three particles of equal masses are placed at the corners of an equilateral triangle as shown in the figure. Now particle $A$ starts with a velocity $v_1$ towards line $A B,$ particle $B$ starts with a velocity $v_2$ towards line $B C$ and particle $C$ starts with velocity $v_3$ towards line $C A.$The displacement of $\mathrm{CM}$ of three particle $A, B$ and $C$ after time $t$ will be $($given if $\left.v_1=v_2=v_3\right)$