A wide bottom cylindrical massless plastic container of height $9 \,cm$ has $40$ identical coins inside it and is floating on water with $3 \,cm$ inside the water. If we start putting more of such coins on its lid, it is observed that after $N$ coins are put, its equilibrium changes from stable to unstable. Equilibrium in floating is stable if the geometric centre of the submerged portion is above the centre of the mass of the object). The value of $N$ is closed to

A block of volume $V$ and of density $\sigma _b$ is placed in liquid of density $\sigma _l(\sigma _l > \sigma _b)$ , then block is moved upward upto a height $h$ and it is still in liquid. The increase in gravitational potential energy of the system is

The spring balance $A$ reads $2$ $kg$ with a block $m $ suspended from it. $A$ balance $B$ reads $5$ $kg$ when a beaker with liquid is put on the pan of the balance. The two balances are now so arranged that the hanging mass is inside the liquid in the beaker as shown in the figure in this situation:

An ice block contains a glass ball when the ice melts within the water containing vessel, the level of water

A uniform rod of density $\rho $ is placed in a wide tank containing a liquid of density ${\rho _0}({\rho _0} > \rho )$. The depth of liquid in the tank is half the length of the rod. The rod is in equilibrium, with its lower end resting on the bottom of the tank. In this position the rod makes an angle $\theta $ with the horizontal