$A$ beaker containing water is placed on the platform of a spring balance. The balance reads $1.5 \, kg$. $A$ stone of mass $0.5 \, kg$ and density $500 \, kg/m^3$ is immersed in water without touching the walls of the beaker. What will be the balance reading now?

$A$ cubical block is floating in a liquid with half of its volume immersed in the liquid. When the whole system accelerates upwards with a net acceleration of $g/3$,the fraction of volume immersed in the liquid will be:

$A$ small metal sphere of density $\varrho$ is dropped from height $h$ into a jar containing liquid of density $\sigma$ $(\sigma > \varrho)$. The maximum depth up to which the sphere sinks is (Neglect damping forces).

$A$ solid sphere of radius $R$ and density $\rho$ is attached to one end of a massless spring of force constant $k$. The other end of the spring is connected to another solid sphere of radius $R$ and density $3\rho$. The complete arrangement is placed in a liquid of density $2\rho$ and is allowed to reach equilibrium. The correct statement$(s)$ is (are):
$(A)$ The net elongation of the spring is $\frac{4 \pi R^3 \rho g}{3 k}$
$(B)$ The net elongation of the spring is $\frac{8 \pi R^3 \rho g}{3 k}$
$(C)$ The light sphere is partially submerged.
$(D)$ The light sphere is completely submerged.

$A$ wide-bottomed cylindrical massless plastic container of height $9 \, cm$ contains $40$ identical coins and is floating on water with $3 \, cm$ of its height submerged. If we start placing additional identical coins on its lid,it is observed that after $N$ coins are added,the equilibrium changes from stable to unstable. Equilibrium in floating is stable if the geometric centre of the submerged portion is above the centre of mass of the object. The value of $N$ is closest to:

An iceberg floats in water with part of it submerged. What is the fraction of the volume of the iceberg submerged,if the density of ice is $\rho_{i} = 0.917 \ g \ cm^{-3}$ and the density of water is $\rho_{w} = 1.00 \ g \ cm^{-3}$?