A cubical block of wood of specific gravity $0.5$ and a chunk of concrete of specific gravity $2.5$ are fastened together. The ratio of the mass of wood to the mass of concrete which makes the combination to float with its entire volume submerged under water is

A solid sphere of density $\eta$ $( > 1)$  times lighter than water is suspended in a water tank by a string tied to its base as shown in fig. If the mass of the sphere is m then the tension in the string is given by

A dumbbell is placed in water of density $\rho$ . It is observed that by attaching a mass $m$ to the rod, the dumbbell floats with the rod horizontal on the surface of water and each sphere exactly half submerged as shown in the figure. The volume of the mass $m$ is negligible. The value of length $l$ is

A rectangular box has water in it. It is being pulled to the right with an acceleration $a$. Which of the following options shows the correct shape of water surface on it ?

A cube of edge length $10 \,cm$ is just balanced at the interface of two liquids $A$ and $B$ as shown in figure. If $A$ and $B$ has specific gravity $0.6$ and $0.4$ respectively, then mass of cube is ................ $g$

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