Two solid spheres $A$ and $B$ of equal volumes but of different densities $d_A$ and $d_B$ are connected by a string. They are fully immersed in a fluid of density $d_F$. They get arranged into an equilibrium state as shown in the figure with a tension in the string. The arrangement is possible only if

$(A)$ $d_Ad_F$  $(B)$ $d_B > d_F$ $(C)$ $d_A>d_F$ $(D)$ $d_A+d_B=2 d_F$

A solid sphere of specific gravity $27$ has a concentric spherical cavity and it just sinks in water. The ratio of cavity radius to that of outer radius of sphere is

Water flows out of the hole on the side of a bucket and follows a parabolic path. If the bucket falls freely under gravity, ignoring air resistance, the water flow

A cubical block of side $0.5\,m$ floats on water with $30\%$ of its volume under water. ……. $kg$ is the maximum weight that can be put on the block without fully submerging it under water ? [Take density of water $= 10^3\,kg/m^3$ ]

A cork is submerged in water by a spring attached to the bottom of a bowl. When the bowl is kept in an elevator moving with acceleration downwards, the length of spring