Fluid Mechanics - Mechanics - Physics Previous Year Questions

A liquid of density $\rho$ in a container weighs $W$. A cubic block of side $L$ and density $\rho_b<\rho$ is pushed by a stick to completely submerge the block in the liquid without touching the bottom. If $g$ is the acceleration due to the gravity and liquid displaced by the stick is negligible, what is the new weight of the container as registered by the weighing machine below?

Two cubes $A$ and $B$ of same dimensions are made of different materials with densities $\rho_A$ and $\rho_{\mathrm{B}}$, respectively. Cube A floats in water with a fraction $\eta$ of its volume immersed. When cube $B$ is placed on top of cube $A$, it is found that cube $A$ is just entirely immersed, while cube $B$ is entirely above the surface of the water. What is the ratio $\rho_{\mathrm{B}} / \rho_{\mathrm{A}}$ ?

A solid bob of a material having density twice that of water is suspended with a massless and inextensible string of length $L$. The whole set-up is placed inside a water-filled tank. The bob is imparted a horizontal velocity $V_0$ at the lowest point A , while the other end of the string is fixed, such that the bob completes a semi-circular trajectory in the vertical plane. The string becomes slack only when the bob reaches the topmost point C. Assume that the effects of viscosity and water currents are negligible. The acceleration due to gravity is $g$. What is the expression for $V_0$ ?