Introduction:
When a rectangular block of mass m and area of cross section A is submerged in a liquid of density rho, it experiences a buoyant force equal to the weight of the liquid displaced by the block. If the block is given a small vertical displacement from equilibrium, it will undergo oscillation with a time period T.

Explanation:
Buoyant Force:
When the block is submerged in the liquid, it displaces a volume of liquid equal to its own volume. This displaced liquid exerts an upward force on the block known as the buoyant force. The buoyant force can be calculated using Archimedes' principle, which states that the buoyant force is equal to the weight of the liquid displaced by the object.

Equilibrium and Displacement:
When the block is in equilibrium, the weight of the block is equal to the buoyant force. However, when the block is given a small vertical displacement from equilibrium, the forces acting on the block are no longer balanced. The block experiences a net force that causes it to move back towards its equilibrium position.

Harmonic Oscillation:
The motion of the block can be described as simple harmonic motion (SHM) because the restoring force acting on the block is proportional to its displacement from equilibrium and is directed towards the equilibrium position. This restoring force is provided by the buoyant force.

Time Period:
The time period of the oscillation, T, can be calculated using the formula T = 2π√(m/k), where m is the mass of the block and k is the effective spring constant. In this case, the effective spring constant is given by k = ρgA, where ρ is the density of the liquid and g is the acceleration due to gravity.

Conclusion:
When a rectangular block of mass m and area of cross section A is submerged in a liquid of density rho, it undergoes oscillation with a time period T when given a small vertical displacement from equilibrium. This oscillation is due to the buoyant force acting as a restoring force, causing the block to move back towards its equilibrium position. The time period of the oscillation can be calculated using the formula T = 2π√(m/k), where k is the effective spring constant given by k = ρgA.