Hydrostatics

Step 1: Boyle’s law.
PV=constant
Step 2: Pressure difference.
Pressure decreases upward.

Step 1: Archimedes' principle.
An ice block floats in the liquid because the weight of the ice is equal to the buoyant force exerted by the displaced liquid. Since the density of the ice is less than water, the volume displaced is more than the volume of the ice.

Step 2: After melting.
When the ice melts, the volume of liquid displaced decreases, leading to a decrease in the liquid level.
Thus, the correct answer is (2).

Step 1: Apply the principle of buoyancy.
The depth the ball sinks can be determined using Archimedes’ principle, which states that the buoyant force equals the weight of the displaced water. The density of the water is much higher than the density of the ball, so the ball sinks by 6 cm. Final Answer:

Step 1: Pressure at depth :
Step 2: Force on an elemental strip:
Step 3: Total torque about hinge:
Step 4: Balancing torque gives required force:

Step 1: Resultant hydrostatic force on the gate: FR = ρ g A hc = ρ g (1)(0.5) = (ρ g)/(2)
Step 2: Centre of pressure from top: h = (2)/(3)m Distance from hinge (mid-point): d = (2)/(3) - (1)/(2) = (1)/(6)
Step 3: Taking moments about hinge: F × (1)/(2) = (ρ g)/(2) × (1)/(6) ⟹ F = (ρ g)/(6)

Step 1: Understanding the Question:
A "just sinking" object has an average density equal to the density of the fluid (water). We need to find the ratio of the cavity volume ( ) to the material volume of the stopper ( ). \

Step 2: Key Formula or Approach:
For an object to just sink, the total weight must equal the buoyancy force: \

Step 3: Detailed Explanation:
Let be the volume of the cavity and be the volume of the glass material. The total volume . The mass of the stopper is purely from the glass material: . Given relative density of glass is 2.5, so . \ For "just sinking": \

Step 4: Final Answer:
The ratio of the volume of the cavity to the material volume of the stopper is 3:2. \