Physics · Mechanical Properties Of Fluids · NEET
Both push up. Any part of the cylinder that is inside a liquid gets an upthrust from that liquid. The lower part sitting in the denser liquid gets a large upthrust, and the upper part sitting in the lighter liquid gets a smaller upthrust. The total buoyant force is the sum of the two. This is why we write weight = (upthrust from bottom liquid) + (upthrust from top liquid).
Floating means the cylinder is in equilibrium and not moving up or down. So the net force is zero. Gravity pulls the whole cylinder down with force weight = d x L x A x g. The two liquids together push it up. For balance, total upthrust must exactly equal the weight. That single balance equation gives you everything you need.
Let total length be L and the length inside the denser liquid be pL (with p less than 1). Then the length inside the lighter liquid is the rest, which is (1 - p)L. Each length times the cross-section area A gives the submerged volume in that liquid. You never need the actual value of A - it cancels out because it appears in every term.
Every term in weight = total buoyancy has the same A and the same g and the same L. Weight = d L A g. Bottom upthrust = (n rho)(pL)(A)(g). Top upthrust = (rho)((1-p)L)(A)(g). Divide the whole equation by (L A g) and A, g, L all cancel, leaving only densities: d = n rho p + rho(1 - p).
Then it would sink to the bottom and rest on the base, so it would not float freely across the boundary. Floating across the boundary only happens when the cylinder's density d lies between the top density and the bottom density. The lighter liquid alone cannot support it, but the denser liquid at the bottom provides the extra push, so it settles with its lower part in the denser layer.
Two non-mixing liquids of densities rho and n rho (n > 1) are put in a container. The height of each liquid is h. A solid cylinder of length L and density d is put in this container. The cylinder floats with its axis vertical and length pL (p < 1) in the denser liquid. The density d is equal to:
Try the real previous-year questions from this chapter — each with the answer and a full solution.
Weight of cylinder = upthrust from top liquid + upthrust from bottom liquid. In symbols: d L A g = rho(top) x (length in top)(A)(g) + rho(bottom) x (length in bottom)(A)(g).
d = rho[1 + (n - 1)p], where rho is the lighter density, n rho is the denser density, and pL is the length inside the denser liquid.
No. As long as the cylinder floats across the boundary with a known length in each liquid, only the densities and the submerged lengths matter. The given h just tells you the layers are deep enough for this to happen.
Its density must be more than the top liquid density rho and less than the bottom liquid density n rho. That is why it settles with part in each layer instead of fully floating or fully sinking.
Because the cylinder has a uniform area A throughout. A multiplies every term equally, so it cancels out when you compare weight with buoyancy. Only lengths and densities survive.
