Shown in the figure are two hollow cubes C₁ and C₂ of negligible mass partially filled (depicted by darkened area) with liquids of densities ρ₁ and ρ₂, respectively, floating in water (density ρ_W). The relationship between ρ₁, ρ₂ and ρ_W is
- (a)ρ₂ < ρ_W < ρ₁
- (b)ρ₂ < ρ₁ < ρ_W
- (c)ρ₁ < ρ₂ < ρ_W
- (d)ρ₁ < ρ_W < ρ₂
Answer
Why
Correct — D, ρ₁ < ρ_W < ρ₂. The cubes themselves have no mass, so all the weight is the liquid inside. Floating means that weight equals the weight of water displaced: ρ · V_filled = ρ_W · V_submerged, so ρ = ρ_W × (submerged fraction ÷ filled fraction). The whole question is therefore a comparison of two fractions read off the figure. In the printed drawing both cubes are filled to about the same level — roughly seven-tenths of their height — but they ride very differently in the water. C₁ sits high, with only about half of it below the surface; C₂ sits low, with about four-fifths of it submerged. For C₁ the submerged fraction is SMALLER than the filled fraction, so ρ₁ < ρ_W — it displaces less water than the liquid it holds would occupy.
For C₂ the submerged fraction is LARGER than the filled fraction, so ρ₂ > ρ_W. Hence ρ₁ < ρ_W < ρ₂.
Why the others are wrong
- (a)ρ₂ < ρ_W < ρ₁ — This reverses the two cubes. C₂ is the one riding low in the water, which marks it as the denser of the pair, not the lighter.
- (b)ρ₂ < ρ₁ < ρ_W — It puts both liquids below the density of water. C₂ is submerged more deeply than it is filled, which is only possible if its liquid is denser than water.
- (c)ρ₁ < ρ₂ < ρ_W — It gets the order of the two liquids right but keeps both under ρ_W. The deep float of C₂ places it above water's density, so ρ_W has to sit between the two, not above both.
Concept
A floating body displaces its own weight of fluid. For a container of negligible mass the weight is entirely that of its contents, so ρ·V_filled = ρ_W·V_submerged. Rearranged, the density of the contained liquid is water's density scaled by the ratio of how deeply the container floats to how full it is.
Comparing how low each cube rides is not enough on its own — a cube that is fuller will naturally sit deeper even with the same liquid. What matters is the submerged depth relative to the fill level. Here that comparison is easy because both cubes are filled to about the same fraction, so the very different float depths translate directly into different densities, and the line ρ_W falls between them.
Key facts
- A floating body displaces a weight of fluid equal to its own weight.
- For a massless container, rho x V_filled = rho_W x V_submerged.
- So rho = rho_W x (submerged fraction / filled fraction).
- Submerged fraction < filled fraction means the liquid is lighter than water.
- Here C1 floats high (lighter than water) and C2 floats low (denser than water).
Compare depth of float against fill level, not against the other cube — option (d).
Study next
Common traps
- Judging density from how low a container floats without allowing for how full it is.
- Assuming both liquids must be lighter than water because both cubes float.
- Forgetting that the container's own mass is stated to be negligible.
NDA prints two floating containers with different fill and float levels — form the ratio of submerged fraction to filled fraction for each and compare it with 1.
Related PYQs
No directly related past PYQ was found.
Practice
- practice — not a real PYQ
A block floats in water with three-quarters of its volume submerged. Its density is
- (a)0·25 times that of water
- (b)0·75 times that of water
- (c)1·33 times that of water
- (d)equal to that of water
Answer(b) 0·75 times that of water — the submerged fraction equals the relative density for a solid block. - practice — not a real PYQ
A hollow container of negligible mass, half filled with a liquid, floats with half its volume submerged. The liquid's density compared with water is
- (a)half
- (b)equal
- (c)double
- (d)one-quarter
Answer(b) equal — submerged fraction and filled fraction are the same, so the ratio is 1.