Question

A metal alloy cylinder floats completely submerged inside a large tank containing water and a liquid (refer figure).

The side length of cylinder present inside water at equilibrium is _______ cm:

Answer 1

40 cm

Answer 2

Let the length of the cylinder in water be $x$ (in cm). Then, the length in the denser liquid is $60 - x$ cm. For a completely submerged floating body, the buoyant force equals the weight of the body.

• Weight of the cylinder:

  $$W = (\text{density of cylinder}) \times g \times (\text{Volume}) = 2\rho \, g\, (A \times 60)$$

• Buoyant force: (contributions from water and liquid)

  $$F_b = A\,g \left( \rho \, x + 4\rho (60 - x)\right)$$

Setting $W = F_b$ and cancelling $A$, $g$, and $\rho$ (since they are common and nonzero), we get:

$$2 \times 60 = x + 4(60 - x)$$ $$120 = x + 240 -4x$$ $$120 = 240 - 3x \quad \Longrightarrow \quad 3x = 240 -120 =120$$ $$x = \frac{120}{3} = 40 \text{ cm.}$$

Thus, the side (or length) of the cylinder present inside water at equilibrium is 40 cm.

Balance buoyant force and weight: $2\rho\,(60) = \rho \,x + 4\rho (60-x)$ leading to $x=40$ cm.