Physics Q25 – NEET-UG 2026

A cylindrical cork of uniform density floats in a liquid of density

ρ_{1}

. If the cork is depressed slightly and released, it oscillates harmonically with time period T. If the same cork floats in another liquid of density

ρ_{2}

, then the similar oscillation has time period 2T. The value of

ρ_{2} / ρ_{1}

is :

Clue 1 of 3: Oscillations of a Floating Body

When a body floats and is slightly displaced, the restoring force due to buoyancy causes it to oscillate in Simple Harmonic Motion (SHM).

🥷

Ninja Strategy•Inverse Square Root Relationship

Recognize that the time period of oscillation for a floating body is inversely proportional to the square root of the liquid density ( $T \propto 1 / \sqrt{ρ_{L}}$ ). This implies $T^{2} \propto 1 / ρ_{L}$ . A doubling of $T$ means $T^{2}$ quadruples, so $1 / ρ_{L}$ must quadruple, meaning $ρ_{L}$ must be $1 / 4$ of its original value.

🧠 Full Solution Path

Step 1: Determine the restoring force and effective spring constant✦ Active

When a cylindrical cork of mass $m$ and cross-sectional area $A$ floats in a liquid of density $ρ_{L}$ , and is depressed by a small distance $y$ , the additional buoyant force acts as a restoring force. The additional volume submerged is $A y$ . The restoring force is $F = (ρ_{L} A y) g$ . Comparing this to $F = k y$ , the effective spring constant is $k = ρ_{L} A g$ .

Step 2: Write the expression for the time period of oscillation○ Expand

The time period of oscillation for SHM is $T = 2 π \sqrt{\frac{m}{k}}$ . Substituting $k = ρ_{L} A g$ , we get:

T = 2 π \sqrt{\frac{m}{ρ_{L} A g}}

From this, we can see that $T \propto \frac{1}{\sqrt{ρ_{L}}}$ , or $T^{2} \propto \frac{1}{ρ_{L}}$ .

💡 Teacher's Secret Hint

Remember that $m$ , $A$ , and $g$ are constants for the given cork and gravitational field.

Step 3: Apply the given conditions and calculate the ratio○ Expand

For the first liquid with density $ρ_{1}$ and time period $T$ : $T^{2} = C \frac{1}{ρ_{1}}$ (where $C = \frac{4 π^{2} m}{A g}$ is a constant for the cork). For the second liquid with density $ρ_{2}$ and time period $2 T$ : $(2 T)^{2} = C \frac{1}{ρ_{2}}$ . Dividing the second equation by the first:

\frac{(2 T)^{2}}{T^{2}} = \frac{C / ρ_{2}}{C / ρ_{1}} ⟹ 4 = \frac{ρ_{1}}{ρ_{2}}

Therefore, the required ratio is:

\frac{ρ_{2}}{ρ_{1}} = \frac{1}{4}

💡 Teacher's Secret Hint

Pay attention to which ratio is asked: $ρ_{2} / ρ_{1}$ not $ρ_{1} / ρ_{2}$ .

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Physics Question 25 – NEET-UG 2026

Clue 1 of 3: Oscillations of a Floating Body

🧠 Full Solution Path