Question

A small metal sphere of density $\rho$ is dropped from height $h$ into a jar containing liquid of density $\sigma (\sigma > \rho)$. The maximum depth up to which the sphere sinks is (Neglect damping forces)

• A. $\frac{\rho}{\rho-\sigma}$
• B. $\frac{\text{h}\sigma}{(\rho-\sigma)}$
• C. $\frac{\sigma}{(\rho-\sigma)}$
• D. $\frac{\text{h}\rho}{(\sigma-\rho)}$

Hint:

Work-Energy Theorem: $W_{net} = \Delta KE$. At max depth, $KE = 0$. $W_g + W_b = 0$.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The sphere falls from height $h$, gaining kinetic energy. Inside the liquid, buoyancy opposes gravity.
Since $\sigma>\rho$, the upward buoyant force is greater than the downward weight, acting as a retarding force until the sphere comes to a momentary halt at maximum depth.
Step 2: Key Formula or Approach:
Use the Work-Energy Theorem: Total work done = Change in Kinetic Energy.
Since it starts from rest and stops at maximum depth $x$, $\Delta K = 0$.
Work done by gravity + Work done by buoyancy = 0.
$W_g = mg(h + x) = (V\rho)g(h + x)$.
$W_b = -F_b x = -(V\sigma g)x$.
Step 3: Detailed Explanation:
Set total work to zero: \[ (V\rho)g(h + x) - (V\sigma g)x = 0 \] Divide by common terms $V$ and $g$: \[ \rho(h + x) - \sigma x = 0 \] Expand the terms: \[ \rho h + \rho x - \sigma x = 0 \] Isolate $x$ terms on one side: \[ \rho h = \sigma x - \rho x \] \[ \rho h = x(\sigma - \rho) \] Solve for $x$: \[ x = \frac{\rho h}{\sigma - \rho} \] Note that since $\sigma>\rho$, the denominator $\sigma - \rho$ is positive, yielding a positive physical depth $x$.
Looking at the options, Option D is written as $\frac{h\rho}{\rho - \sigma}$. This has a flipped sign in the denominator relative to standard conventions, implying a negative result if strictly evaluated. However, in multiple choice contexts, this usually represents a typo in the option's sign conventions by the author while preserving the correct algebraic structure.
Matching the structure (numerator $h\rho$, denominator elements $\rho, \sigma$), option (D) is the intended choice.
Step 4: Final Answer:
The intended formula structurally matches $\frac{\text{h}\rho}{(\rho-\sigma)}$.