The Question is from neet-physics mechanical-properties-of-fluids variation-of-pressure-upthrust, and it was created by the author SUBHANI

Question

A solid cylinder of height $h$ and density $\rho_c$ floats vertically in a liquid of density $\rho_l$. A small disturbance causes it to oscillate vertically. Neglecting viscous effects, the time period of oscillation is:

SUBHANI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: A
## CONCEPT FRAMEWORK
 **Primary Concept**: Simple Harmonic Motion (SHM) of a floating body
 **Related Topics**: Buoyancy, Archimedes' principle, restoring force
 **Prerequisites**: Understanding of SHM, knowledge of buoyancy
## SOLUTION PATHWAY
**Given Information**:
  Height of cylinder = $h$
  Density of cylinder = $ho_c$
  Density of liquid = $ho_l$
  Viscous effects are neglected.
**Method & Steps**:
 1. Determine the restoring force.
  Reasoning: When the cylinder is displaced slightly downwards, the buoyant force increases, creating a restoring force proportional to the displacement.
  Formula/Rule:  Restoring force $F = -A ho_l g x$, where $A$ is the cross-sectional area of the cylinder, $g$ is acceleration due to gravity, and $x$ is the displacement from equilibrium.
 2. Relate the restoring force to the mass of the cylinder and its acceleration.
  Working: The mass of the cylinder is $m = A h ho_c$.  Newton's second law gives $F = ma$, so $-A ho_l g x = A h ho_c \frac{d^2x}{dt^2}$. This simplifies to $\frac{d^2x}{dt^2} = -\frac{ho_l g}{h ho_c} x$.
  Key Point: The equation of motion is of the form $\frac{d^2x}{dt^2} = -\omega^2 x$, which represents SHM with angular frequency $\omega$.
 3. Find the time period.
  Result: Comparing the equation of motion to the standard SHM equation, we get $\omega^2 = \frac{ho_l g}{h ho_c}$. The time period $T$ is given by $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{h ho_c}{g ho_l}}$.
  Verification: The units of the expression for $T$ are seconds, which is consistent with the time period.
 ## OPTION ANALYSIS
A ✓: $2\pi \sqrt{\frac{h ho_c}{g ho_l}}$ is the correct expression derived from the SHM analysis.
B :x: This option has the densities inverted.
C :x: This option ignores the densities of the cylinder and the liquid.
D :x: This option is half the correct value.
 ## LEARNING AIDS
 **Common Mistakes**:
  1.Incorrectly applying Archimedes' principle: Make sure to understand that the restoring force is proportional to the displaced volume of liquid.
  2.Mixing up the densities: Carefully distinguish between the density of the cylinder and the density of the liquid.
 **Quick Tips**:
  1.Remember the standard SHM equation $\frac{d^2x}{dt^2} = -\omega^2 x$ and relate it to the forces acting on the cylinder.
  2.Use dimensional analysis to check the correctness of your final expression for the time period.

SUBHANI · Answer

The correct Option is "$2\pi \sqrt{\frac{h \rho_c}{g \rho_l}}$"

SUBHANI · Answer

$2\pi \sqrt{\frac{h \rho_l}{g \rho_c}}$

SUBHANI · Answer

$2\pi \sqrt{\frac{h}{g}}$

SUBHANI · Answer

$\pi \sqrt{\frac{h \rho_c}{g \rho_l}}$

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