If rod is displaced slightly in vertical direction, find the...

If rod is displaced slightly in vertical direction, find the time period of small oscillation. (A) $\pi \sqrt{\frac{l}{g}}$ (B) $2 \pi \sqrt{\frac{l}{g}}$ (C) $\frac{\pi}{2} \sqrt{\frac{l}{g}}$ (D) None of thesePassage 4 Figure $8.47$ shows a container having ideal liquid of variable density. The density of liquid varies as $\rho=\rho_{0}\left(4-\frac{3 h}{h_{0}}\right)$, where $h_{0}$ is height of liquid in container, $\rho_{0}$ is constant and $h$ is height from bottom. A solid block of small dimensions whose density $\frac{5}{2} \rho_{0}$ and mass $m$ is released from bottom of tank.

If rod is displaced slightly in vertical direction, find the time period of small oscillation.
(A) $\pi \sqrt{\frac{l}{g}}$
(B) $2 \pi \sqrt{\frac{l}{g}}$
(C) $\frac{\pi}{2} \sqrt{\frac{l}{g}}$
(D) None of thesePassage 4 Figure $8.47$ shows a container having ideal liquid of variable density. The density of liquid varies as $\rho=\rho_{0}\left(4-\frac{3 h}{h_{0}}\right)$, where $h_{0}$ is height of liquid in container, $\rho_{0}$ is constant and $h$ is height from bottom. A solid block of small dimensions whose density $\frac{5}{2} \rho_{0}$ and mass $m$ is released from bottom of tank.

Key Concepts

Archimedes’ Principle

Archimedes’ Principle states that an object submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by it. This principle is fundamental in determining the net force acting on the block when it is released in a fluid, particularly when the fluid’s density is not uniform.

Simple Harmonic Motion (SHM)

SHM is the principle that small displacements from an equilibrium position (where the restoring force is proportional to the displacement) result in oscillatory motion with a constant time period. This concept is used to analyze the rod’s small oscillations as well as oscillatory behavior in buoyant systems when displaced from equilibrium.

Moment of Inertia

In rotational dynamics, the moment of inertia quantifies how a body's mass is distributed relative to an axis of rotation. It plays a crucial role in determining the time period of a physical pendulum, such as a rod undergoing small oscillations, by affecting the system's angular acceleration under a given torque.

Physical Pendulum

A physical pendulum involves a rigid body swinging about a fixed pivot rather than a simple point mass on a string. Its period of oscillation is determined by both the gravitational torque and the moment of inertia about the pivot, making it essential for analyzing the dynamics of a displaced rod.

Buoyancy in Variable Density Fluids

When the density of a fluid varies with position, the buoyant force on an object becomes a function of the local density distribution. Analyzing such a scenario requires integrating the density variation effects to understand the equilibrium condition and the nature of any resulting oscillations when the object is displaced.

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