Question

A cylindrical block of mass $M$ and area of cross section $A$ is floating in a liquid of density $\rho$ with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is ___.

• A. $2\pi\sqrt{\dfrac{\rho A}{Mg}}$
• B. $\pi\sqrt{\dfrac{\rho A}{Mg}}$
• C. $2\pi\sqrt{\dfrac{M}{\rho A g}}$
• D. $\pi\sqrt{\dfrac{2M}{\rho A g}}$

Hint:

Small vertical oscillations of floating bodies always execute simple harmonic motion due to buoyant restoring force.

Answer 1

The Correct Option is C

To solve the problem of determining the period of oscillation for a cylindrical block floating in a liquid, we begin by analyzing the principles involved. The block undergoes simple harmonic motion (SHM) due to being slightly depressed and released in the liquid.

The frequency of oscillation for a floating body is determined by the balance between gravitational force and buoyant force. According to Archimedes' principle, the buoyant force acting on the submerged block is equal to the weight of the liquid displaced.

Derivation

1. The buoyant force \(F_b\) when the block is deeper than its equilibrium position by a depth \(h\) is given by: \(F_b = \rho \cdot A \cdot g \cdot h\) where:
   • \(\rho\) is the liquid's density
   • \(A\) is the area of cross-section of the block
   • \(g\) is the acceleration due to gravity
   • \(h\) is the submerged depth
2. The restoring force \(F_r\) due to this buoyant force acting on the cylindrical block is: \(F_r = \rho \cdot A \cdot g \cdot h\)
3. Using Hooke's Law, the force in simple harmonic motion is also defined as \(F = -k \cdot x\), where \(k\) is the force constant and \(x\) is the displacement.
4. Comparing forces, we equate: \(-k \cdot h = \rho \cdot A \cdot g \cdot h\) Thus, \(k = \rho \cdot A \cdot g\)
5. The time period \(T\) for SHM is given by the formula: \(T = 2\pi\sqrt{\dfrac{m}{k}}\) where \(m\) is the mass of the block. Substituting the value of \(k\), we get: \(T = 2\pi\sqrt{\dfrac{M}{\rho \cdot A \cdot g}}\)

Therefore, the correct answer is \(2\pi\sqrt{\dfrac{M}{\rho A g}}\), which matches the correct option given.

This option accurately represents the period of oscillation for the system described in the question.