Physics: Deriving the Period of Oscillation
A cylindrical block of mass M and cross-sectional area A floats in a liquid (density ρ). When displaced vertically and released, it oscillates. We need to find the oscillation period.
Equilibrium Analysis
At equilibrium, the block's weight (Mg) equals the buoyant force. Let the submerged depth be h. The buoyant force is $ F_B = A h \rho g $. Equating forces:
$ M g = A h \rho g $
Restoring Force for Oscillation
If the block is depressed by a small distance x, the submerged depth increases to (h + x). The new buoyant force is $ F_B' = A(h+x)\rho g $. The net upward force, acting as the restoring force, is the difference between the new buoyant force and the weight:
$ F_{restore} = F_B' - Mg $ $ F_{restore} = A(h+x)\rho g - A h \rho g $ $ F_{restore} = A x \rho g $ This force opposes the displacement x, so the restoring force is $ F_{restore} = - A x \rho g $.
Simple Harmonic Motion (SHM)
Using Newton's second law ($ F=Ma $, where $ a = \frac{d^2x}{dt^2} $):
$ M a = - A x \rho g $ $ M \frac{d^2x}{dt^2} = - A x \rho g $ The equation for SHM is $ \frac{d^2x}{dt^2} = -\omega^2 x $. Comparing, the square of the angular frequency ($ \omega^2 $) is:
$ \omega^2 = \frac{A \rho g}{M} $
Period Calculation
The angular frequency is $ \omega = \sqrt{\frac{A \rho g}{M}} $. The period (T) is related to angular frequency by $ T = \frac{2\pi}{\omega} $.
$ T = \frac{2\pi}{\sqrt{\frac{A \rho g}{M}}} $ $ T = 2\pi \sqrt{\frac{M}{A \rho g}} $
Result
The derived period of oscillation is $ T = 2\pi \sqrt{\frac{M}{\rho A g}} $. This expression matches Option 3. The provided correct answer is Option A.
Similar Questions
- The displacement of a particle, executing simple harmonic motion with time period $T$, is expressed as $x(t) = A\sin\omega t$, where $A$ is the amplitude. The maximum value of potential energy of this oscillator is found at $t = T/2\beta$. The value of $\beta$ is ________.
A simple pendulum has a bob with mass $m$ and charge $q$. The pendulum string has negligible mass. When a uniform and horizontal electric field $\vec{E}$ is applied, the tension in the string changes. The final tension in the string, when pendulum attains an equilibrium position is _________.
(g: acceleration due to gravity)In an open organ pipe $\nu_3$ and $\nu_6$ are $3^{\text{rd}}$ and $6^{\text{th}}$ harmonic frequencies, respectively. If $\nu_6 - \nu_3 = 2200 \text{ Hz}$ then length of the pipe is _________ mm.
(Take velocity of sound in air is $330 \text{ m/s}$.)- Using a simple pendulum experiment g is determind by measuring its time period T. Which of the following plots represent the correct relation between the pendulum length L and time period T ?
- The kinetic energy of a simple harmonic oscillator is oscillating with angular frequency of $176\text{ rad/s}$. The frequency of this simple harmonic oscillator is __________ $\text{Hz}$. $\left[ \text{take } \pi = \frac{22}{7} \right]$
- Two tuning forks $A$ and $B$ are sounded together giving rise to $8$ beats in $2\text{ s}$. When fork $A$ is loaded with wax, the beat frequency is reduced to $4$ beats in $2\text{ s}$. If the original frequency of tuning fork $B$ is $380\text{ Hz}$ then original frequency of tuning fork $A$ is _________ $\text{Hz}$.
- A simple pendulum of string length 30 cm performs 20 oscillations in 10 s. The length of the string required for the pendulum to perform 40 oscillations in the same time duration is ___________ cm. [Assume that the mass of the pendulum remains same.]
- The velocity of sound in air is doubled when the temperature is raised from $0^\circ\text{C}$ to $\alpha^\circ\text{C}$. The value of $\alpha$ is ________.
- A particle is executing simple harmonic motion. Its amplitude is $A$ and time period is 5 sec. The time required by it to move from $x = A$ to $x = \frac{A}{\sqrt{2}}$ is _________ sec.
- Match List - I with List - II.
List - I List - II A. $\sin^{2} \omega t$ I. Periodic with time period $T=\frac{\pi}{\omega}$ but not simple harmonic motion (SHM) B. $\sin^{3} (2\omega t)$ II. Periodic with time period $T=\frac{2\pi}{\omega}$ but Not SHM C. $\sin (\omega t) + \cos(\pi \omega t)$ III. Periodic with time period $T=\frac{\pi}{\omega}$ and SHM D. $\cos \omega t + \cos 2\omega t$ IV. Non-periodic
Choose the correct answer from the options given below :
Important Questions from Oscillations and Waves
- The displacement of a particle, executing simple harmonic motion with time period $T$, is expressed as $x(t) = A\sin\omega t$, where $A$ is the amplitude. The maximum value of potential energy of this oscillator is found at $t = T/2\beta$. The value of $\beta$ is ________.
A simple pendulum has a bob with mass $m$ and charge $q$. The pendulum string has negligible mass. When a uniform and horizontal electric field $\vec{E}$ is applied, the tension in the string changes. The final tension in the string, when pendulum attains an equilibrium position is _________.
(g: acceleration due to gravity)In an open organ pipe $\nu_3$ and $\nu_6$ are $3^{\text{rd}}$ and $6^{\text{th}}$ harmonic frequencies, respectively. If $\nu_6 - \nu_3 = 2200 \text{ Hz}$ then length of the pipe is _________ mm.
(Take velocity of sound in air is $330 \text{ m/s}$.)- Using a simple pendulum experiment g is determind by measuring its time period T. Which of the following plots represent the correct relation between the pendulum length L and time period T ?
- The kinetic energy of a simple harmonic oscillator is oscillating with angular frequency of $176\text{ rad/s}$. The frequency of this simple harmonic oscillator is __________ $\text{Hz}$. $\left[ \text{take } \pi = \frac{22}{7} \right]$
