Understanding Simple Harmonic Motion
The position $x$ of a particle in Simple Harmonic Motion (SHM) can be described by the equation: $x(t) = A \cos(\omega t + \phi)$ where $A$ is the amplitude, $\omega$ is the angular frequency, $t$ is time, and $\phi$ is the phase constant.
Calculating Angular Frequency
The angular frequency $\omega$ is related to the time period $T$ by: $\omega = \frac{2\pi}{T}$ Given the time period $T = 5$ sec, the angular frequency is: $\omega = \frac{2\pi}{5}$ rad/sec.
Determining Phase and Time
We assume the motion starts from the extreme position $x = A$ at $t=0$. This implies the phase constant $\phi = 0$. The equation of motion simplifies to: $x(t) = A \cos(\omega t)$
We need to find the time $t$ when the particle moves from $x = A$ to $x = \frac{A}{\sqrt{2}}$.
- At $t=0$, $x = A \cos(0) = A$. This is the starting point.
- We need to find the time $t$ when $x = \frac{A}{\sqrt{2}}$. $\frac{A}{\sqrt{2}} = A \cos(\omega t)$ $\frac{1}{\sqrt{2}} = \cos(\omega t)$
For the first time this occurs after $t=0$, the angle $\omega t$ must be $\frac{\pi}{4}$ radians.
Final Time Calculation
Now, we solve for $t$: $\omega t = \frac{\pi}{4}$ Substitute the value of $\omega = \frac{2\pi}{5}$: $\left(\frac{2\pi}{5}\right) t = \frac{\pi}{4}$ $t = \frac{\pi}{4} \times \frac{5}{2\pi}$ $t = \frac{5}{8}$ sec.
Therefore, the time required to move from $x = A$ to $x = \frac{A}{\sqrt{2}}$ is $\frac{5}{8}$ sec.
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 ?
- A cylindrical block of mass $M$ and area of cross section $A$ is floating in a liquid of density $\rho$ and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is ______.
- 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 ________.
- 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 ?
- A cylindrical block of mass $M$ and area of cross section $A$ is floating in a liquid of density $\rho$ and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is ______.
