Understanding Kinetic Energy Oscillation Frequency in SHO
The kinetic energy of a Simple Harmonic Oscillator (SHO) oscillates at twice the frequency of the oscillator itself. This is because kinetic energy depends on the square of the velocity ($KE = \frac{1}{2}mv^2$), and velocity in SHO is proportional to $\sin(\omega t)$ or $\cos(\omega t)$. Squaring this results in terms like $\sin^2(\omega t)$ or $\cos^2(\omega t)$, which have a frequency of $2\omega$.
Relating Angular Frequencies
Let $\omega_{KE}$ be the angular frequency of the kinetic energy oscillation and $\omega_{SHO}$ be the angular frequency of the simple harmonic oscillator.
- Given: $\omega_{KE} = 176 \text{ rad/s}$.
- Relationship: $\omega_{KE} = 2 \times \omega_{SHO}$.
- Therefore, the oscillator's angular frequency is $\omega_{SHO} = \frac{\omega_{KE}}{2} = \frac{176 \text{ rad/s}}{2} = 88 \text{ rad/s}$.
Calculating Oscillator Frequency (f)
The relationship between angular frequency ($\omega$) and frequency ($f$) is given by $\omega = 2\pi f$. We need to find $f$ in Hertz (Hz).
We have $\omega_{SHO} = 88 \text{ rad/s}$ and we are given $\pi = \frac{22}{7}$.
Rearranging the formula to solve for $f$: $f = \frac{\omega_{SHO}}{2\pi}$.
Step-by-Step Calculation
- Substitute the values into the formula: $f = \frac{88 \text{ rad/s}}{2 \times \frac{22}{7}} $
- Simplify the denominator: $2 \times \frac{22}{7} = \frac{44}{7}$
- Divide the angular frequency by the simplified denominator: $f = \frac{88}{\frac{44}{7}} = \frac{88 \times 7}{44}$
- Perform the division and multiplication: $f = 2 \times 7 = 14 \text{ Hz}$
Conclusion
The frequency of the simple harmonic oscillator is $14 \text{ Hz}$.
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 ______.
- 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 ?
- 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 ______.
