To solve this problem, we need to determine the new string length required for the pendulum to perform 40 oscillations in 10 seconds, given that it currently performs 20 oscillations in the same duration with a 30 cm string.
- The formula for the time period \((T)\) of a simple pendulum, assuming small oscillations, is given by: \(T = 2\pi \sqrt{\frac{L}{g}}\)
where \(L\) is the length of the pendulum and \(g\) is the acceleration due to gravity. - The time period \((T)\) is the time for one complete oscillation. Thus, the frequency \((f)\) of oscillation, which is the number of oscillations per unit time, is given by: \(f = \frac{1}{T}\)
- Let \(T_1\) and \(f_1\) be the time period and frequency for the initial conditions: \(f_1 = \frac{20}{10} = 2 \text{ Hz}\)
- The relationship between frequency and length for pendulum oscillations is: \(f \propto \frac{1}{\sqrt{L}}\)
- Let \(L_2\) be the new length required for 40 oscillations in 10 seconds, giving us \(f_2 = \frac{40}{10} = 4 \text{ Hz}\)
- The proportionality between the frequencies and the square roots of the lengths is: \(\frac{f_1}{f_2} = \sqrt{\frac{L_2}{L_1}}\)
- Substitute known values to find \(L_2\):
- \(\frac{2}{4} = \sqrt{\frac{L_2}{30}}\)
- Simplify: \(\frac{1}{2} = \sqrt{\frac{L_2}{30}}\)
- Square both sides: \(\frac{1}{4} = \frac{L_2}{30}\)
- Arrange to find \(L_2\): \(L_2 = 30 \times \frac{1}{4} = 120 \text{ cm}\)
- Thus, the required string length for the pendulum to perform 40 oscillations in 10 seconds is 120 cm.
Hence, the correct answer is 120 cm.
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 _________ $\text{Hz}$.
- 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 ______.
