/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 261 In forced oscillation of a parti... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 261

In forced oscillation of a particle the amplitude is maximum for a frequency \(\omega_{1}\) of the force while the energy is maximum for a frequency \(\omega_{2}\) of the force; then (A) \(\omega_{1}<\omega_{2}\) when damping is small and \(\omega_{1}>\omega_{2}\) when damping is large (B) \(\omega_{1}>\omega_{2}\) (C) \(\omega_{1}=\omega_{2}\) (D) \(\omega_{1}<\omega_{2}\)

Short Answer

Expert verified
The correct answer is (A) \(\omega_{1}<\omega_{2}\) when damping is small and \(\omega_{1}>\omega_{2}\) when damping is large. This conclusion is based on the analysis of underdamped and overdamped systems, taking into account that the amplitude is maximum at frequency \(\omega_1\) and the energy is maximum at frequency \(\omega_2\).

Step by step solution

01

Understanding the given information

In the given problem, we are told that amplitude is maximum for a frequency \(\omega_{1}\) while energy is maximum for a frequency \(\omega_{2}\). We need to find the relationship between these two frequencies for different damping conditions.
02

Analyzing the problem with respect to damping

Let us consider the damping coefficient as "b" and mass as "m". The damping ratio, \(\zeta\), can be expressed as follows: \[ \zeta = \frac{b}{2m\omega_{0}} \] where \(\omega_{0}\) is the natural frequency of the system. Depending on this damping ratio, we have three cases: 1. If \(\zeta < 1\), the system is underdamped. 2. If \(\zeta = 1\), the system is critically damped. 3. If \(\zeta > 1\), the system is overdamped. Now, we need to analyze the amplitude and energy of the system for these cases.
03

Amplitude and Energy for Underdamped system

For an underdamped system, \(\zeta < 1\). In this case, the resonance frequency (\(\omega_{res}\)) is given by: \[ \omega_{res} = \omega_{0}\sqrt{1 - \zeta^{2}} \] Since the amplitude is maximum at \(\omega_1\), we have: \(\omega_{1} = \omega_{res}\). The energy of the system is given by: \[ E(\omega) = \frac{1}{2} m A^{2}\omega^{2} \] where A is the amplitude at frequency \(\omega\). For an underdamped system, \(\omega_2\) is greater than \(\omega_{res}\) and therefore, \(\omega_1 < \omega_2\) as damping is small.
04

Amplitude and Energy for Overdamped system

For an overdamped system, \(\zeta > 1\). In this case, the resonance frequency does not exist. However, the amplitude is still maximum at \(\omega_1\). As damping increases, \(\omega_{1}\) shifts towards higher frequencies while \(\omega_{2}\) remains at the same value. Therefore, for large damping, \(\omega_1 > \omega_2\).
05

Conclusion

Combining our analysis for underdamped and overdamped systems, we conclude that the correct answer is: (A) \(\omega_{1}<\omega_{2}\) when damping is small and \(\omega_{1}>\omega_{2}\) when damping is large

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

An organ pipe of length \(3.9 \pi \mathrm{m}\) open at both ends is driven to third harmonic standing wave pattern. If the maximum amplitude of pressure oscillations is \(1 \%\) of mean atmospheric pressure \(\left(P_{0}=105 \mathrm{~N} / \mathrm{m}^{2}\right)\) the maximum displacement of the particle from mean position will be (Velocity of sound \(=200 \mathrm{~m} / \mathrm{s}\) and density of air \(=1.3 \mathrm{~kg} / \mathrm{m}^{3}\) ) (A) \(2.5 \mathrm{~cm}\) (B) \(5 \mathrm{~cm}\) (C) \(1 \mathrm{~cm}\) (D) \(2 \mathrm{~cm}\)

A stationary source of sound is emitting waves of frequency \(30 \mathrm{~Hz}\) towards a stationary wall. There is an observer standing between the source and the wall. If the wind blows from the source to the wall with a speed \(30 \mathrm{~m} / \mathrm{s}\) then the number of beats heard by the observer is (velocity of sound with respect to wind is \(330 \mathrm{~m} / \mathrm{s})\) (A) 10 (B) 3 (C) 6 (D) Zero

A body executes simple harmonic motion. The potential energy (PE), the kinetic energy (KE) and total energy (TE) are measured as a function of displacement \(x\). Which of the following statements is true ? (A) \(\mathrm{KE}\) is maximum when \(x=0\). (B) TE is zero when \(x=0\) (C) \(\mathrm{KE}\) is maximum when \(x\) is maximum (D) \(\mathrm{PE}\) is maximum when \(x=0\)

For a certain stretched string, three consecutive resonance frequencies are observed as \(105,175,245 \mathrm{~Hz}\), respectively. Then select the correct alternatives (A) The string is fixed at both ends. (B) The string is fixed at one end only. (C) The fundamental frequency is \(35 \mathrm{~Hz}\). (D) The fundamental frequency is \(52.5 \mathrm{~Hz}\).

A \(3.6 \mathrm{~m}\) long vertical pipe is filled completely with a liquid. A small hole is drilled at the base of the pipe due to which liquids starts leaking out. This pipe resonates with a tuning fork. The first two resonances occur when height of water column is \(3.22 \mathrm{~m}\) and \(2.34 \mathrm{~m}\), respectively. The area of cross-section of pipe is (A) \(25 \pi \mathrm{cm}^{2}\) (B) \(100 \pi \mathrm{cm}^{2}\) (C) \(200 \pi \mathrm{cm}^{2}\) (D) \(400 \pi \mathrm{cm}^{2}\)
See all solutions

Recommended explanations on Physics Textbooks

Kinematics Physics

Read Explanation

Waves Physics

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation

Turning Points in Physics

Read Explanation

Space Physics

Read Explanation

Work Energy and Power

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.