Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 10: Problem 1370

Which of the equation given below represents a S.H.M.? (A) acceleration \(=-\mathrm{k}(\mathrm{x}+\mathrm{a})\) (B) acceleration \(=\mathrm{k}(\mathrm{x}+\mathrm{a})\) (C) acceleration \(=\mathrm{kx}\) (D) acceleration \(=-\mathrm{k}_{0} \mathrm{x}+\mathrm{k}_{1} \mathrm{x}^{2}\) \\{Here \(\mathrm{k}, \mathrm{k}_{0}\) and \(\mathrm{k}_{1}\) are force constants and units of \(\mathrm{x}\) and a is meter \(\\}\)

Short Answer

Expert verified
The short answer is: Only equation (C), which is \( a = kx \), represents a Simple Harmonic Motion, as it satisfies the condition where the acceleration is proportional to the displacement and directed towards the mean position.

Step by step solution

01

Equation (A)

The acceleration in equation (A) is given by: \( a = -k(x+a) \) Expanding this equation, we get: \( a = -kx - ka \) This equation represents a motion with the acceleration being proportional to the displacement and has an additional constant term. The presence of this constant term, however, means that this equation does not satisfy the condition of S.H.M. Therefore, equation (A) does not represent a S.H.M. #Step 2: Check Equation (B)#
02

Equation (B)

The acceleration in equation (B) is given by: \( a = k(x+a) \) Expanding this equation, we get: \( a = kx + ka \) Similar to equation (A), the acceleration is proportional to the displacement, and it also has a constant term. Moreover, the acceleration acts in the same direction as the displacement, which does not satisfy the condition of S.H.M. Therefore, equation (B) does not represent a S.H.M. #Step 3: Check Equation (C)#
03

Equation (C)

The acceleration in equation (C) is given by: \( a = kx \) This equation represents a motion with acceleration directly proportional to the displacement and directed oppositely. Since it satisfies the condition of S.H.M, equation (C) represents a S.H.M. #Step 4: Check Equation (D)#
04

Equation (D)

The acceleration in equation (D) is given by: \( a = -k_{0}x + k_{1}x^{2} \) This equation represents a motion with acceleration that is not only proportional to the displacement but also depends on the square of the displacement. This additional dependency does not satisfy the condition of S.H.M. Therefore, equation (D) does not represent a S.H.M. #Conclusion#: Out of the given equations, only equation (C) represents a Simple Harmonic Motion, since the acceleration is proportional to the displacement and directed towards the mean position.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

If \((1 / 4)\) of a spring having length \(\ell\) is cutoff, then what will be the spring constant of remaining part? (A) \(\mathrm{k}\) (B) \(4 \mathrm{k}\) (C) \((4 \mathrm{k} / 3)\) (D) \((3 \mathrm{k} / 4)\)

For the following questions, statement as well as the reason(s) are given. Each questions has four options. Select the correct option. (a) Statement \(-1\) is true, statement \(-2\) is true; statement \(-2\) is the correct explanation of statement \(-1\). (b) Statement \(-1\) is true, statement \(-2\) is true but statement \(-2\) is not the correct explanation of statement \(-1\) (c) Statement \(-1\) is true, statement \(-2\) is false (d) Statement \(-1\) is false, statement \(-2\) is true (A) a (B) \(b\) (C) \(\mathrm{c}\) (D) d Statement \(-1:\) For a particle executing S.H.M. with an amplitude of $0.01 \mathrm{~m}\( and frequency \)30 \mathrm{hz}\(, the maximum acceleration is \)36 \pi^{2} \mathrm{~m} / \mathrm{s}^{2}$. Statement \(-2:\) The maximum acceleration for the above particle is $\pm \omega 2 \mathrm{~A}\(, where \)\mathrm{A}$ is amplitude. (A) a (B) \(\mathrm{b}\) (C) \(c\) (D) \(\mathrm{d}\)

Two waves are represented by $\mathrm{y}_{1}=\mathrm{A} \sin \omega \mathrm{t}\( and \)\mathrm{y}_{2}=\mathrm{A}\( cos \)\omega \mathrm{t}$. The phase of the first wave, \(\mathrm{w}\). r. t. to the second wave is (A) more by radian (B) less by \(\pi\) radian (C) more by \(\pi / 2\) (D) less by \(\pi / 2\)

The periodic time of a S.H.O. oscillating about a fixed point is $2 \mathrm{~s}$. After what time will the kinetic energy of the oscillator become \(25 \%\) of its total energy? (A) \(1 / 12 \mathrm{~s}\) (B) \(1 / 6 \mathrm{~s}\) (C) \(1 / 4 \mathrm{~s}\) (D) \(1 / 3 \mathrm{~s}\).

The wave number for a wave having wavelength \(0.005 \mathrm{~m}\) is \(\ldots \ldots \mathrm{m}^{-1}\) (A) 5 (B) 50 (C) 100 (D) 200
See all solutions

Recommended explanations on English Textbooks

Language Analysis

Read Explanation

Lexis and Semantics Summary

Read Explanation

Pragmatics

Read Explanation

Prosody

Read Explanation

Multiple Choice Questions

Read Explanation

Linguistic Terms

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.

Sign-up for free