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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.

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Most popular questions from this chapter

The equation for displacement of a particle at time \(t\) is given by the equation \(\mathrm{y}=3 \cos 2 \mathrm{t}+4 \sin 2 \mathrm{t}\). The frequency of the particle is \(\ldots \ldots \mathrm{s}^{-1}\). (A) \((1 / \pi)\) (B) \(\pi\) (C) \((1 / 2 \pi)\) (D) \((\pi / 2)\)

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) \(\mathrm{d}\)

A particle having mass \(1 \mathrm{~kg}\) is executing S.H.M. with an amplitude of \(0.01 \mathrm{~m}\) and a frequency of \(60 \mathrm{hz}\). The maximum force acting on this particle is \(\ldots \ldots . . \mathrm{N}\) (A) \(144 \pi^{2}\) (B) \(288 \pi^{2}\) (C) \(188 \pi^{2}\) (D) None of these. (A) \(x=a \sin 2 p \sqrt{(\ell / g) t}\) (B) \(x=a \cos 2 p \sqrt{(g / \ell) t}\) (C) \(\mathrm{x}=\mathrm{a} \sin \sqrt{(\mathrm{g} / \ell) \mathrm{t}}\) (D) \(\mathrm{x}=\mathrm{a} \cos \sqrt{(\mathrm{g} / \ell) \mathrm{t}}\)

If two almost identical waves having frequencies \(\mathrm{n}_{1}\) and \(\mathrm{n}_{2}\), produced one after the other superposes then the time interval to obtain a beat of maximum intensity is \(\ldots \ldots \ldots .\) (A) \(\left\\{1 /\left(\mathrm{n}_{1}-\mathrm{n}_{2}\right)\right\\}\) (B) \(\left(1 / \mathrm{n}_{1}\right)-\left(1 / \mathrm{n}_{2}\right)\) (C) \(\left(1 / \mathrm{n}_{1}\right)+\left(1 / \mathrm{n}_{2}\right)\) (D) \(\left\\{1 /\left(\mathrm{n}_{1}+\mathrm{n}_{2}\right)\right\\}\)

The equation for displacement of a particle at time \(\mathrm{t}\) is given by the equation \(\mathrm{y}=3 \cos 2 \mathrm{t}+4 \sin 2 \mathrm{t}\). The amplitude of oscillation is \(\ldots \ldots \ldots . \mathrm{cm}\). (A) 1 (B) 3 (C) 5 (D) 7
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