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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

If the maximum frequency of a sound wave at room temperature is $20,000 \mathrm{~Hz}\( then its minimum wavelength will be approximately \)\ldots \ldots\left(\mathrm{v}=340 \mathrm{~ms}^{-1}\right)$ (A) \(0.2 \AA\) (B) \(5 \AA\) (C) \(5 \mathrm{~cm}\) to \(2 \mathrm{~m}\) (D) \(20 \mathrm{~mm}\)

A string of linear density \(0.2 \mathrm{~kg} / \mathrm{m}\) is stretched with a force of \(500 \mathrm{~N}\). A transverse wave of length \(4.0 \mathrm{~m}\) and amplitude \(1 / 1\) meter is travelling along the string. The speed of the wave is \(\ldots \ldots \ldots \ldots \mathrm{m} / \mathrm{s}\) (A) 50 (B) \(62.5\) (C) 2500 (D) \(12.5\)

Two wires made up of same material are of equal lengths but their radii are in the ratio \(1: 2\). On stretching each of these two strings by the same tension, the ratio between their fundamental frequency is \(\ldots \ldots \ldots .\) (A) \(1: 2\) (B) \(2: 1\) (C) \(1: 4\) (D) \(4: 1\)

The amplitude for a S.H.M. given by the equation $\mathrm{x}=3 \sin 3 \mathrm{pt}+4 \cos 3 \mathrm{pt}\( is \)\ldots \ldots \ldots \ldots \mathrm{m}$ (A) 5 (B) 7 (C) 4 (D) \(3 .\)

As shown in figure, a block A having mass \(M\) is attached to one end of a massless spring. The block is on a frictionless horizontal surface and the free end of the spring is attached to a wall. Another block B having mass ' \(\mathrm{m}\) ' is placed on top of block A. Now on displacing this system horizontally and released, it executes S.H.M. What should be the maximum amplitude of oscillation so that B does not slide off A? Coefficient of static friction between the surfaces of the block's is \(\mu\). (A) \(A_{\max }=\\{(\mu \mathrm{mg}) / \mathrm{k}\\}\) (B) \(A_{\max }=[\\{\mu(m+M) g\\} / k]\) (C) \(A_{\max }=[\\{\mu(M-\mathrm{m}) g\\} / \mathrm{k}]\) (D) \(A_{\max }=[\\{2 \mu(M+m)\\} / k]\)
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