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Chapter 10: Problem 1349

A particle executing S.H.M. has an amplitude \(\mathrm{A}\) and periodic time \(\mathrm{T}\). The minimum time required by the particle to get displaced by \((\mathrm{A} / \sqrt{2})\) from its equilibrium position is $\ldots \ldots \ldots \mathrm{s}$. (A) \(\mathrm{T}\) (B) \(\mathrm{T} / 4\) (C) \(\mathrm{T} / 8\) (D) \(\mathrm{T} / 16\)

Short Answer

Expert verified
The minimum time required by the particle to get displaced by A/√2 from its equilibrium position is \(t = \frac{T}{8}\).

Step by step solution

01

Write the equation of SHM

A particle executing SHM can be represented by the equation: \(x(t) = A\sin(\omega t)\), where x(t) is the displacement from the equilibrium position at time t, A is the amplitude, and ω is the angular frequency. For a particle with a periodic time T, the angular frequency ω can be calculated as: \(\omega = \frac{2\pi}{T}\).
02

Write the displacement equation for the given displacement

Now we want to find the time when the displacement x(t) reaches A/√2. So we can set up the equation: \(\frac{A}{\sqrt{2}} = A\sin(\omega t)\).
03

Solve for t

To solve for t, we can first divide both sides of the equation by A and then take the inverse sine of both sides: \[\frac{1}{\sqrt{2}} = \sin(\omega t) \implies \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) = \omega t\] Now substitute the expression for ω and solve for t: \[\sin^{-1}\left(\frac{1}{\sqrt{2}}\right) = \frac{2\pi}{T}t \implies t = \frac{T}{2\pi}\sin^{-1}\left(\frac{1}{\sqrt{2}}\right)\] Use the fact that \(\sin^{-1}\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4}\), then solve for t: \[t = \frac{T}{2\pi}\times\frac{\pi}{4} = \frac{T}{8}\]
04

Choose the correct answer

Now we can see that the minimum time required by the particle to get displaced by A/√2 from its equilibrium position is T/8. Therefore, the correct answer is: (C) T/8

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Most popular questions from this chapter

When a mass \(M\) is suspended from the free end of a spring, its periodic time is found to be \(\mathrm{T}\). Now, if the spring is divided into two equal parts and the same mass \(\mathrm{M}\) is suspended and oscillated, the periodic time of oscillation is found to be \(\mathrm{T}\) '. Then \(\ldots \ldots \ldots\) (A) \(\mathrm{T}<\mathrm{T}^{\prime}\) (B) \(\mathrm{T}=\mathrm{T}^{\prime}\) (C) \(\mathrm{T}>\mathrm{T}^{\prime}\) (D) Nothing can be said.

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 periodic time of oscillation is \(\ldots \ldots \ldots \ldots\) (A) \(2 \mathrm{~s}\) (B) \(\pi \mathrm{s}\) (C) \((\pi / 2) \mathrm{s}\) (D) \(2 \pi \mathrm{s}\)

An listener is moving towards a stationary source of sound with a speed (1/4) times the speed of sound. What will be the percentage increase in the frequency of sound heard by the listener? (A) \(20 \%\) (B) \(25 \%\) (C) \(2.5 \%\) (D) \(5 \%\)

A block having mass \(\mathrm{M}\) is placed on a horizontal frictionless surface. This mass is attached to one end of a spring having force constant \(\mathrm{k}\). The other end of the spring is attached to a rigid wall. This system consisting of spring and mass \(\mathrm{M}\) is executing SHM with amplitude \(\mathrm{A}\) and frequency \(\mathrm{f}\). When the block is passing through the mid-point of its path of motion, a body of mass \(\mathrm{m}\) is placed on top of it, as a result of which its amplitude and frequency changes to \(\mathrm{A}^{\prime}\) and \(\mathrm{f}\). The ratio of frequencies \((\mathrm{f} / \mathrm{f})=\ldots \ldots \ldots\) (A) \(\sqrt{\\{} \mathrm{M} /(\mathrm{m}+\mathrm{M})\\}\) (B) \(\sqrt{\\{\mathrm{m} /(\mathrm{m}+\mathrm{M})\\}}\) (C) \(\sqrt{\\{\mathrm{MA} / \mathrm{mA}}\\}\) (D) \(\sqrt{[}\\{(\mathrm{M}+\mathrm{m}) \mathrm{A}\\} / \mathrm{mA}]\)

The speed of a particle executing motion changes with time according to the equation $\mathrm{y}=\mathrm{a} \sin \omega \mathrm{t}+\mathrm{b} \cos \omega \mathrm{t}\(, then \)\ldots \ldots \ldots$ (A) Motion is periodic but not a S.H.M. (B) It is a S.H.M. with amplitude equal to \(\mathrm{a}+\mathrm{b}\) (C) It is a S.H.M. with amplitude equal to \(\mathrm{a}^{2}+\mathrm{b}^{2}\)
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