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Chapter 10: Problem 1362

A particle is executing S.H.M. between \(\mathrm{x}=-\mathrm{A}\) and \(\mathrm{x}=+\mathrm{A}\). If the time taken by the particle to travel from \(\mathrm{x}=0\) to \(\mathrm{A} / 2\) is \(\mathrm{T}_{1}\) and that taken to travel from \(\mathrm{x}=\mathrm{A} / 2\) to \(\mathrm{x}=\mathrm{A}\) is \(\mathrm{T}_{2}=\) then \(\ldots .\) (A) \(\mathrm{T}_{1}<\mathrm{T}_{2}\) (B) \(\mathrm{T}_{1}>\mathrm{T}_{2}\) (C) \(\mathrm{T}_{1}=2 \mathrm{~T}_{2}\) (D) \(\mathrm{T}_{1}=\mathrm{T}_{2}\)

Short Answer

Expert verified
(A) \(T_{1}<T_{2}\)

Step by step solution

01

Review the Equation for SHM Position

In Simple Harmonic Motion, the position of a particle is given by the equation \(x(t) = A \sin{(\omega t)}\), where \(A\) is the amplitude, \(\omega\) is the angular frequency, and \(t\) is time. By knowing that the time taken to travel from an initial position to a final position can be found by comparing the corresponding sin values, we can proceed to the next step.
02

Calculate the Phase Angles for the Different Position

We need to find the phase angles that correspond to the positions \(x = 0, A/2\), and \(A\). For this, we can use the position equation of SHM. 1. For \(x = 0\), \(0 = A \sin{(\omega t_0)}\), \(\omega t_0 = 0\). 2. For \(x = A/2\), \(A/2 = A \sin{(\omega t_1)}\), \(\omega t_1 = \frac{\pi}{6}\) or \(\frac{5\pi}{6}\). 3. For \(x = A\), \(A = A \sin{(\omega t_2)}\), \(\omega t_2 = \frac{\pi}{2}\). For this exercise, we choose the smallest positive values \(\omega t_1'\) and \(\omega t_2'\) such that \(t_1\) < \(\omega t_1'\) < \(t_2\) < \( \omega t_2'\).
03

Compare T1 and T2

From the previous step, we can find the time \(T_1 = \frac{\pi}{6\omega} - \frac{0}{\omega} = \frac{\pi}{6\omega}\) and the time \(T_2 = \frac{\pi}{2\omega} - \frac{\pi}{6\omega} = \frac{\pi}{3\omega}\). Now, we can compare these times: Since \(\frac{\pi}{6\omega} < \frac{\pi}{3\omega}\), it means that \(T_1 < T_2\). Therefore, the correct answer is (A) \(\ T_{1}<\ T_{2}\).

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Most popular questions from this chapter

A rectangular block having mass \(\mathrm{m}\) and cross sectional area A is floating in a liquid having density \(\rho\). If this block in its equilibrium position is given a small vertical displacement, its starts oscillating with periodic time \(\mathrm{T}\). Then in this case \(\ldots \ldots\) (A) \(\mathrm{T} \propto(1 / \sqrt{\mathrm{m}})\) (B) \(T \propto \sqrt{\rho}\) (C) \(\mathrm{T} \propto(1 / \sqrt{\mathrm{A}})\) (D) \(\mathrm{T} \propto(1 / \sqrt{\rho})\)

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) \(A_{\max }=[\\{2 \mu(M+m)\\} / k]\)

A simple pendulum is executing S.H.M. around point \(\mathrm{O}\) between the end points \(B\) and \(C\) with a periodic time of \(6 \mathrm{~s}\). If the distance between \(\mathrm{B}\) and \(\mathrm{C}\) is \(20 \mathrm{~cm}\) then in what time will the bob move from \(C\) to \(D\) ? Point \(D\) is at the mid-point of \(C\) and \(\mathrm{O}\). (A) \(1 \mathrm{~s}\) (B) \(2 \mathrm{~s}\) (C) \(3 \mathrm{~s}\) (D) \(4 \mathrm{~s}\)

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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