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Chapter 10: Problem 1408

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) \(\mathrm{d}\) Statement \(-1:\) Periodic time of a simple pendulum is independent of the mass of the bob. Statement \(-2:\) The restoring force does not depend on the mass of the bob. (A) a (B) \(b\) (C) \(\mathrm{c}\) (D) \(\mathrm{d}\)

Short Answer

Expert verified
(C) Statement \(-1\) is true, statement \(-2\) is false

Step by step solution

01

Analyze Statement 1

The periodic time of a simple pendulum is given by the formula \(T = 2\pi \sqrt{\frac {L}{g}}\), where \(T\) is the period, \(L\) is the length of the pendulum, and \(g\) is the acceleration due to gravity. As we can see from the formula, the period does not depend on the mass of the bob. Therefore, Statement 1 is true.
02

Analyze Statement 2

The equation for the restoring force acting on a pendulum bob in a simple pendulum system is \(F = -mgsin(\theta)\), where \(F\) is the restoring force, \(m\) is the mass of the bob, \(g\) is the acceleration due to gravity, and \(\theta\) is the angular displacement of the pendulum from its equilibrium position. As we can see from the equation, the restoring force does depend on the mass of the bob. Therefore, Statement 2 is false.
03

Check Explanation

As Statement 1 is true and Statement 2 is false, there is no need to check if Statement 2 explains Statement 1 since Statement 2 is not the correct explanation for Statement 1. Based on our analysis, the correct option is: (C) Statement \(-1\) is true, statement \(-2\) is false

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

When a block of mass \(\mathrm{m}\) is suspended from the free end of a massless spring having force constant \(\mathrm{k}\), its length increases by y. Now when the block is slightly pulled downwards and released, it starts executing S.H.M with amplitude \(\mathrm{A}\) and angular frequency \(\omega\). The total energy of the system comprising of the block and spring is \(\ldots \ldots \ldots\) (A) \((1 / 2) \mathrm{m} \omega^{2} \mathrm{~A}^{2}\) (B) \((1 / 2) m \omega^{2} A^{2}+(1 / 2) \mathrm{ky}^{2}\) (C) \((1 / 2) \mathrm{ky}^{2}\) (D) \((1 / 2) m \omega^{2} A^{2}-(1 / 2) k y^{2}\)

If the equation for a transverse wave is $\mathrm{y}=\mathrm{A} \operatorname{Sin} 2 \pi\( \)\\{(1 / \mathrm{T})-(\mathrm{x} / \lambda)\\}$. then for what wavelength will the maximum velocity of the particle be double the wave velocity ? (A) \((\pi \mathrm{A} / 4)\) (B) \((\pi \mathrm{A} / 2)\) (C) \(\pi \mathrm{A}\) (D) \(2 \pi \mathrm{A}\)

The ratio of force constants of two springs is \(1: 5\). The equal mass suspended at the free ends of both springs are performing S.H.M. If the maximum acceleration for both springs are equal, the ratio of amplitudes for both springs is \(\ldots \ldots\) (A) \((1 / \sqrt{5})\) (B) \((1 / 5)\) (C) \((5 / 1)\) (D) \((\sqrt{5} / 1)\)

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) \(3 .\)
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