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Chapter 10: Problem 1392

A simple pendulum having length \(\ell\) is suspended at the roof of a train moving with constant acceleration 'a' along horizontal direction. The periodic time of this pendulum is.... (A) \(\mathrm{T}=2 \pi \sqrt{(\ell / \mathrm{g})}\) (B) \(\mathrm{T}=2 \pi \sqrt{\\{\ell /(\mathrm{g}+\mathrm{a})\\}}\) (C) \(\mathrm{T}=2 \pi \sqrt{\\{\ell /(\mathrm{g}-\mathrm{a})\\}}\) (D) $\left.\mathrm{T}=2 \pi \sqrt{\\{\ell} /\left(\mathrm{g}^{2}+\mathrm{a}^{2}\right)\right\\}$

Short Answer

Expert verified
The short answer based on the given step-by-step solution is: (D) \(\mathrm{T}=2 \pi \sqrt{\\{\ell}/\left(\mathrm{g}^{2}+\mathrm{a}^{2}\right)\\}\)

Step by step solution

01

Analyze the Situation in the Non-Inertial Frame of Reference

In the train's frame of reference (which is not an inertial frame due to acceleration 'a'), there is a pseudo force acting on the pendulum opposite to the direction of the train's acceleration. This force is equal to \(-m\mathrm{a}\) where m is the mass of the pendulum bob.
02

Determine the Effective Gravity Acting on the Pendulum

In addition to the actual gravitational force \(\downarrow m\mathrm{g}\) acting on the pendulum, we must consider the pseudo force \(\leftarrow (-m\mathrm{a})\) acting in the train's frame of reference. These combined forces form a resultant force making an angle with the vertical direction. The effective gravity 'g_eff' acting on the pendulum is the total force divided by the mass of the bob: \( g_{\mathrm{eff}} = \sqrt{g^{2} + a^{2}} \)
03

Apply the Simple Pendulum Formula

Now that we know the effective gravity acting on the pendulum, we can apply the formula for the period of a simple pendulum: \( \mathrm{T} = 2\pi\sqrt{\frac{\ell}{g_{\mathrm{eff}}}} \)
04

Substitute the Effective Gravity into the Formula

Replace g_eff in the formula with the expression found in Step 2: \( \mathrm{T} = 2\pi\sqrt{\frac{\ell}{\sqrt{g^{2} + a^{2}}}} \) This matches the Option (D), so the correct answer is: (D) \(\mathrm{T}=2 \pi \sqrt{\\{\ell}/\left(\mathrm{g}^{2}+\mathrm{a}^{2}\right)\\}\)

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Most popular questions from this chapter

A person standing in a stationary lift measures the periodic time of a simple pendulum inside the lift to be equal to \(\mathrm{T}\). Now, if the lift moves along the vertically upward direction with an acceleration of $(\mathrm{g} / 3)$, then the periodic time of the lift will now be \((\mathrm{A}) \sqrt{3} \mathrm{~T}\) (B) \(\sqrt{(3 / 2) \mathrm{T}}\) (C) \((\mathrm{T} / 3)\) (D) \((\mathrm{T} / \sqrt{3})\)

Which of the following functions represents a travelling wave? (A) \((\mathrm{x}-\mathrm{vt})^{2}\) (B) in \((\mathrm{x}+\mathrm{vt})\) (C) \(\mathrm{e}^{-(\mathrm{x}+\mathrm{vt}) 2}\) (D) \(\\{1 /(\mathrm{x}+\mathrm{vt})\\}\)

A simple pendulum having length \(\ell\) is given a small angular displacement at time \(t=0\) and released. After time \(t\), the linear displacement of the bob of the pendulum is given by \(\ldots \ldots \ldots \ldots\) (A) \(x=a \sin 2 p \sqrt{(\ell / g) t}\) (B) \(\mathrm{x}=\mathrm{a} \cos 2 \mathrm{p} \sqrt{(\mathrm{g} / \ell) t}\) (C) \(\mathrm{x}=\mathrm{a} \sin \sqrt{(\mathrm{g} / \ell) \mathrm{t}}\) (D) \(\mathrm{x}=\mathrm{a} \cos \sqrt{(\mathrm{g} / \ell) \mathrm{t}}\)

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) \(3 .\)

When an elastic spring is given a displacement of \(10 \mathrm{~mm}\), it gains an potential energy equal to \(\mathrm{U}\). If this spring is given an additional displacement of \(10 \mathrm{~mm}\), then its potential energy will be.............. (A) \(\mathrm{U}\) (B) \(2 \mathrm{U}\) (C) \(4 \mathrm{U}\) (D) \(\mathrm{U} / 4\).
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