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Chapter 10: Problem 1375

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) \(\mathrm{x}=\mathrm{a} \cos \sqrt{(\mathrm{g} / \ell) \mathrm{t}}\)

Short Answer

Expert verified
The short version of the answer is: \(x(t) = a \cos(\sqrt{\frac{g}{\ell}t})\) The correct option is (D).

Step by step solution

01

Establish the relationship between angular displacement and linear displacement.

The linear displacement (x) of the bob of the pendulum is related to the angular displacement (θ) as follows: x = ℓ * θ
02

Write the equation for angular displacement as a function of time.

For a simple pendulum with small angular displacements, the angular displacement θ(t) as a function of time can be expressed using the equation: θ(t) = θ₀ * cos(ωt) where θ₀ is the initial angular displacement, ω is the angular frequency, and t is the time.
03

Calculate the angular frequency (ω).

The angular frequency (ω) of a simple pendulum is given by the formula: ω = √(g/ℓ) where g is the acceleration due to gravity and ℓ is the length of the pendulum.
04

Substitute the angular frequency (ω) and angular displacement (θ) into the equation for linear displacement (x).

We now have everything needed to calculate the linear displacement x(t) as a function of time. Substitute θ(t) and ω into the equation for x: x(t) = ℓ * θ₀ * cos(√(g/ℓ)t)
05

Simplify the equation for linear displacement.

To simplify the equation for linear displacement, let a = ℓ * θ₀: x(t) = a cos(√(g/ℓ)t)
06

Compare with given options.

The final equation for linear displacement as a function of time is: x(t) = a cos(√(g/ℓ)t) Comparing this equation with the given options, we find that the correct answer is: (D) x = a cos √((g/ℓ)t)

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Most popular questions from this chapter

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) \(\mathrm{b}\) (C) \(\mathrm{c}\) (D) \(\mathrm{d}\) Statement \(-1:\) The periodic time of a S.H.O. depends on its amplitude and force constant. Statement \(-2:\) The elasticity and inertia decides the frequency of S.H.O. (A) a (B) \(b\) (C) c (D) \(\mathrm{d}\)

As shown in figure, a spring attached to the ground vertically has a horizontal massless plate with a \(2 \mathrm{~kg}\) mass in it. When the spring (massless) is pressed slightly and released, the \(2 \mathrm{~kg}\) mass, starts executing S.H.M. The force constant of the spring is \(200 \mathrm{Nm}^{-1}\). For what minimum value of amplitude, will the mass loose contact with the plate? (Take \(\left.\mathrm{g}=10 \mathrm{~ms}^{-2}\right)\) (A) \(10.0 \mathrm{~cm}\) (B) \(8.0 \mathrm{~cm}\) (C) \(4.0 \mathrm{~cm}\) (D) For any value less than \(12.0 \mathrm{~cm}\).

A tuning fork arrangement produces 4 beats/second with one fork of frequency \(288 \mathrm{~Hz}\). A little wax is applied on the unknown fork and it then produces 2 beats/s. The frequency of the unknown fork is $\ldots \ldots \ldots . \mathrm{Hz}$. (A) 286 (B) 292 (C) 294 (D) 288

If the equation for a particle performing S.H.M. is given by $\mathrm{y}=\sin 2 \mathrm{t}+\sqrt{3} \cos 2 \mathrm{t}\(, its periodic time will be \)\ldots \ldots .$ s. (A) 21 (B) \(\pi\) (C) \(2 \pi\) (D) \(4 \pi\).

Equation for a progressive harmonic wave is given by $\mathrm{y}=8 \sin 2 \pi(0.1 \mathrm{x}-2 \mathrm{t})\(, where \)\mathrm{x}\( and \)\mathrm{y}$ are in \(\mathrm{cm}\) and \(\mathrm{t}\) is in seconds. What will be the phase difference between two particles of this wave separated by a distance of \(2 \mathrm{~cm} ?\) (A) \(18^{\circ}\) (B) \(36^{\circ}\) (C) \(72^{\circ}\) (D) \(54^{\circ}\)
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