Physics: Asinusoidally varying driving force is applied to a damped harmonic oscillator of force constant k and mass m. if the damping constant has a value b1, the amplitude is a1 when the driving angular frequency equals k/m√.

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Asinusoidally varying driving force is applied

Ответ:

shayshayyy41

08.01.2022

Physics

The new amplitude of the harmonic oscillator when the damping constant is $3{b_1}$ will be $\boxed{\frac{{{A_1}}}{3}}$ .

Further Explanation:

Given:

The amplitude of the harmonic oscillator is ${A_1}$ .

The damping constant of the spring is ${b_1}$ .

The angular frequency of operation of the harmonic oscillator is $\sqrt {\dfrac{k}{m}}$ .

Concept:

The expression for the amplitude of a harmonic oscillator is given by:

$A = \dfrac{{{F_{\max }}}}{{\sqrt {{{\left({k - m{\omega ^2}}\right)}^2} + {b^2}{\omega ^2}} }}$

Here, is the amplitude of the harmonic oscillator, ${F_{\max }}$ is the maximum force, is the force constant of oscillator, $\omega$ is the driving angular frequency and is the damping constant of the oscillator.

The angular frequency of the harmonic oscillator is $\sqrt {\dfrac{k}{m}}$ .

Substitute ${A_1}$ for , ${b_1}$ for and $\sqrt {\dfrac{k}{m}}$ for $\omega$ in above expression.

$\begin{aligned}{A_1} &= \frac{{{F_{\max }}}}{{\sqrt {{{\left({k - m{{\left({\sqrt {\frac{k}{m}} }\right)}^2}}\right)}^2} + b_1^2{{\left({\frac{k}{m}}\right)}^2}}}}\\&= \frac{{{F_{\max }}}}{{\sqrt {\left( {k - m\frac{k}{m}} \right) + \frac{{b_1^2k}}{m}} }}\\&=\frac{{{F_{\max }}}}{{{b_1}\sqrt {\frac{k}{m}}}}\\\end{aligned}$

Now, if the damping constant of the harmonic oscillator becomes $3{b_1}$ .

Substitute for and $3{b_1}$ for in above expression of amplitude.

$\begin{aligned}A' =\frac{{{F_{\max }}}}{{3{b_1}\sqrt {\frac{k}{m}}}}\\=\frac{{{A_1}}}{3}\\\end{aligned}$

Therefore, the new amplitude of the harmonic oscillator for the damping constant of is $\boxed{\frac{{{A_1}}}{3}}$

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Answer Details:

Grade: Senior School

Subject: Physics

Chapter: Oscillations

Keywords: Harmonic oscillator, damping constant, force constant, A1, b1, 3b1, sinusoidally, varying driving force, amplitude, driving angular frequency.

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Asinusoidally varying driving force is applied to a damped harmonic oscillator of force constant k and mass m. if the damping constant has a value b1, the amplitude is a1 when the driving angular frequency equals k/m√.

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