Question

A sinusoidally 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−−−−√.

Answer 1

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, $A$ is the amplitude of the harmonic oscillator, ${F_{\max }}$ is the maximum force, $k$ 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 $A$,  ${b_1}$ for $b$ 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 $A'$ for $A$ and $3{b_1}$ for $b$ 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 $3b$ 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.

Answer 2

We are given
damped harmonic oscillation force = k
mass = m
damping constant = b1
amplitude = a1
driving angular frequency = k/m

I think we asked for the amplitude of the force at different damping constant
The formula to use is
A = (F/ (√(k - m w²)² + (b² w²)) 
Simply substitute and solve for A in terms of a1