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### Fundamentals Of Physics

Book edition 10th Edition
Author(s) David Halliday
Pages 1328 pages
ISBN 9781118230718

# What is the phase constant for the harmonic oscillator with the velocity function ${\mathbf{v}}\left(t\right)$ given in Figure if the position function ${\mathbf{x}}\left(t\right)$ has the form ${\mathbf{x}}{\mathbf{=}}{{\mathbf{x}}}_{{\mathbf{m}}}{\mathbf{cos}}\left(\mathrm{\omega t}+\varphi \right)$? The vertical axis scale is set by ${{\mathbf{v}}}_{{\mathbf{s}}}{\mathbf{=}}{\mathbf{4}}{\mathbf{.}}{\mathbf{0}}{\mathbf{}}{\mathbf{cm}}{\mathbf{/}}{\mathbf{s}}$.

The phase constant for the harmonic oscillator with the velocity function $\mathrm{v}\left(\mathrm{t}\right)$ , if the position function has the form $\mathrm{x}\left(\mathrm{t}\right)={\mathrm{x}}_{\mathrm{m}}\mathrm{cos}\left(\mathrm{\omega t}+\mathrm{\varphi }\right)$, is $-0.927\mathrm{rad}$.

See the step by step solution

## Step 1: Stating the given data

Vertical axis scale is set by ${\mathrm{v}}_{\mathrm{s}}=4.0\mathrm{cm}/\mathrm{s}$.

## Step 2: Understanding the concept of displacement equation

Using the formula of velocity, we can find the phase constant for the harmonic oscillator from the position function of form, ${\mathbf{x}}\left(t\right){\mathbf{=}}{{\mathbf{x}}}_{{\mathbf{m}}}{\mathbf{cos}}\left(\mathrm{\omega t}+\varphi \right)$.

Formulae:

The velocity of a body in motion

$\mathrm{v}=\frac{d\mathrm{x}}{d\mathrm{t}}$ (i)

Equation of displacement of the motion

$\mathrm{x}\left(\mathrm{t}\right)={\mathrm{x}}_{\mathrm{m}}\mathrm{cos}\left(\mathrm{\omega t}+\mathrm{\varphi }\right)$ (ii)

## Step 3: Calculation of phase constant of the harmonic oscillator

Differentiating equation (ii), we get

$\mathrm{v}=-{\mathrm{v}}_{\mathrm{m}}\mathrm{sin}\left(\mathrm{\omega t}+\mathrm{\varphi }\right)$ (iii)

Using the values $\mathrm{t}=0\mathrm{s},{\mathrm{v}}_{\mathrm{m}}=5\mathrm{cm}/\mathrm{s},\mathrm{v}=4\mathrm{cm}/\mathrm{s}$ in equation (iii), we get

$\begin{array}{rcl}4\mathrm{cm}/\mathrm{s}& =& -5\mathrm{cm}/\mathrm{s}\mathrm{sin}\left(\mathrm{\omega }\left(0\right)+\mathrm{\varphi }\right)\\ \mathrm{\varphi }& =& {\mathrm{sin}}^{-1}\left(\frac{4}{5}\right)\\ & =& -0.927\mathrm{rad}\end{array}$

Therefore, the phase constant for the harmonic oscillator with the velocity function $\mathrm{v}\left(\mathrm{t}\right)$, if the position function has the form $\mathrm{x}\left(\mathrm{t}\right)={\mathrm{x}}_{\mathrm{m}}\mathrm{cos}\left(\mathrm{\omega t}+\mathrm{\varphi }\right)$, is $-0.927\mathrm{rad}$.