Chapter 15: 12P (page 436)

What is the phase constant for the harmonic oscillator with the velocity function $v (t)$ given in Figure if the position function $x (t)$ has the form $x = x_{m} \cos (ωt + ϕ)$ ? The vertical axis scale is set by $v_{s} = 4.0 cm / s$ .

Short Answer

Expert verified

The phase constant for the harmonic oscillator with the velocity function $v (t)$ , if the position function has the form $x (t) = x_{m} \cos (ωt + ϕ)$ , is $- 0.927 rad$ .

Step by step solution

Stating the given data

Vertical axis scale is set by $v_{s} = 4.0 cm / s$ .

Understanding the concept of displacement equation

Using the formula of velocity, we can find the phase constant for the harmonic oscillator from theposition function of form, $x (t) = x_{m} \cos (ωt + ϕ)$ .

Formulae:

The velocity of a body in motion

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

Equation of displacement of the motion

$x (t) = x_{m} \cos (ωt + ϕ)$ (ii)

Calculation of phase constant of the harmonic oscillator

Differentiating equation (ii), we get

$v = - v_{m} \sin (ωt + ϕ)$ (iii)

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

$\begin{array}{rcl} 4 cm / s & = & - 5 cm / s \sin (ω (0) + ϕ) \\ ϕ & = & \sin^{- 1} (\frac{4}{5}) \\ = & - 0.927 rad \end{array}$

Therefore, the phase constant for the harmonic oscillator with the velocity function $v (t)$ , if the position function has the form $x (t) = x_{m} \cos (ωt + ϕ)$ , is $- 0.927 rad$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

What is the phase constant for the harmonic oscillator with the velocity function $v (t)$ given in Figure if the position function $x (t)$ has the form $x = x_{m} \cos (ωt + ϕ)$ ? The vertical axis scale is set by $v_{s} = 4.0 cm / s$ .

Short Answer

Step by step solution

Stating the given data

Understanding the concept of displacement equation

Calculation of phase constant of the harmonic oscillator

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Kinematics Physics

Engineering Physics

Particle Model of Matter

Torque and Rotational Motion

Wave Optics

Rotational Dynamics

Study anywhere. Anytime. Across all devices.

What is the phase constant for the harmonic oscillator with the velocity functionv(t)given in Figure if the position function x(t)has the form x=xmcos(ωt+ϕ)? The vertical axis scale is set by vs=4.0cm/s.

Short Answer

Step by step solution

Stating the given data

Understanding the concept of displacement equation

Calculation of phase constant of the harmonic oscillator

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Kinematics Physics

Engineering Physics

Particle Model of Matter

Torque and Rotational Motion

Wave Optics

Rotational Dynamics

Study anywhere. Anytime. Across all devices.

What is the phase constant for the harmonic oscillator with the velocity function $v (t)$ given in Figure if the position function $x (t)$ has the form $x = x_{m} \cos (ωt + ϕ)$ ? The vertical axis scale is set by $v_{s} = 4.0 cm / s$ .