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Chapter 15: 12P (page 436)

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

Expert verified

The phase constant for the harmonic oscillator with the velocity function vt, if the position function has the form xt=xmcosωt+ϕ, is -0.927rad.

Step by step solution

01

Stating the given data

Vertical axis scale is set by vs=4.0cm/s.

02

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)=xmcos(ωt+ϕ).

Formulae:

The velocity of a body in motion

v=dxdt (i)

Equation of displacement of the motion

xt=xmcosωt+ϕ (ii)

03

Calculation of phase constant of the harmonic oscillator

Differentiating equation (ii), we get

v=-vmsinωt+ϕ (iii)

Using the valuest=0s,vm=5cm/s,v=4cm/sin equation (iii), we get

4cm/s=-5cm/ssinω0+ϕϕ=sin-145=-0.927rad

Therefore, the phase constant for the harmonic oscillator with the velocity function vt, if the position function has the form xt=xmcosωt+ϕ, is -0.927rad.

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