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Fundamentals Of Physics
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Short Answer

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=xmcos(ωt+ϕ)? The vertical axis scale is set by vs=4.0 cm/s.

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

See the step by step solution

Step by Step Solution

Step 1: Stating the given data

Vertical axis scale is set by vs=4.0 cm/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, 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)

Step 3: Calculation of phase constant of the harmonic oscillator

Differentiating equation (ii), we get

v=-vmsinωt+ϕ (iii)

Using the values t=0 s, vm=5 cm/s, v=4 cm/s in equation (iii), we get

4 cm/s=-5 cm/s sinω0+ϕϕ=sin-145=-0.927 rad

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

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