Physics for Scientists and Engineers: Foundations and Connections
Physics for Scientists and Engineers: Foundations and Connections
1st Edition
ISBN: 9781133939146
Author: Katz, Debora M.
Publisher: Cengage Learning
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Chapter 16, Problem 82PQ
To determine

To prove the relation for system.

Expert Solution & Answer
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Answer to Problem 82PQ

The solution to the equation is Fmaxcos(ωdrvt)=m(d2y/dt2)+b(dy/dt)+ky_.

Explanation of Solution

Write the expression for the given solution of the equation.

  y(t)=Adrvcos(ωdrvt+ϕdrv)                                                                                  (I)

Here, y(t) is the instantaneous displacement, Adrv is the amplitude, ωdrv is the angular frequency, t is the time and ϕdrv is the phase angle.

Write the expression for the equation.

  Fmaxcos(ωdrvt)=m(d2y/dt2)+b(dy/dt)+ky                                                 (II)

Here, Fmax is the maximum force, (d2y/dt2) is the second derivative and (dy/dt) is the first derivative.

Write the expression for the angular frequency.

  ωdrv=km                                                                                                         (III)

Here, k is wave vector and m is mass.

Conclusion:

Taking first and second derivative from equation (I).

  dydt=ωdrvAdrvsin(ωdrvt+ϕdrv)                                                                        (IV)

  d2ydt2=ωdrv2Adrvcos(ωdrvt+ϕdrv)                                                                       (V)

Substitute ωdrvAdrvsin(ωdrvt+ϕdrv) for dydt, ωdrv2Adrvcos(ωdrvt+ϕdrv) for d2ydt2 and Adrvcos(ωdrvt+ϕdrv) for y(t) in Equation (II) to find Fmaxcos(ωdrvt).

  Fmaxcos(ωdrvt)=(m[ωdrv2Adrvcos(ωdrvt+ϕdrv)]+b[ωdrvAdrvsin(ωdrvt+ϕdrv)]+k[Adrvcos(ωdrvt+ϕdrv)])                             (VI)

Using trigonometric identity and comparing both side the components of sine and cosine terms.

  Fmax=Adrv[(kmωdrv2)cos(ϕdrv)bωdrvsinϕdrv]                                             (VII)

  0=Adrv[(kmωdrv2)sin(ϕdrv)bωdrvcosϕdrv]                                              (VIII)

Write the expression for the trigonometric identities by using equation (VIII).

  tan(ϕdrv)=bωdrvkmωdrv2                                                                                       (IX)

  sin(ϕdrv)=bωdrv(bωdrv)2+(kmωdrv2)2                                                                 (X)

  cos(ϕdrv)=(kmωdrv2)(bωdrv)2+(kmωdrv2)2                                                               (XI)

Substitute bωdrv(bωdrv)2+(kmωdrv2)2 for sin(ϕdrv) and (kmωdrv2)(bωdrv)2+(kmωdrv2)2 for cos(ϕdrv) in equation (VII) to find Adrv.

  Fmax=Adrv[(kmωdrv2)[(kmωdrv2)(bωdrv)2+(kmωdrv2)2]bωdrv[bωdrv(bωdrv)2+(kmωdrv2)2]]=Adrv(bωdrv)2+m2(ω2ωdrv2)Adrv=Fmax(bωdrv)2+m2(ω2ωdrv2)

Thus, the solution is satisfying both side of the equation.

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Chapter 16 Solutions

Physics for Scientists and Engineers: Foundations and Connections

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