Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University
Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University
9th Edition
ISBN: 9781305372337
Author: Raymond A. Serway | John W. Jewett
Publisher: Cengage Learning
bartleby

Concept explainers

Question
Book Icon
Chapter 16, Problem 61AP

(a)

To determine

The power P(x) carried by this wave at a point.

(a)

Expert Solution
Check Mark

Answer to Problem 61AP

The power P(x) carried by this wave at a point is 12kμω3A02e2bx .

Explanation of Solution

Given Info:

The wave function for string is given as,

y(x,t)=(A0ebx)sin(kxωt)

Formula to calculate the velocity of wave for small segment of string is,

v=dydt

Here,

dy is the distance travel by wave in small segment.

dt is the small time interval.

Substitute (A0ebx)sin(kxωt) for y .

v=d(A0ebx)sin(kxωt)dtv=ω(A0ebx)cos(kxωt)

Assume the mass presents in the small segment dx of string is,

dm=μdx

Here,

μ is the linear mass density.

dx is the small length of the string.

Formula to calculate the kinetic energy for small segment is,

dK=12dmv2 (1)

Here,

dK is the kinetic energy for small segment.

dm is the mass of small segment of string.

Substitute μdx for m and ω(A0ebx)cos(kxωt) for v in equation (1).

dK=12μdx(ω(A0ebx)cos(kxωt))2=12μω2(A0ebx)2cos2(kxωt)dx (2)

Integrate the equation (2) over all the string elements in the wavelength of the waves for total kinetic energy.

dK=0λ12μω2(A0ebx)2cos2(kxωt)dxK=14μω2(A0ebx)2λ

Formula to calculate the potential energy for string in small segment is,

dp=12dmωy2 (3)

Here,

dp is the potential energy present in the small segment of string.

Substitute μdx for m and (A0ebx)sin(kxωt) for y in equation (3).

dp=12μdxω((A0ebx)sin(kxωt))2=12μdxω(A0ebx)2sin2(kxωt) (4)

Integrate the equation (4) over all the string elements in the wavelength of the waves for total potential energy.

dp=0λ12μω(A0ebx)2sin2(kxωt)dxp=14μω2(A0ebx)2λ

Formula to calculate the power P(x) is,

P(x)=K+p (5)

Substitute 14μω2(A0ebx)2λ for p and 14μω2(A0ebx)2λ for K in equation (5).

P(x)=14μω2(A0ebx)2λ+14μω2(A0ebx)2λ=12μω2(A0ebx)2λ

Formula to calculate the wave length is,

λ=ωk

Here,

k is the wave number.

λ is the wave number.

Substitute ωk for λ in equation (5) to get P(x) .

P(x)=12μω2(A0ebx)2ωk=12kμω3A02e2bx

Conclusion:

Therefore, the power P(x) carried by this wave at a point is 12kμω3A02e2bx .

(b)

To determine

The power P(0) carried by this wave at the origin.

(b)

Expert Solution
Check Mark

Answer to Problem 61AP

The power P(0) carried by this wave at the origin is 12kμω3A02 .

Explanation of Solution

Given Info:

The wave function for string is given as,

y(x,t)=(A0ebx)sin(kxωt)

Formula to calculate the power for wave in the string is,

P(x)=12kμω2(A0ebx)2λ (6)

Substitute 0 for x in equation (6).

P(0)=12kμω2(A0eb×0)2λP(0)=12kμω2A02λ (7)

Formula to calculate the wave length is,

λ=ωk

Substitute ωk for λ in equation (7) to get P(0) .

P(0)=12kμω2A02ωk=12kμω3A02

Conclusion:

Therefore, the power P(0) carried by this wave at the origin is 12kμω3A02 .

(c)

To determine

The ratio of P(x)P(0) .

(c)

Expert Solution
Check Mark

Answer to Problem 61AP

The ratio of P(x)P(0) is e2bx .

Explanation of Solution

Given Info:

The wave function for string is given as,

y(x,t)=(A0ebx)sin(kxωt)

The ratio of power given as,

P(x)P(0)

Substitute 12kμω2A02 for P(0) and 12kμω2(A0ebx)2 for P(x) .

P(x)P(0)=12kμω3(A0ebx)212kμω3A02=e2bx

Conclusion:

Therefore, the ratio of P(x)P(0) is e2bx .

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Students have asked these similar questions
19:39 · C Chegg 1 69% ✓ The compound beam is fixed at Ę and supported by rollers at A and B. There are pins at C and D. Take F=1700 lb. (Figure 1) Figure 800 lb ||-5- F 600 lb بتا D E C BO 10 ft 5 ft 4 ft-—— 6 ft — 5 ft- Solved Part A The compound beam is fixed at E and... Hình ảnh có thể có bản quyền. Tìm hiểu thêm Problem A-12 % Chia sẻ kip 800 lb Truy cập ) D Lưu of C 600 lb |-sa+ 10ft 5ft 4ft6ft D E 5 ft- Trying Cheaa Những kết quả này có hữu ích không? There are pins at C and D To F-1200 Egue!) Chegg Solved The compound b... Có Không ☑ ||| Chegg 10 וח
No chatgpt pls will upvote
No chatgpt pls will upvote

Chapter 16 Solutions

Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University

Ch. 16 - Prob. 6OQCh. 16 - Prob. 7OQCh. 16 - Prob. 8OQCh. 16 - Prob. 9OQCh. 16 - Prob. 1CQCh. 16 - Prob. 2CQCh. 16 - Prob. 3CQCh. 16 - Prob. 4CQCh. 16 - Prob. 5CQCh. 16 - Prob. 6CQCh. 16 - Prob. 7CQCh. 16 - Prob. 8CQCh. 16 - Prob. 9CQCh. 16 - A seismographic station receives S and P waves...Ch. 16 - Prob. 2PCh. 16 - Prob. 3PCh. 16 - Two points A and B on the surface of the Earth are...Ch. 16 - Prob. 5PCh. 16 - Prob. 6PCh. 16 - Prob. 7PCh. 16 - Prob. 8PCh. 16 - Prob. 9PCh. 16 - When a particular wire is vibrating with a...Ch. 16 - Prob. 11PCh. 16 - Prob. 12PCh. 16 - Prob. 13PCh. 16 - Prob. 14PCh. 16 - Prob. 15PCh. 16 - Prob. 16PCh. 16 - Prob. 17PCh. 16 - A sinusoidal wave traveling in the negative x...Ch. 16 - Prob. 19PCh. 16 - Prob. 20PCh. 16 - Prob. 21PCh. 16 - Prob. 22PCh. 16 - Prob. 23PCh. 16 - Prob. 24PCh. 16 - An Ethernet cable is 4.00 m long. The cable has a...Ch. 16 - Prob. 26PCh. 16 - Prob. 27PCh. 16 - Prob. 28PCh. 16 - Tension is maintained in a string as in Figure...Ch. 16 - Prob. 30PCh. 16 - Prob. 31PCh. 16 - Prob. 32PCh. 16 - Transverse waves are being generated on a rope...Ch. 16 - Prob. 34PCh. 16 - Prob. 35PCh. 16 - Prob. 36PCh. 16 - Prob. 37PCh. 16 - A horizontal string can transmit a maximum power...Ch. 16 - Prob. 39PCh. 16 - A two-dimensional water wave spreads in circular...Ch. 16 - Prob. 41PCh. 16 - Prob. 42PCh. 16 - Show that the wave function y = eb(x vt) is a...Ch. 16 - Prob. 44PCh. 16 - Prob. 45APCh. 16 - Prob. 46APCh. 16 - Prob. 47APCh. 16 - Prob. 48APCh. 16 - Prob. 49APCh. 16 - Prob. 50APCh. 16 - A transverse wave on a string is described by the...Ch. 16 - A sinusoidal wave in a string is described by the...Ch. 16 - Prob. 53APCh. 16 - Prob. 54APCh. 16 - Prob. 55APCh. 16 - Prob. 56APCh. 16 - Prob. 57APCh. 16 - Prob. 58APCh. 16 - A wire of density is tapered so that its...Ch. 16 - Prob. 60APCh. 16 - Prob. 61APCh. 16 - Prob. 62APCh. 16 - Prob. 63APCh. 16 - Prob. 64CPCh. 16 - Prob. 65CPCh. 16 - Prob. 66CPCh. 16 - Prob. 67CP
Knowledge Booster
Background pattern image
Physics
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.
Similar questions
SEE MORE QUESTIONS
Recommended textbooks for you
Text book image
Physics for Scientists and Engineers: Foundations...
Physics
ISBN:9781133939146
Author:Katz, Debora M.
Publisher:Cengage Learning
Text book image
University Physics Volume 1
Physics
ISBN:9781938168277
Author:William Moebs, Samuel J. Ling, Jeff Sanny
Publisher:OpenStax - Rice University
Text book image
Principles of Physics: A Calculus-Based Text
Physics
ISBN:9781133104261
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
Text book image
Physics for Scientists and Engineers, Technology ...
Physics
ISBN:9781305116399
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
Text book image
Physics for Scientists and Engineers
Physics
ISBN:9781337553278
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
Text book image
Physics for Scientists and Engineers with Modern ...
Physics
ISBN:9781337553292
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning