Energy in each cycle of transverse wave?

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SUMMARY

The discussion focuses on calculating the average power and energy in a transverse wave using the formula Pavg = (1/2) μvω2A2. The user calculates the average power as 15.1 W with parameters including a linear mass density (μ) of 0.075 kg/m, wave speed (v) of 10/3 m/s, amplitude (A) of 0.35 m, and angular frequency (ω) of 10π rad/s. The user expresses uncertainty about constructing integrals for kinetic and potential energy over one period and seeks guidance on this calculation.

PREREQUISITES
  • Understanding of transverse wave mechanics
  • Familiarity with the concepts of kinetic and potential energy in wave motion
  • Knowledge of integral calculus for energy calculations
  • Proficiency in using wave equations and parameters such as amplitude and angular frequency
NEXT STEPS
  • Study the derivation of energy equations for transverse waves
  • Learn how to set up and evaluate integrals for kinetic and potential energy in wave systems
  • Explore the relationship between wave parameters and energy transfer
  • Investigate the application of Fourier series in analyzing waveforms
USEFUL FOR

Students in physics, particularly those studying wave mechanics, as well as educators and tutors looking to enhance their understanding of energy calculations in transverse waves.

Sho Kano
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Homework Statement


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Homework Equations

The Attempt at a Solution


a. ##{ P }_{ avg }=\frac { 1 }{ 2 } \mu v{ \omega }^{ 2 }{ A }^{ 2 }=\frac { 1 }{ 2 } 0.075\frac { 10 }{ 3 } { 0.35 }^{ 2 }{ (10\pi ) }^{ 2 }=15.1W##
b. Not sure how to calculate that... I'm guessing it'd be some integral over 1 period for the kinetic energy involved in the up down movement of the string plus some integral for the potential energy involved with the string's tension over 1 wavelength. How do I construct these integrals?
 
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Never mind! ##\omega =10\pi =\frac { 2\pi }{ T } \\ T=\frac { 1 }{ 5 } s\\ PT=\frac { J }{ s } s=J=15\frac { 1 }{ 5 } =3J##
 
I know I can do this using an integral. But how?
 

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