How the wave equation relates to Newton's Second Law of Motion

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SUMMARY

The discussion centers on the relationship between the wave equation and Newton's Second Law of Motion, specifically in the context of a small string segment experiencing different tension forces at each end. Participants highlight the challenge of reconciling the definition of force (F) as the tension in the rope, which is aligned with the string, against the vertical acceleration (a) of an element of the string. The conclusion emphasizes the necessity of understanding how to apply Newton's Second Law in orthogonal directions when analyzing wave motion in strings.

PREREQUISITES
  • Understanding of Newton's Second Law of Motion
  • Familiarity with wave equations in physics
  • Knowledge of vector components in physics
  • Basic principles of tension in strings
NEXT STEPS
  • Study the derivation of the wave equation from Newton's Second Law
  • Explore vector decomposition in physics problems
  • Learn about tension forces in strings and their effects on wave propagation
  • Investigate the relationship between acceleration and force in multi-dimensional systems
USEFUL FOR

Students of physics, educators teaching mechanics, and anyone interested in the mathematical modeling of wave phenomena in strings.

Rich667
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Homework Statement


Open Question 3.bmp

Homework Equations




The Attempt at a Solution



Open Answer 3.bmp

Any help with this would be greatly appreciated
 

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consider a small sting segment with (different) tension force vectors acting on each end, and write down Newtons second law in two orthogonal directions
 
I think the problem lies in the definition of F and acceleration a. In answer 3.bmp the F is defined as the tension in the rope, which is coaxial to the string, while the second derivative of y , the acceleration of an element with coordinate (x,y), is vertical to the string. Surely you can't say F=ma with F and a pointing to 2 different directions.
 

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