Work/Line Integrals

  • #1
1,235
1
Why does work equal: [tex] W = \int^{P_{2}}_{P_{1}} F\cos\phi dl = \int_{P_{1}}^{P_{2}} F_{||} dl = \int^{P_{2}}_{P_{1}} F\bullet dl [/tex]?
Thanks
 

Answers and Replies

  • #2
BobG
Science Advisor
Homework Helper
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Are you asking how they got from the first equation to the last? That comes from the definition for a dot product:
[tex]cos \phi = \frac{F \cdot dl}{F * dl}[/tex]
If you substitute that for [tex]cos \phi[/tex] in the first, the problem simplifies to the last.
 
  • #3
3,768
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plugpoint said:
Why does work equal: [tex] W = \int^{P_{2}}_{P_{1}} F\cos\phi dl = \int_{P_{1}}^{P_{2}} F_{||} dl = \int^{P_{2}}_{P_{1}} F\bullet dl [/tex]?
Thanks
Work is defined as the line integral over some path L of the scalar product of r and F. Now a scalar product of two vectors r and F is equal to
[tex]\vec{r} \cdot \vec{F} = ||\vec{r}||*||\vec {F}||* cos( \theta)[/tex]


The product of F with the cosine of the angle theta between r and F denotes the component of F parallel with vector r.


marlon
 

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