Understanding Work and Line Integrals in Physics: Explained with Formulas

[courtrigrad]

Why does work equal: $$W = \int^{P_{2}}_{P_{1}} F\cos\phi dl = \int_{P_{1}}^{P_{2}} F_{||} dl = \int^{P_{2}}_{P_{1}} F\bullet dl$$?
Thanks

 

[BobG]

Are you asking how they got from the first equation to the last? That comes from the definition for a dot product:
$$cos \phi = \frac{F \cdot dl}{F * dl}$$
If you substitute that for $$cos \phi$$ in the first, the problem simplifies to the last.

 

[marlon]

Why does work equal: $$W = \int^{P_{2}}_{P_{1}} F\cos\phi dl = \int_{P_{1}}^{P_{2}} F_{||} dl = \int^{P_{2}}_{P_{1}} F\bullet dl$$?
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
$$\vec{r} \cdot \vec{F} = ||\vec{r}||*||\vec {F}||* cos( \theta)$$


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