Vector Calculus: Calculate Work via Line Integral from (-1,2) to (1,3)

[mikeyrichster]

Homework Statement

A force is applied to a particle, defined by:

F(x,y)= (y^2, 2xy) << This is a verticle bracket with the y^2 ontop of the 2xy

The path of the particle is straight. The particle moves from (-1,2) to (1,3)

i) Calculate the work that the force F does as the particle moves along the path C by evaluating the appropreate line integral directly


I don't know where to start with this. I know that work: w = fd so to get the distance i must do some integration, but how do you integrate a vector like that?


Can somone please help!? - would be much appreciated


Many thanks

 

[LCKurtz]

Homework Statement

A force is applied to a particle, defined by:

F(x,y)= (y^2, 2xy) << This is a verticle bracket with the y^2 ontop of the 2xy

What do you mean by that? Force is a vector, not a fraction?

The path of the particle is straight. The particle moves from (-1,2) to (1,3)

i) Calculate the work that the force F does as the particle moves along the path C by evaluating the appropreate line integral directly


I don't know where to start with this. I know that work: w = fd so to get the distance i must do some integration, but how do you integrate a vector like that?

w = fd only works for constant forces along straight lines. Perhaps you mean

$$W = \int_C \vec F \cdot d\vec R$$

 

[mikeyrichster]

Hi there! thanks for your help

sorry about the fraction thing.. what i ment was that it is a vector but written vertically not horizontally... so still a vector but not (y^2, 2xy) .. I am damn useless at LaTeX so i can't make it with a large bracket and the y^2 at the top and the 2xy at the bottom.. hope that makes sense

Also i was only making a guess with the w=fd thing, I am sure your expression is what I am looking for as it uses vectors.

What would be the next logical thing to do with it?

Thanks again

 

[vela]

Click on the formula below to see the LaTeX for writing the column vector.

$$\textbf{F} = \begin{pmatrix} y^2 \\ 2xy \end{pmatrix}$$

 

[LCKurtz]

What would be the next logical thing to do with it?

Thanks again

Parameterize the line as

$$\vec R(t) = \langle x(t), y(t), z(t)\rangle$$

and use

$$\int_C \vec F \cdot d\vec R = \int_a^b \vec F(t)\cdot \frac {d\vec R}{dt}\, dt$$