Find work done using force in two dimensions

[shanepitts]

Homework Statement

Homework Equations

∫F⋅dr=W

The Attempt at a Solution

 

[Chestermiller]

The problem statement asks you to find the amount of work done along each of the three legs of the path, and then to add them up. The first leg of the path is particularly tricky because both x and y are changing simultaneously along this leg. Let s be the distance along this leg of the path measured from the initial point at the origin. The total length of this leg is $\sqrt{4^2+2^2}=2\sqrt{5}$. What are x and y expressed as functions of s along this leg of the path?

Chet

 

[SteamKing]

Homework Statement

View attachment 89256

Homework Equations

∫F⋅dr=W

The Attempt at a Solution

View attachment 89257

If r² = x² + y² + z², then what is dr? In general, dr ≠ dx + dy + dz

I believe a better definition for the work performed is W = ∫_C F ⋅ ds, where the path C is the triangle specified in the OP.

 

[shanepitts]

Thank you,

But how can I calculate the work down on each leg of the triangle?

Shall I integrate ∫_C F⋅dr along each line, using the limits as the length of each leg, and then sum them up?

 

[SteamKing]

Thank you,

But how can I calculate the work down on each leg of the triangle?

Shall I integrate ∫C F⋅dr along each line, using the limits as the length of each leg, and then sum them up?

Well, generally finding the work performed involves evaluating a path integral, which is a little different from evaluating a "regular" definite integral.

For example, two of the legs of the triangle C are aligned with the x and y coordinate axes, making for some simplifications in evaluating ∫ F ⋅ ds on these paths.

The link below discusses and illustrates methods for evaluating path integrals:

http://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsVectorFields.aspx

 

[Chestermiller]

If f is the fraction of the distance between (0,0,0) and (4,-2,0) along the path, then x = 4f, y = -2 f, and z = 0f. In terms of the fractional distance f, what is the force vector F? In terms of the fractional distance f, what is the differential vector dr along the path? What is F dotted with dr?

Chet