Calculating Work with Vector Components and Variable Force

[Chansu]

Hello all!

I usually don't like to ask for help... But this is the first week of courses and I'm already stumped on a homework question...

1. Homework Statement
So the question states: Find the work by the force F = x^i + xy^j. If the object starts from the origin (0,0), moves along the x-axis to the point (1,0) then moves in the y direction to point (1,2). Find the work if instead, the object moves diagonally from the origin to (1,2).

F = x^i + xy^j

2. Homework Equations

W = ∫ F ⋅ ds

rise = Slope
run

3. The Attempt at a Solution

Part 1:
First I found the work from the origin to (1,0):

W₁ = ∫ (x^i + xy^j) ⋅ (dx^i + 0^j)

= ∫ x dx + 0

= ½ x² (from 0 to 1)

= ½ (1)² - ½ (0)²

= ½



Then from (1,0) to (1,2):

W₂ = ∫ (x^i + xy^j) ⋅ (0^i + dy^j)

= ∫ (xy) dy + 0

This is where things get hard for me... I've talked to my professor a couple times but I forgot what he said about this part. He said to either solve for x in terms of y and slope; or to set the x to 1 since it is already at (1,0) and does not move on the x-axis. So I'll just use the one i went with on my paper (x = y/2)

= ∫ (y²/2) dy + 0

= ½ (⅓ y³ (from 0 to 2)

= ½ (⅓ 2³ - ½ (⅓ 0³

= 4/3

W_total = ½ + 4/3

For part 2:
From origin (0,0) to final position (1,2)

W = ∫ (x^i + xy^j) ⋅ (dx^i + dy^j)

= ∫ x dx + ∫ (xy) dy

= ½ x² (from 0 to 1) + ½ (⅓ y³ (from 0 to 2)

= ½ + ¾Is this correct? What should I change? Thank you for the help!

 

[NFuller]

This is where things get hard for me... I've talked to my professor a couple times but I forgot what he said about this part. He said to either solve for x in terms of y and slope; or to set the x to 1 since it is already at (1,0) and does not move on the x-axis. So I'll just use the one i went with on my paper (x = y/2)

You should just set $x=1$ since that is the equation of a line pointing straight up. What you did in the second part looks right though.

 

[Chansu]

You should just set $x=1$ since that is the equation of a line pointing straight up. What you did in the second part looks right though.

Thank you!

I've mainly had trouble deciding which way to proceed because I've been comparing the two answers I get (the total from part 1 and the answer from part two). I doubted using x = 1 because then the work for part 1 wouldn't be the same as the work for part 2... I recently came to the conclusion that that is not a fluke, right? If the object were to move diagonally, the force in the y (^j) direction would rely on BOTH x and y thus causing a different work value when compared to taking the route in part 1 (along x-axis then along y-axis)?

 

[ehild]

Thank you!

I've mainly had trouble deciding which way to proceed because I've been comparing the two answers I get (the total from part 1 and the answer from part two). I doubted using x = 1 because then the work for part 1 wouldn't be the same as the work for part 2... I recently came to the conclusion that that is not a fluke, right? If the object were to move diagonally, the force in the y (^j) direction would rely on BOTH x and y thus causing a different work value when compared to taking the route in part 1 (along x-axis then along y-axis)?

You get the same work along any different routes between two points if the force is conservative. The force in the problem is not.

 

[Chansu]

You get the same work along any different routes between two points if the force is conservative. The force in the problem is not.

Alright, thank you guys!