Work done by a force with a given equation

[trogdor5]

Homework Statement

A force acts on a particle and is given by the following expression:
F(x,y)=2x^3 y^2 i+3xy^3 j
What is the work done by this force in moving the particle from a position (x,y) = (0,0) to (4,2) along the path given by the curve y=√x ?

Homework Equations

I know the work is the integral of the Force

The Attempt at a Solution

Honestly, very confused. I don't know how to deal with the i and j values and I have no clue how to handle the work done along the path of a different curve. Any help is appreciated!

 

[haruspex]

Yes, work is the integral of the force wrt distance, each being a vector. So the integrand and differential element look like F.dp, the dot being the scalar product of the two vectors. (I've used p for position vector to avoid confusion with the x scalar in the question.) So you need to express the vector distance element along the path in i and j coordinates. When the particle moves a distance dy in the y direction, how far does it go in the x direction?

 

[trogdor5]

I kind of understand what you're saying, but not really. I don't even think I understand what the question is asking to be honest. I don't understand how there can be x and y but also i and j coordinate systems.

 

[haruspex]

The i and j are unit vectors representing the x and y directions respectively. X and y themselves are magnitudes of position in those directions. I.e. the position vector of the particle at time t is x(t)i + y(t)j.

 

[trogdor5]

I'm honestly trying to do it but since my math isn't that strong and I've never seen a problem like this I'm just having problems. Can you do the first step for me or walk me through a bit more step-by-step? I'm just having extreme difficulty

 

[haruspex]

y² = x; 2ydy = dx
A small step in position = idx + jdy = 2iydy + jdy
You know the vector form of F. Take the dot product of this with position differential above.