How to Calculate Work on an Object with Varying Position-dependent Force?

[mcmuffinpopper]

Homework Statement

An object has several forces acting on it. One of these forces is F=alpha xy ihat, a force in the x-direction whose magnitude depends on the position of the object, with alpha (not zero). Calculate the work done on the object by this force for the following displacements of the object.

1.The object starts at point x=x₀, y=0 and moves in the y-direction to the point x=x₀ and y=y₁

2. The object starts at the origin (x,y)=(0,0) and moves on the line y = 1.5x to the point x = x₂, y = y₂.

Homework Equations

W = F*s

The Attempt at a Solution

I don't really know how to solve this as I just started with vectors and I still have problems understanding them. I assumed for the 1st one that there was no work done because x stays zero as it is moving parallell to the x-axis while staying 0, so F*s= 0. I don't know how to solve the second one.

 

[haruspex]

1.The object starts at the point x = 0, and moves in the y-direction to the point x = x_0 and y = y_1.

Are you sure you have stated that correctly? x=0 is not a point. If it starts at point where x=0 and moves in the y direction then, as you say, x will still be 0, so it seems odd to say it finishes at (x₀,y₁). But if that is what it does then yes, the force is always zero so the work is zero.

For the second part, you need to write your W=Fs equation in vectors, and allowing for the displacement vector to be infinitesimal. Do you know what I mean by that, and how to do it?

 

[mcmuffinpopper]

Are you sure you have stated that correctly? x=0 is not a point. If it starts at point where x=0 and moves in the y direction then, as you say, x will still be 0, so it seems odd to say it finishes at (x₀,y₁). But if that is what it does then yes, the force is always zero so the work is zero.

For the second part, you need to write your W=Fs equation in vectors, and allowing for the displacement vector to be infinitesimal. Do you know what I mean by that, and how to do it?

I'm sorry I have indeed stated that incorrectly. It is:
1. The object starts at point x=x₀, y=0 and moves in the y-direction to the point x=x₀ and y=y₁

And I think you mean that I need to integrate W=F*s? I know that if you want to write it in vectors you should write it with arrows on top of them, I don't really know how I should proceed further, so no I don't know how to do it.

 

[haruspex]

I'm sorry I have indeed stated that incorrectly. It is:
1. The object starts at point x=x₀, y=0 and moves in the y-direction to the point x=x₀ and y=y₁

And I think you mean that I need to integrate W=F*s? I know that if you want to write it in vectors you should write it with arrows on top of them, I don't really know how I should proceed further, so no I don't know how to do it.

Ok, the vectorial calculus form is $\int \vec F.\vec {ds}$ where the dot is the dot product of the vectors. $\vec {ds}$ is the vector (dx,dy), or in the hat notation $\hat i dx+\hat j dy$. Plug in the vector form of the force you are given, and see where that takes you.

 

[mcmuffinpopper]

Is this correct?

 

[haruspex]

That image took ages to load and came out sideways. Parts are also hard ro read. Please take the trouble to type in your algebra.
It was ok down to where you wrote out the force as a vector in hat notation You need to do the same for $\vec {ds}$ and execute the dot product.
Edit: if you cannot write LaTeX, don't worry about the arrows and hats.
It's late here... off to bed.