- #1
whatdoido
- 48
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Hello, I picked up a challenging problem (at least to me) and I'm having difficulties.
1. Homework Statement
An object moves in xy-plane from point O = (0; 0) to point A = (1 m; 0) and from there to point B = (1 m; 2 m). All this time when the object moves a force [itex]\vec F[/itex] = ax2[itex]\vec i[/itex] + by[itex]\vec j[/itex] affects the object, where a = 3,0 N/m2 and b = 1,5 N/m2. Calculate the work done by this force in this path.
[itex]W = \int_{x_0}^x F_x(x) \, dx[/itex]
After trying with different methods and failing, I thought the solution should be obtained by using a vector from O to B.
So I created vectors:
[itex]\vec {OA}[/itex] = a(1 m - 0 m)2[itex]\vec i[/itex] + b(0 m - 0 m)2[itex]\vec j[/itex] = a * 1 m2[itex]\vec i[/itex]
[itex]\vec {AB}[/itex] = a(1 m - 1 m)2[itex]\vec i[/itex] + b(2 m - 0 m)2[itex]\vec j[/itex] = b * 2 m[itex]\vec j[/itex]
[itex]\vec {OB}[/itex] = [itex]\vec {OA}[/itex] + [itex]\vec {AB}[/itex] = a * 1 m2[itex]\vec i[/itex] + b * 2 m[itex]\vec j[/itex] = [itex]\vec s[/itex]
After this I'm stuck. I think I need the dot product of [itex]\vec F[/itex] * [itex]\vec s[/itex] ([itex]\vec i[/itex] and [itex]\vec j[/itex] would disappear then) which could be integrated, but by what? dx? Or maybe somehow by dx and dy..? I found some similar problems of line integrals and vector fields, but I don't know apply those ideas to my problem if it is even possible.
There is also one other difficulty. By using the equation [itex]W = \int_{x_0}^x F_x(x) \, dx[/itex], I need to know x0 and x. x0 = 0 m, but do I use the length of |[itex]\vec s[/itex]| as ending point?
Am I completely off track or what? Any hints? Thanks for reading
1. Homework Statement
An object moves in xy-plane from point O = (0; 0) to point A = (1 m; 0) and from there to point B = (1 m; 2 m). All this time when the object moves a force [itex]\vec F[/itex] = ax2[itex]\vec i[/itex] + by[itex]\vec j[/itex] affects the object, where a = 3,0 N/m2 and b = 1,5 N/m2. Calculate the work done by this force in this path.
Homework Equations
[itex]W = \int_{x_0}^x F_x(x) \, dx[/itex]
The Attempt at a Solution
After trying with different methods and failing, I thought the solution should be obtained by using a vector from O to B.
So I created vectors:
[itex]\vec {OA}[/itex] = a(1 m - 0 m)2[itex]\vec i[/itex] + b(0 m - 0 m)2[itex]\vec j[/itex] = a * 1 m2[itex]\vec i[/itex]
[itex]\vec {AB}[/itex] = a(1 m - 1 m)2[itex]\vec i[/itex] + b(2 m - 0 m)2[itex]\vec j[/itex] = b * 2 m[itex]\vec j[/itex]
[itex]\vec {OB}[/itex] = [itex]\vec {OA}[/itex] + [itex]\vec {AB}[/itex] = a * 1 m2[itex]\vec i[/itex] + b * 2 m[itex]\vec j[/itex] = [itex]\vec s[/itex]
After this I'm stuck. I think I need the dot product of [itex]\vec F[/itex] * [itex]\vec s[/itex] ([itex]\vec i[/itex] and [itex]\vec j[/itex] would disappear then) which could be integrated, but by what? dx? Or maybe somehow by dx and dy..? I found some similar problems of line integrals and vector fields, but I don't know apply those ideas to my problem if it is even possible.
There is also one other difficulty. By using the equation [itex]W = \int_{x_0}^x F_x(x) \, dx[/itex], I need to know x0 and x. x0 = 0 m, but do I use the length of |[itex]\vec s[/itex]| as ending point?
Am I completely off track or what? Any hints? Thanks for reading