Question

An object travels from (4,0) to (0,4) along path defined by x

2
+y
2
=16 under the applied force is
F
=(
x
2
+y
2

1
​
)i+(
x
2
+y
2

4
​
)j. Find the work done using the differential form of the work integral. Clearly show the work integral.

Answer 1

The work done on an object moving from position (x1, y1) to (x2, y2) under an applied force F is given by the work integral:

W = ∫ F ⋅ dr

where dr is the displacement vector from position (x1, y1) to (x2, y2).

For the given scenario, the applied force is F = (x^2 + y^2)i + (x^2 + y^2)^(1/2)j, and the displacement is from (4,0) to (0,4), which is given by dr = (-4, 4)i + (0, 4)j.

So, the work done is:

W = ∫ F ⋅ dr = ∫ ((x^2 + y^2)i + (x^2 + y^2)^(1/2)j) ⋅ ((-4)i + (4)j)

= ∫ -4x^2 - 4y^2 + 4(x^2 + y^2)^(1/2) dx

This work integral cannot be solved in closed form and will require numerical methods to evaluate the definite integral.