For F(x, y) = 2i + 2j the work done by the force field on a particle moving along an arbitrary smooth curve from P(0, 0) to Q(8, 5) is
A) 10 Nm
B) 16 Nm
C) 26 Nm
D) 0 Nm
E) 30 Nm

Question

Quizplus · Accepted Answer

The work done by a force field along a curve C from P to Q is the line integral:

W = ∫C F ⋅ dr

where F is the force field and dr is the differential displacement along the curve C.

In this case, F(x,y) = 2i + 2j, so F ⋅ dr = 2dx + 2dy. We can parameterize the curve C as:

x = 8t, y = 5t, 0 ≤ t ≤ 1

Then, dr = (8dt)i + (5dt)j, and the line integral becomes:

W = ∫C F ⋅ dr = ∫0^1 (2dx + 2dy) = ∫0^1 (2(8dt) + 2(5dt)) = ∫0^1 26dt = 26

Therefore, the work done by the force field on a particle moving from P(0,0) to Q(8,5) along an arbitrary smooth curve is 26 Nm.

For F(x, Y) = 2i + 2j the Work Done