Get the solution to "Given the force field F, find the work..."

kdgg0909gn

Answered question

2021-11-20

Given the force field F, find the work required to move an object on the given oriented curve.

F = ⟨ y, x ⟩

on the parabola

y = 2 x^{2} o m P S K (0, 0)

(2, 8)

Supoilign1964

Beginner2021-11-21Added 19 answers

F = ⟨ y, x ⟩

parabola

y = 2 x^{2}

(0, 0)

(2, 8)

parameterizing

r (t) = ⟨ t, 2 t^{2} ⟩, 0 \leq t \leq 2

therefore

r^{'} (t) = ⟨ 1, 4 t ⟩

the work is

w = \int_{0}^{2} ⟨ 2 t^{2}, t ⟩ \cdot ⟨ 1, 4 t ⟩ d t

simplify

w = \int_{0}^{2} (2 t^{2} + 4 t^{2}) d t = \int_{0}^{2} 6 t^{2} d t

w = {[2 t^{3}]}_{{0}}^{2} = 2 {(2)}^{3} = 16

Forneadil

Beginner2021-11-22Added 18 answers

Step 1
The force field is

F (x, y) = (y, x)

.
The curve is

r (t) = (t, 2 t^{2}), \forall t \in [0, 2] \Rightarrow r^{'} (t) = (1, 4 t) .

Thus, the integral is

\int_{C} F \cdot T d s = \int_{0}^{2} F (r (t)) r^{'} (t) d t

= \int_{0}^{2} (2 t^{2}, t) \cdot (1, 4 t) d t

\int_{0}^{2} 6 t^{2} d t

= 16

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