Determine o trabalho realizado pelo campo de força F(x, y) = (x sen y, y em uma partícula que se move sobre a parábola y = x^2 de (-1, 1) a 2,4.

Passo 1

Beleza, vamos entender o que o exercício está pedindo. Temos um campo de força e uma partícula que se move ao longo da parábola , indo do ponto até . Nosso objetivo é calcular o trabalho realizado por esse campo de força durante esse movimento.
Para resolver isso, vamos precisar usar conceitos de cálculo vetorial, especificamente o cálculo do trabalho realizado por um campo de força ao longo de uma trajetória. Vamos precisar parametrizar a curva e integrar o produto escalar entre o campo de força e o diferencial de deslocamento ao longo dessa curva.

Passo 2

Primeiro, vamos parametrizar a trajetória. Sabemos que a partícula se move ao longo da parábola . Podemos usar x como nosso parâmetro. Portanto, podemos expressar a posição da partícula como:
y = sen(x^2)
Aqui, y = sen(x^2) varia de x a -1, que são as coordenadas iniciais e finais.

Passo 3

Agora, precisamos encontrar o vetor diferencial de deslocamento . Para isso, derivamos em relação a x:
y = sen(x^2)
Então,
y = sen(x^2)

Passo 4

O trabalho W realizado pelo campo de força ao longo da trajetória é dado pela integral do produto escalar entre F e de a x = 2:
y = sen(x^2)
Substituindo y = sen(x^2) e , temos:
Simplificando a expressão dentro da integral:
y = sen(x^2)

Passo 5

Vamos resolver a integral separadamente. Primeiro, a parte que envolve :
Para , fazemos uma substituição:
  • Seja
  • Então, ou
A integral se torna:
y = sen(x^2)
Agora, a segunda parte da integral, y = sen(x^2):

Passo 6

Agora, combinando as duas integrais e avaliando de a x = 2:
y = sen(x^2)
Calculando em y = sen(x^2):
x = 2
E em y = sen(x^2):
y = sen(x^2)
Subtraindo:
Simplificando:
y = sen(x^2)
y = sen(x^2)

Resposta

O trabalho realizado pelo campo de força é:
y = sen(x^2)

Exercícios relacionados

Calcule as integrais de linha:
© . dr, onde e .
Enuncie e escreva a demonstração do Teorema Fundamental das Integrais de Linha.
Calcule as integrais de linha:
(h) . dr, em que F(x, y, z) = (e^y, x e^y, (z+1) e^z) e .
<html><head></head><body><div>A força exercida pela carga elétrica colocada na origem sobre uma partícula carregada com vetor posição <span class="math-inline ">
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<div><span class="math-block ">
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<div>onde <span class="math-inline ">
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(a) Mostre que o campo <span class="math-inline " data-overflow=“visible”>
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