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ections 17.2, 17.3) Find the work done by F(x, y) = −2yî – 2xĵ in moving a particle along a line segment from (0, 4) to -4) by: ) using the parametrization of the curve to evaluate the work integral. ) showing that the vector field is conservative, finding the potential function f so that Vƒ = F, and using f to calculate the work done by F.

ections 17.2, 17.3) Find the work done by F(x, y) = −2yî – 2xĵ in moving a particle along a line segment from (0, 4) to -4) by: ) using the parametrization of the curve to evaluate the work integral. ) showing that the vector field is conservative, finding the potential function f so that Vƒ = F, and using f to calculate the work done by F.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Topic: Calculating Work Done by a Force Field** In this exercise, we are tasked with finding the work done by the vector field \(\vec{F}(x, y) = -2y\hat{\imath} - 2x\hat{\jmath}\) in moving a particle along a line segment from \((0, 4)\) to \((2, -4)\) using two different methods: ### (a) Parametrization of the Curve - Use the parametrization of the curve to evaluate the work integral. - This involves finding a suitable parameter \(t\) such that the positions of the particle as a function of \(t\) move from the initial point \((0, 4)\) to the final point \((2, -4)\). - Calculate the work integral along the path of the particle by integrating the dot product of \(\vec{F}\) and the derivative of the position vector with respect to \(t\). ### (b) Verifying Conservative Vector Field - Show that the vector field is conservative by verifying if the curl of \(\vec{F}\) is zero. - Find the potential function \(f\) so that \(\nabla f = \vec{F}\). - Use the potential function \(f\) to calculate the work done, leveraging the properties of conservative vector fields where the work done is path-independent and can be calculated using the change in potential function values between the two points. By completing these tasks, we explore how different mathematical techniques can be utilized to solve problems in vector calculus and the concepts of work and energy in physics.
Transcribed Image Text:**Topic: Calculating Work Done by a Force Field** In this exercise, we are tasked with finding the work done by the vector field \(\vec{F}(x, y) = -2y\hat{\imath} - 2x\hat{\jmath}\) in moving a particle along a line segment from \((0, 4)\) to \((2, -4)\) using two different methods: ### (a) Parametrization of the Curve - Use the parametrization of the curve to evaluate the work integral. - This involves finding a suitable parameter \(t\) such that the positions of the particle as a function of \(t\) move from the initial point \((0, 4)\) to the final point \((2, -4)\). - Calculate the work integral along the path of the particle by integrating the dot product of \(\vec{F}\) and the derivative of the position vector with respect to \(t\). ### (b) Verifying Conservative Vector Field - Show that the vector field is conservative by verifying if the curl of \(\vec{F}\) is zero. - Find the potential function \(f\) so that \(\nabla f = \vec{F}\). - Use the potential function \(f\) to calculate the work done, leveraging the properties of conservative vector fields where the work done is path-independent and can be calculated using the change in potential function values between the two points. By completing these tasks, we explore how different mathematical techniques can be utilized to solve problems in vector calculus and the concepts of work and energy in physics.
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