4. Find the work done by the force field F(x, y) = (2y, x) acting on an object as it moves along the parabola y 2x² from (3, 18) to (0, 0) is given by: [2ydx + xdy. C
4. Find the work done by the force field F(x, y) = (2y, x) acting on an object as it moves along the parabola y 2x² from (3, 18) to (0, 0) is given by: [2ydx + xdy. C
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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![### Problems on Work Done by Force Fields
**Problem 4:**
Find the work done by the force field \( \vec{F}(x, y) = \langle 2y, x \rangle \) acting on an object as it moves along the parabola \( y = 2x^2 \) from \( (3, 18) \) to \( (0, 0) \). The work done is given by the line integral:
\[
\int_C 2y \, dx + x \, dy.
\]
**Problem 5:**
Find the work done by the force field \( \vec{F}(x, y) = \langle x^2, xy \rangle \) acting on an object that is moving along in the direction of the vector-valued function:
\[
\vec{r}(t) = \langle 3t, 2t \rangle, \quad 0 \leq t \leq 1.
\]
In these problems, you will apply techniques of vector calculus to find the work done by force fields on objects moving along specified paths. Both problems involve calculating line integrals, which require substituting the given paths into the integrals and evaluating them over the defined intervals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca92bf5f-8e35-4ea7-bd85-d600dcb720b2%2Fe499b331-ab04-4df5-bb9c-d67a9a19c598%2Fxrxmz4q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problems on Work Done by Force Fields
**Problem 4:**
Find the work done by the force field \( \vec{F}(x, y) = \langle 2y, x \rangle \) acting on an object as it moves along the parabola \( y = 2x^2 \) from \( (3, 18) \) to \( (0, 0) \). The work done is given by the line integral:
\[
\int_C 2y \, dx + x \, dy.
\]
**Problem 5:**
Find the work done by the force field \( \vec{F}(x, y) = \langle x^2, xy \rangle \) acting on an object that is moving along in the direction of the vector-valued function:
\[
\vec{r}(t) = \langle 3t, 2t \rangle, \quad 0 \leq t \leq 1.
\]
In these problems, you will apply techniques of vector calculus to find the work done by force fields on objects moving along specified paths. Both problems involve calculating line integrals, which require substituting the given paths into the integrals and evaluating them over the defined intervals.
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