Q 55. Find the work done by the given ... [FREE SOLUTION]

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Chapter 14: Q 55. (page 1107)

Find the work done by the given vector field $F$ F along the specified curve $C$ .
$F (x, y) = y i - x j$ where $C$ is the portion of the circle of radius $2$ , centered at the origin, and in the first quadrant, traversed counterclockwise from $(2, 0) - (0, 2)$ .

Short Answer

Expert verified

The required work done is ${鈭�}_{C} F (x, y) 路 d r = - \frac{蟺}{2}$ .

Step by step solution

Given Information

The vector field is $F (x, y) = y i - x j$

Curve is portion of circle of radius $2$ , centered at origin in first quadrant, traversed counterclockwise from $(2, 0) - (0, 2)$

Parametrization of curve

The curve is parametrized as

$r (t) = 鉄� \cos t, \sin t 鉄� for 0 鈮� t 鈮� \frac{蟺}{2}$

$鈬� r^{'} (t) = \frac{d}{d t} r (t)$

$= \frac{d}{d t} 鉄� \cos t, \sin t 鉄�$

$= 鉄� - \sin t, \cos t 鉄�$

Rewrite the vector field

It can be written as

$F (x, y) = y i - x j$

$F (x (t), y (t)) = \sin t i - \cos t j$

$= 鉄� \sin t, - \cos t 鉄�$

Calculating Work done

The required integral is ${鈭�}_{C} F (x, y) 路 d r$

$= {鈭�}_{C} F (x (t), y (t)) 路 r^{'} (t) d t$

$= {鈭�}_{0}^{蟺 / 2} 鉄� \sin t, - \cos t 鉄� 路鉄� - \sin t, \cos t 鉄� d t$

$= {鈭�}_{0}^{蟺 / 2} [(\sin t) (- \sin t) + (- \cos t) (\cos t)] d t$

Applying values

$= {鈭�}_{0}^{蟺 / 2} [- \sin^{2} t - \cos^{2} t] d t$

$= {鈭�}_{0}^{蟺 / 2} [- (\sin^{2} t + \cos^{2} t)] d t$

$= {鈭�}_{0}^{蟺 / 2} (- 1) d t$

$= - 1 路 [t]_{0}^{蟺 / 2}$

$= - 1 路 [\frac{蟺}{2} - 0]$

$= - \frac{蟺}{2}$

Hence, work done is ${鈭�}_{C} F (x, y) 路 d r = - \frac{蟺}{2}$

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91影视

Find the work done by the given vector field $F$ F along the specified curve $C$ .
$F (x, y) = y i - x j$ where $C$ is the portion of the circle of radius $2$ , centered at the origin, and in the first quadrant, traversed counterclockwise from $(2, 0) - (0, 2)$ .

Short Answer

Step by step solution

Given Information

Parametrization of curve

Rewrite the vector field

Calculating Work done

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Most popular questions from this chapter

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91影视

Find the work done by the given vector fieldFF along the specified curve C.F(x,y)=yi-xjwhere Cis the portion of the circle of radius 2, centered at the origin, and in the first quadrant, traversed counterclockwise from2,0-0,2.

Short Answer

Step by step solution

Given Information

Parametrization of curve

Rewrite the vector field

Calculating Work done

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Discrete Mathematics

Applied Mathematics

Logic and Functions

Mechanics Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Find the work done by the given vector field $F$ F along the specified curve $C$ .
$F (x, y) = y i - x j$ where $C$ is the portion of the circle of radius $2$ , centered at the origin, and in the first quadrant, traversed counterclockwise from $(2, 0) - (0, 2)$ .