   Chapter 16.2, Problem 40E

Chapter
Section
Textbook Problem

Find the work done by the force field F(x, y) = x2 i + yex j on a particle that moves along the parabola x = y2 + 1 from (1, 0) to (2, 1).

To determine

To find: The work done by the force field F(x,y)=x2i+yexj .

Explanation

Given data:

The force field is F(x,y)=x2i+yexj .

The particle moves along the parabola x=y2+1 from the point (1,0) to (2,1) .

Formula used:

Write the expression to find the work done by the force field F(x,y) along the arch of the cycloid r(y) .

CFdr=abF(r(y))r(y)dy (1)

Here,

r(y) is the vector function,

a is the lower limit of curve, and

b is the upper limit of the curve.

As the particle moves along the parabola x=y2+1 from the point (1,0) to (2,1) , choose y as parameter and parameterize the curve as follows.

x=y2+1,y=y,0y1

Then, the vector function of the curve is written as follows.

r(y)=(y2+1)i+yj,0y1

Write the point (x,y) from the vector function as follows.

(x,y)=(y2+1,y)

Write the force field as follows.

F(x,y)=x2i+yexj (2)

Calculation of F(r(y)) :

Substitute (y2+1) for x and y for y in equation (2),

F(r(y))=(y2+1)2i+ye(y2+1)j=(y4+1+2y2)i+ye(y2+1)j

Calculation of r(y) :

To find the derivative of the vector function, differentiate each component of the vector function.

Differentiate each component of the vector function r(y)=(y2+1)i+yj as follows.

ddy[r(y)]=ddy[(y2+1)i+yj,0y1]

Rewrite the expression as follows.

r(y)=ddy(y2+1)i+ddy(y)j=(2y+0)i+(1)j=2yi+j

Calculation of CFdr :

Substitute [(y4+1+2y2)i+ye(y2+1)j] for F(r(y)) , (2yi+j) for r(y) , 0 for a , and 1 for b in equation (1),</

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