   Chapter 16.2, Problem 32E

Chapter
Section
Textbook Problem

(a) Find the work done by the force field F(x, y) = x2 i + xy j on a particle that moves once around the circle x2 + y2 = 4 oriented in the counterclockwise direction.(b) Use a computer algebra system to graph the force field and circle on the same screen. Use the graph to explain your answer to part (a).

(a)

To determine

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

Explanation

Given data:

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

The particle moves once around the circle x2+y2=4 oriented in the counter clock wise direction.

Formula used:

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

CFdr=abF(r(t))r(t)dt (1)

Here,

r(t) is the position vector of the particle,

a is the lower limit of curve, and

b is the upper limit of the curve.

Parameterize the circle x2+y2=4 as follows.

x=2cost,y=2sint,0t2π

Consider the position vector as follows.

r(t)=xi+yj (2)

Substitute 2cost for x , and 2sint for y in equation (2),

r(t)=2costi+2sintj

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

(x,y)=(2cost,2sint)

Write the force field as follows.

F(x,y)=x2i+xyj (3)

Calculation of F(r(t)) :

Substitute 2cost for x and 2sint for y in equation (3),

F(r(t))=(2cost)2i+(2cost)(2sint)j=4cos2ti+4costsintj

Calculation of r(t) :

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

Differentiate each component of the vector function r(t)=2costi+2sintj as follows.

ddt[r(t)]=ddt[2costi+2sintj]

Rewrite the expression as follows

(b)

To determine

To find: The work done by the force field F(x,y)=x2i+xyj with the use of computer algebra system.

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