   Chapter 16.2, Problem 47E

Chapter
Section
Textbook Problem

(a) Show that a constant force field does zero work on a particle that moves once uniformly around the circle x2 + y2 = 1.(b) Is this also true for a force field F(x) = kx, where k is a constant and x = ⟨x, y⟩?

(a)

To determine

To show: The constant force field does zero work on a particle.

Explanation

Given data:

The particle moves once uniform around the circle x2+y2=1 .

Formula used:

Write the expression to find the work done on the object by the force field F(x,y) .

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

Here,

F(t) is the force field on the object,

r(t) is the velocity vector of the object, which is derivative of the vector r(t) ,

a is the lower limit of the scalar parameter, and

b is the upper limit of the scalar parameter.

Write the expression to find r(t) of the object.

r(t)=ddt[r(t)] (2)

Here,

r(t) is the position vector of the object.

Consider the constant force vector as follows.

F(t)=k1,k2

Here,

k1 and k2 are the constants.

As the particle moves around the circle x2+y2=1 , parameterize the equation such that the parameters must satisfy the equation of the circle.

x=cost,y=sint,0t2π

Write the position vector from the parametric equations as follows.

r(t)=cost,sint

Calculation of r(t) :

Substitute cost,sint for r(t) in equation (2),

r(t)=ddt[cost,sint]=ddt(cost),ddt(sint)=<

(b)

To determine

To verify: The work done by a force field F(x,y)=kx on an object is zero.

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