   Chapter 16, Problem 10RE

Chapter
Section
Textbook Problem

Find the work done by the force field F(x, y, z) = z i + x j+y k in moving a particle from the point (3, 0, 0) to the point (0, π/2, 3) along(a) a straight line(b) the helix x = 3 cos t, y = t, z = 3 sin t

(a)

To determine

The work done by the force field F(x,y,z)=zi+xj+yk in a moving particle from the point (3,0,0) to the point (0,π2,3) along a straight line.

Explanation

Given data:

The moving particle points are as follows.

(3,0,0) , to (0,π2,3) .

Formula used:

Write the expression vector representation of line segment.

r(t)=(1t)r0+tr1 0t1 (1)

Here,

r0 is starting point of line segment, and

r1 is ending point of line segment.

Write the expression to find the work done.

W=CFdr (2)

Write the required differential and integration formulae to evaluate the given integral.

ddxxn=nxn1[f(x)]ndx=[f(x)]n+1n+1

Find parametric representation of line segment for x-coordinate using equation (1).

Substitute x for r(t) , 3 for r0 and 0 for r1 in equation (1),

x=(1t)(3)+t(0)=33t

Differentiate x with respect to t .

dxdt=03dx=3dt

Find parametric representation of line segment for y-coordinate using equation (1).

Substitute y for r(t) , 0 for r0 and π2 for r1 in equation (1),

y=(1t)(0)+t(π2)=tπ2

Differentiate y with respect to t .

dydt=π2dy=π2dt

Find parametric representation of line segment for z-coordinate using equation (1).

Substitute z for r(t) , 0 for r0 and 3 for r1 in equation (1),

z=(1t)(0)+t(3)=0+3t=3t

Differentiate z with respect to t .

dzdt=3dz=3dt

Write the expression for r(t) in terms of x , y and z coordinates.

r(t)=xi+yj+zk (3)

Differentiate equation (3) with respect to t

(b)

To determine

The work done by the force field F(x,y,z)=zi+xj+yk in the helix.

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