   # 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 ### Calculus (MindTap Course List)

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285740621

#### Solutions

Chapter
Section ### Calculus (MindTap Course List)

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285740621
Chapter 16.R, Problem 10E
Textbook Problem
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## 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

To determine

(a)

To find:

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

Solution:

W=123π-9

Explanation:

1) Concept:

i) The vector representation of the line segment that starts at r0 and ends at r1 is given by

rt=1-tr0+tr1, 0t1

ii) The work done by the force field F in a moving the partical along C, where C is given by the vector function rt, atb is

W=CF·dr=abF(rt)·r'tdt

2) Given:

Fx, y, z=zi+xj+yk and  the particle moving from the point (3, 0, 0) to the point 0,π2, 3 along a straight line path.

3) Calculations:

The force field is Fx, y, z=zi+xj+yk

As the particle moving from the point (3, 0, 0) to the point 0,π2, 3 along a straight line path.

By using concept i),

rt=1-t(3, 0, 0)+t0,π2, 3, 0t1

=3-3t, 0, 0+0,π2t, 3t, 0t1

=3-3t, π2t, 3t, 0t1

That is,

rt=3-3ti+π2tj+3tk

Or parametric form as,

x=3-3t, y= π2t,z=3 t,   0t1

Frt=3 ti+3-3tj+π2tk

r't=-3i+π2j+3k

Frt·r't)=3 ti+3-3tj+π2tk ·-3i+π2j+3k

=-9t+3π2-3πt2+3πt2

=-9t+3π2

By using conceptii),

W=01F(rt)·r'tdt

01Frt·r'tdt=01-9t+3π2dt

Integrate with respect to t, it gives

=-92t2+3π2t10

By using the limits of integration,

=-9212+3π21-0

Simplifying,

=-92+3π2

=123π-9

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

W=123π-9

Conclusion:

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

W=CF·dr=123π-9

(b)

To find:

The work done by the force field Fx, y, z=zi+xj+yk in moving a particle from the point (3, 0, 0) to the point 0,π2, 3 along the helix x=3cost, y=t, z=3sint.

Solution:

-3π4

Explanation:

1) Concept:

The work done by the force field F in a moving the partical along C, where C is given by the vector function rt, atb is

W=CF·dr=abF(rt)·r'tdt

2) Given:

Fx, y, z=zi+xj+yk and the particle moving from the point (3, 0, 0) to the point 0,π2, 3 alongthe helix x=3cost, y=t, z=3sint.

3) Calculations:

The force field is Fx, y, z=zi+xj+yk

As the particle moving from the point 3, 0, 0 to the point 0,π2, 3 along the helix x=3cost, y=t, z=3sint

Therefore,

rt=3costi+ tj+3sintk, 0tπ2

Frt=3sinti+3costj+tk

r't=-3sinti+j+3costk

Frt·r't)=3sinti+3costj+tk ·-3sinti+j+3costk

=-9sin2t+3cost+3tcost

By using concept, the work done by the given force field is

W=0π2F(rt)·r'tdt

0π2Frt·r'tdt=0π2-9sin2t+3cost+3tcostdt

=0π2-921-cos2t+3cost+3tcostdt

……………. since sin2t=1-cos2t2

Consider u=t and v'=cost in last part then apply integration by parts rule, it gives

0π2Frt·r'tdt=-92t-12sin2t+3sint+3(tsint+cost)π20

By using the limits of integration,

=-92π2-12sin2π2+3sinπ2+3π2sinπ2+cosπ2--92(0)-12sin0+3sin0+30+cos0

Simplifying,

=-9π4+3+3π2-3

=-3π4

Thus, the work done by the force field Fx, y, z=zi+xj+yk in moving a particle from the point (3, 0, 0) to the point 0,π2, 3 along the helix x=3cost, y=t, z=3sint is

W=CF·dr=-3π4

Conclusion:

The work done by the force field Fx, y, z=zi+xj+yk in moving a particle from the point (3, 0, 0) to the point 0,π2, 3 along the helix x=3cost, y=t, z=3sint is

W=CF·dr=-3π4

### Explanation of Solution

1) Concept:

i) The vector representation of the line segment that starts at r0 and ends at r1 is given by

rt=1-tr0+tr1, 0t1

ii) The work done by the force field F in a moving the partical along C, where C is given by the vector function rt, atb is

W=CF·dr=abF(rt)·r'tdt

2) Given:

Fx, y, z=zi+xj+yk and  the particle moving from the point (3, 0, 0) to the point 0,π2, 3 along a straight line path.

3) Calculations:

The force field is Fx, y, z=zi+xj+yk

As the particle moving from the point (3, 0, 0) to the point 0,π2, 3 along a straight line path.

By using concept i),

rt=1-t(3, 0, 0)+t0,π2, 3, 0t1

=3-3t, 0, 0+0,π2t, 3t, 0t1

=3-3t, π2t, 3t, 0t1

That is,

rt=3-3ti+π2tj+3tk

Or parametric form as,

x=3-3t, y= π2t,z=3 t,   0t1

Frt=3 ti+3-3tj+π2tk

r't=-3i+π2j+3k

Frt·r't)=3 ti+3-3tj+π2tk ·-3i+π2j+3k

=-9t+3π2-3πt2+3πt2

=-9t+3π2

By using conceptii),

W=01F(rt)·r'tdt

01Frt·r'tdt=01-9t+3π2dt

Integrate with respect to t, it gives

=-92t2+3π2t10

By using the limits of integration,

=-9212+3π21-0

Simplifying,

=-92+3π2

=123π-9

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

W=123π-9

Conclusion:

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

W=CF·dr=123π-9

(b)

To find:

The work done by the force field Fx, y, z=zi+xj+yk in moving a particle from the point (3, 0, 0) to the point 0,π2, 3 along the helix x=3cost, y=t, z=3sint.

Solution:

-3π4

1) Concept:

The work done by the force field F in a moving the partical along C, where C is given by the vector function rt, atb is

W=CF·dr=abF(rt)·r'tdt

2) Given:

Fx, y, z=zi+xj+yk and the particle moving from the point (3, 0, 0) to the point 0,π2, 3 alongthe helix x=3cost, y=t, z=3sint

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