Answers and Solutions to Section 13.2 Homework Problems 1"27 ...

31. The work done by the force …eld
F (x; y) = xi+ (y + 2) j
in moving an object along an arch of the cycloid
r (t) = (t sin (t)) i+ (1 cos (t)) j
(corresponding to the positive t direction for 0 t 2 ) is
Z Z 2
F dr = F (r (t)) r 0 (t) dt.
Since
and
we have
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0
F (r (t)) = (t sin (t)) i+ ((1 cos (t)) + 2) j
= (t sin (t)) i+ (3 cos (t)) j
r 0 (t) = (1 cos (t)) i+ sin (t) j,
F (r (t)) r 0 (t) = ((t sin (t)) i+ (3 cos (t)) j) ((1 cos (t)) i+ sin (t) j)
= (t sin (t)) (1 cos (t)) + (3 cos (t)) sin (t)
= t t cos (t) + 2 sin (t) .
Therefore, the work done is
Z Z 2
F dr = (t t cos (t) + 2 sin (t)) dt = 2 2 .
C
33. The work done by the force …eld
0
F (x; y; z) = xzi+yxj+zyk
on a particle that moves along the curve
is Z
r (t) = t 2 i t 3 j+t 4 k, 0 t 1
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F dr =
Z 1
0
10
F (r (t)) r 0 (t) dt.