Math 2263 Multivariable Calculus
Homework 24: 16.2 #32a, 42
July 19, 2011
16.2#32a
Find the work done by the force field F (x, y) = x2 i + xyj on a particle that moves once
around the circle x2 + y 2 = 4 oriented in the counterclockwise direction.
We may parametrize the curve by
r(t) = h2 cos t, 2 sin ti, 0 ≤ t ≤ 2π.
2π 2
2 cos t(−2 sin t) + (2 cos t)(2 sin t)(2 cos t) dt
Z
Z
F · dr =
0
C
2π
Z
0 dt
=
0
=0
16.2#42
The force exerted on an electric charge at the origin on a charged particle at a point (x, y, z)
with position vector r = hx, y, zi is F (r) = Kr/|r|3 where K is a constant. Find the work
done as the particle moves along a straight line from (2, 0, 0) to (2, 1, 5).
The curve C can be parametrized by
r(t) = (1 − t)h2, 0, 0i + th2, 1, 5i = h2, t, 5ti, 0 ≤ t ≤ 1.
Z
Z
F · dr =
C
Z
=
0
1
F (r(t)) · r0 (t) dt
0
1
t2
(4 +
Z 1
=
K
h2, t, 5ti · h0, 1, 5i dt
+ 25t2 )3/2
K
(t + 25t) dt
2 3/2
0 (4 + 26t )
Z 1
26Kt
=
dt
2 3/2
0 (4 + 26t )
Z 30
1
=
Ku−3/2 du
2
4
30
−1/2 = −Ku
4
1
1
= −K √ −
30 2
√ 15 − 30
=K
30