Download Quiz 17 Given the force field F: F(x, y) = 〈 −y, x 〉. Find the work done

Department of Mathematical Sciences
Instructor: Daiva Pucinskaite
Calculus III
July 27, 2016
Quiz 17
Given the force field F:
F(x, y) = h −y, x i.
Find the work done by the force field F(x, y) = hf (x, y), g(x, y)i on a particle that moves
along the given oriented curve C:
Recall: Let F be a force field in a region of R2 , and r(t) = hx(t), y(t)i for a ≤ t ≤ b, be a curve. The work done in
moving a particle along C in the positive direction is
Z b
Z b
W =
F(r(t)) · r′ (t) dt =
f x(t), y(t) , g x(t), y(t) • x′ (t), y ′ (t) dt
a
a
(1) C is the upper half of the unit circle centered at the origin oriented counterclockwise.
A parametric description of the curve C is
+
*
r(t) =
π
0 ≤ t ≤ |{z}
cos
sin t , for |{z}
|{z}t, |{z}
x(t)
a
y(t)
at the time t = t the particle is at position
b
cos(t), sin(t) .
t = π
2
1
t = π
4
t = 3π
2
t = π
t = 0
-1
0
1
-1
The work done by the force field F(x, y) =
W =
=
=
Z
Z
π
0
π
t|π0
−y , x
|{z} |{z}
+
on a particle that moves along C
g(x,y)
f (x,y)
−
sin }t , cos
• −
|{z}t
| {zx} dt
{z x}, cos
| {z
| sin
f (x(t),y(t)) g(x(t),y(t))
2
0
*
2
sin
+ sin }t dt =
| t {z
1
= π.
Z
x′ (t)
π
1dt
0
y ′ (t)
(2) C is the upper half of the unit circle centered at the origin oriented clockwise.
A parametric description of the curve C is
+
*
π
sin t , for |{z}
0 ≤ t ≤ |{z}
−
cos }t, |{z}
| {z
r(t) =
a
y(t)
x(t)
at the time t = t the particle is at position
b
− cos(t), sin(t) .
t = π
2
1
t = π
4
t = 3π
2
t = π
t = 0
-1
0
1
-1
The work done by the force field F(x, y) =
W =
=
Z
Z
π
0
π
−y , x
|{z} |{z}
f (x,y)
+
on a particle that moves along C
g(x,y)
−
sin }t , −
cos }t • sin
x, cos
| {zx} dt
|{z}
| {z
| {z
x′ (t)
f (x(t),y(t)) g(x(t),y(t))
2
0
*
2
|− sin t{z− sin }t dt = −
−1
= −t|π0 = −π.
Z
π
1dt
0
y ′ (t)