The work done by the force $\mathbf { F } ( x , y ) = ( 2 x + y ) \mathbf { i } + ( x - 2 y ) \mathbf { j }$ moving along $\mathbf { r } ( t ) = 3 \cos t \mathbf { i } + 3 \sin t \mathbf { j }$ with $0 \leq t \leq 2 \pi$ is

A) $\frac { 5 } { 2 }$
B) $\frac { 8 } { 3 }$
C) $\frac { 4 } { 3 }$
D)0
E) $\frac { 4 } { 5 }$

Question

The work done by the force

\mathbf { F } ( x , y ) = ( 2 x + y ) \mathbf { i } + ( x - 2 y ) \mathbf { j }

moving along

\mathbf { r } ( t ) = 3 \cos t \mathbf { i } + 3 \sin t \mathbf { j }

with

0 \leq t \leq 2 \pi

is

A) $\frac { 5 } { 2 }$
B) $\frac { 8 } { 3 }$
C) $\frac { 4 } { 3 }$
D)0
E) $\frac { 4 } { 5 }$

Quizplus · Accepted Answer

The work done by a force field $\mathbf{F}$ along a curve $\mathbf{r}(t)$ is given by the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$. In this case, the curve is a circle with radius 3 centered at the origin, completing a full loop. The force field $\mathbf{F}(x, y) = (2x + y)\mathbf{i} + (x - 2y)\mathbf{j}$ is conservative, as it can be derived from a potential function. The work done by a conservative force around any closed loop is zero, hence the answer is D.

The Work Done by the Force F(x,y)=(2x+y)i+(x−2y)j\mathbf { F } ( x , y ) = ( 2 x + y ) \mathbf { i } + ( x - 2 y ) \mathbf { j }F(x,y)=(2x+y)i+(x−2y)j

The Work Done by the Force $\mathbf { F } ( x , y ) = ( 2 x + y ) \mathbf { i } + ( x - 2 y ) \mathbf { j }$