Tutorial Exercise

Find the work done by the force field

F$($x$,$ y$)=$ xi $+($y $+8)$j

in moving an object along an arch of the cycloid

r$($t$)=($t $$ sin$($t$))$i $+(1$ cos$($t$))$j$,0$ t $2.$

Step $1$

We know that the work done by the force field F in moving an object along the path C which is parameterized by the vector function r$($t$)$ is found by the following equation.

W $=$

$$C

F $$ dr

For

F$($x$,$ y$)=$ xi $+($y $+8)$j

and

r$($t$)=($t $$ sin$($t$))$i $+(1$ cos$($t$))$j$,$

we have

F$($t$)=$

t $$ sin$($t$),$

$$

and

dr $=$

$1$

$$

$,$ sin$($t$)$

$.$

Step $2$

Therefore,

F $$ dr$=$

t $$ sin$($t$),9$ cos$($t$)$

$$

$1$ cos$($t$),$ sin$($t$)$

$$

$=$t $$ t cos$($t$)$ sin$($t$)+$ sin$($t$)$ cos$($t$)+$

$$

$$ sin$($t$)$ cos$($t$)$

$=$t $$ t cos$($t$)+$

$$

$.$

Step $3$

Substituting this back into the integral, we have the following.

W$=$

$$F $$ drC

$$

$=$

$($t $$ t cos$($t$)+8$ sin$($t$))$ dtC

Using integration by parts for the second term, we find that

t cos$($t$)$ dt

$=$ t

$+$ cos$($t$).$