Section 5.3 Work

5.3 Work

This section, we work on line integral over a vector field. The motivation for this is to find the work done by a force. Recall that if a force

$f(x)$ is applying on an object and move the object along a line from $x=a$ to $x=b$ then the work done by the force is ${\int }_a^{b}f(x)dx$. At here, we wish to compute the work done by a force field $\mathbf{F}=<P(x,y,z),Q(x,y,z),R(x,y,z)>$ along the curve $\overrightarrow{r(t)}=<x(t),y(t),z(t)>$. 

Definition 

The vector line integral of vector field $\mathbf{F}$ along oriented smooth curve $C$ is

$${\int }_C\mathbf{F}\cdot \overrightarrow{T(t)}ds=\underset{n\to \mathrm{\infty }}{lim}\sum _{i=1}^n\mathbf{F}(P_i^{\ast })\cdot \overrightarrow{T(P_i^{\ast })}\mathrm{△}{s}_i$$

if this limit exists. 

Theorem: Work 

The work $W$ done by the force field $\mathbf{F}=<P,Q,R>$ along $C$ with parameterization $\overrightarrow{r(t)}=<x(t),y(t),z(t)>$, $a\le t\le b$ is 

$${\int }_C\mathbf{F}\cdot \overrightarrow{T(t)}ds={\int }_a^b\mathbf{F}(\overrightarrow{r(t)})\cdot \overrightarrow{r^{\prime }(t)}dt={\int }_C\mathbf{F}\cdot d\overrightarrow{r}={\int }_{C}Pdx+{\int }_{C}Qdy+{\int }_{C}Rdz.$$

Example 1: Evaluate ${\int }_C\mathbf{F}\cdot d\overrightarrow{r}$ where $C$ is the curve $\overrightarrow{r(t)}=<t^2,-t,t^3>$ with $0\le t\le 2$ and $\mathbf{F}(x,y,z)=<x^2,xy,z>$.

Exercise 1: Evaluate ${\int }_C\mathbf{F}\cdot d\overrightarrow{r}$ where $C$ is the curve $\overrightarrow{r(t)}=<t,t^2,-3t>$ with $0\le t\le 1$ and $\mathbf{F}(x,y,z)=<xy,z,y>$.

Example 2: Evaluate ${\int }_C\mathbf{F}\cdot d\overrightarrow{r}$ where $C$ is the curve $\overrightarrow{r(t)}=<t^2,-t^3,t>$ with $0\le t\le 1$ and $\mathbf{F}(x,y,z)=<\text{cos}(x),\text{sin}(y),xz>$.

Exercise 2: Evaluate ${\int }_C\mathbf{F}\cdot d\overrightarrow{r}$ where $C$ is the curve $\overrightarrow{r(t)}=<t,t^2,-3t>$ with $0\le t\le 1$ and $\mathbf{F}(x,y,z)=<xy,\text{sin}(y),z>$.

Example 3: Evaluate ${\int }_C\mathbf{F}\cdot d\overrightarrow{r}$ where $C$ is the arc of $x^2+y^2=9$ traversed clockwise from $(0,3)$ to $(3,0)$ and $\mathbf{F}(x,y)=<x,x+y>$.

Exercise 3: Evaluate ${\int }_C\mathbf{F}\cdot d\overrightarrow{r}$ where $C$ is the arc of $x^2+y^2=1$ traversed counter counterclockwise from $(0,1)$ to $(-1,0)$ and $\mathbf{F}(x,y)=<y-x,y>$.

Example 4: Find the work done by $\mathbf{F}(x,y)=<xy,y^2>$ on a particle that moves once around the circle $x^2+y^2=9$ oriented in counter clockwise direction. 

Exercise 4: Find the work done by $\mathbf{F}(x,y)=<y^2,2xy>$ on a particle that moves once around the circle $x^2+y^2=4$ oriented in counter clockwise direction. 

Example 5: Find the work done by $\mathbf{F}(x,y,z)=<x^2-y,z-y^2,x-z^2>$ on a particle that moves along a line segment from $(1,0,0)$ to $(1,0,2)$.

Exercise 5: Find the work done by $\mathbf{F}(x,y,z)=<y+z,z-x^2,x+y^2>$ on a particle that moves along a line segment from $(0,1,0)$ to $(-1,0,1)$.

Example 6: Find the work done by $\mathbf{F}(x,y)=<y+1,x>$ on a particle that moves along an arch of the cycloid $\overrightarrow{r(t)}=<1-\text{cos}(t),t+\text{sin}(t)>$ for $0\le t\le 2\pi$.

Group work:

1. Find the work done by $\mathbf{F}(x,y)=<2x,y>$ on a particle that moves along an arch of the cycloid $\overrightarrow{r(t)}=<1+\text{sin}(t),1-\text{sin}(t)>$ for $0\le t\le \pi$.

2. Find the work done by $\mathbf{F}(x,y)=<x{e}^y,y^2>$ on a particle that moves along the parabola $y=x^2-1$ from $(1,0)$ to $(2,3).$

3. Find the work done by $\mathbf{F}(x,y,z)=<x^2+z,x+y^2,x{z}^2>$ on a particle that moves along a line segment from $(0,0,1)$ to $(1,0,-1)$.

4. Find the work done by $\mathbf{F}(x,y)=<x+y,x^2>$ on a particle that moves once around the circle $x^2+y^2=4$ oriented in counter clockwise direction.