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

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

Definition 

The vector line integral of vector field [latex]\mathbf{F}[/latex] along oriented smooth curve [latex]C[/latex] is

\[\int_{C}\mathbf{F}\cdot\overrightarrow{T(t)}ds=\lim_{n\rightarrow\infty}\sum_{i=1}^{n}\mathbf{F}(P_{i}^{*})\cdot\overrightarrow{T(P_{i}^{*})}\triangle s_{i}\]

if this limit exists. 

Theorem: Work 

The work [latex]W[/latex] done by the force field [latex]\mathbf{F}=\lt P,Q,R\gt[/latex] along [latex]C[/latex] with parameterization [latex]\overrightarrow{r(t)}=\lt x(t),y(t),z(t)\gt[/latex], [latex]a\le t\le b[/latex] is 

\[\int_{C}\mathbf{F}\cdot\overrightarrow{T(t)}ds=\int_{a}^{b}\mathbf{F}(\overrightarrow{r(t)})\cdot\overrightarrow{r'(t)}dt=\int_{C}\mathbf{F}\cdot d\overrightarrow{r}=\int_{C}Pdx+\int_{C}Qdy+\int_{C}Rdz.\]

Example 1: Evaluate [latex]\int_{C}\mathbf{F}\cdot d\overrightarrow{r}[/latex] where [latex]C[/latex] is the curve [latex]\overrightarrow{r(t)}=\lt t^{2},-t,t^{3}\gt[/latex] with [latex]0\leq t\leq2[/latex] and [latex]\mathbf{F}(x,y,z)=\lt x^{2},xy,z\gt[/latex].

Exercise 1: Evaluate [latex]\int_{C}\mathbf{F}\cdot d\overrightarrow{r}[/latex] where [latex]C[/latex] is the curve [latex]\overrightarrow{r(t)}=\lt t,t^{2},-3t\gt[/latex] with [latex]0\leq t\leq1[/latex] and [latex]\mathbf{F}(x,y,z)=\lt xy,z,y\gt[/latex].

Example 2: Evaluate [latex]\int_{C}\mathbf{F}\cdot d\overrightarrow{r}[/latex] where [latex]C[/latex] is the curve [latex]\overrightarrow{r(t)}=\lt t^{2},-t^{3},t\gt[/latex] with [latex]0\leq t\leq1[/latex] and [latex]\mathbf{F}(x,y,z)=\lt \text{cos}(x),\text{sin}(y),xz\gt[/latex].

Exercise 2: Evaluate [latex]\int_{C}\mathbf{F}\cdot d\overrightarrow{r}[/latex] where [latex]C[/latex] is the curve [latex]\overrightarrow{r(t)}=\lt t,t^{2},-3t\gt[/latex] with [latex]0\leq t\leq1[/latex] and [latex]\mathbf{F}(x,y,z)=\lt xy,\text{sin}(y),z\gt[/latex].

Example 3: Evaluate [latex]\int_{C}\mathbf{F}\cdot d\overrightarrow{r}[/latex] where [latex]C[/latex] is the arc of [latex]x^{2}+y^{2}=9[/latex] traversed clockwise from [latex](0,3)[/latex] to [latex](3,0)[/latex] and [latex]\mathbf{F}(x,y)=\lt x,x+y\gt[/latex].

Exercise 3: Evaluate [latex]\int_{C}\mathbf{F}\cdot d\overrightarrow{r}[/latex] where [latex]C[/latex] is the arc of [latex]x^{2}+y^{2}=1[/latex] traversed counter counterclockwise from [latex](0,1)[/latex] to [latex](-1,0)[/latex] and [latex]\mathbf{F}(x,y)=\lt y-x,y\gt[/latex].

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

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

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

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

Example 6: Find the work done by [latex]\mathbf{F}(x,y)=\lt y+1,x\gt[/latex] on a particle that moves along an arch of the cycloid [latex]\overrightarrow{r(t)}=\lt 1-\text{cos}(t),t+\text{sin}(t)\gt[/latex] for [latex]0\leq t\leq2\pi[/latex].

Group work:

1. Find the work done by [latex]\mathbf{F}(x,y)=\lt 2x,y\gt[/latex] on a particle that moves along an arch of the cycloid [latex]\overrightarrow{r(t)}=\lt 1+\text{sin}(t),1-\text{sin}(t)\gt[/latex] for [latex]0\leq t\leq\pi[/latex].

2. Find the work done by [latex]\mathbf{F}(x,y)=\lt xe^{y},y^{2}\gt[/latex] on a particle that moves along the parabola [latex]y=x^{2}-1[/latex] from [latex](1,0)[/latex] to [latex](2,3).[/latex]

3. Find the work done by [latex]\mathbf{F}(x,y,z)=\lt x^{2}+z,x+y^{2},xz^{2}\gt[/latex] on a particle that moves along a line segment from [latex](0,0,1)[/latex] to [latex](1,0,-1)[/latex].

4. Find the work done by [latex]\mathbf{F}(x,y)=\lt x+y,x^{2}\gt[/latex] on a particle that moves once around the circle [latex]x^{2}+y^{2}=4[/latex] oriented in counter clockwise direction.