Question

Use Green’s Theorem to evaluate around the boundary curve C of the region R, where R is the triangle formed by the point (0, 0), (1, 1) and (1, 3). Find the work done by the force field F(x,y)=4yi+2xj in moving a particle along a circle x^2+y^2=1 from(0,1)to(1,0).

Use Green’s Theorem to evaluate around the boundary curve C of the region R, where R is the triangle formed by the point (0, 0), (1, 1) and (1, 3).
Find the work done by the force field F(x,y)=4yi+2xj in moving a particle along a circle \(\displaystyle{x}^{{2}}+{y}^{{2}}={1}\) from(0,1)to(1,0).

Answers (1)

2021-01-06
Step 1
According to the given information, it is required to calculate the work done by the force field
F(x,y)=4yi + 2xj
moving along the particle along a circle \(\displaystyle{x}^{{2}}+{y}^{{2}}={1}\)
from(0,1)to(1,0)
Step 2
The work done can be calculated as:
work done = \(\displaystyle\int_{{C}}{F}.{d}{r}\)
Step 3
Solving further to get:
\(\displaystyle\int_{{C}}{F}.{d}{r}\)
\(\displaystyle{r}{\left({t}\right)}={\left({\sin{{t}}}\right)}{i}+{\left({\cos{{t}}}\right)}{j}{f}{\quad\text{or}\quad}{0}\le{t}\le\frac{\pi}{{2}}\)
\(\displaystyle{r}'{\left({t}\right)}={\left({\cos{{t}}}\right)}{i}+{\left(-{\sin{{t}}}\right)}{j}\)
\(\displaystyle\int_{{C}}{F}.{d}{r}={\int_{{0}}^{{\frac{\pi}{{2}}}}}{\left({\left({4}{\cos{{t}}}\right)}{i}+{\left({2}{\sin{{t}}}\right)}{j}\right)}.{\left({\left({\cos{{t}}}\right)}{i}+{\left(-{\sin{{t}}}\right)}{j}\right)}{\left.{d}{t}\right.}\)
\(\displaystyle={\int_{{0}}^{{\frac{\pi}{{2}}}}}{\left({4}{{\cos}^{{2}}{t}}-{2}{{\sin}^{{2}}{t}}\right)}{\left.{d}{t}\right.}\)
\(\displaystyle={\int_{{0}}^{{\frac{\pi}{{2}}}}}{\left({6}{{\cos}^{{2}}{t}}-{2}\right)}{\left.{d}{t}\right.}\)
\(\displaystyle=\frac{\pi}{{2}}\)
Therefore, the total work done by the force field is \(\displaystyle\frac{\pi}{{2}}\).
0
 
Best answer

expert advice

Have a similar question?
We can deal with it in 3 hours

Relevant Questions

asked 2020-10-27
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The work required to move an object around a closed curve C in the presence of a vector force field is the circulation of the force field on the curve.
b. If a vector field has zero divergence throughout a region (on which the conditions of Green’s Theorem are met), then the circulation on the boundary of that region is zero.
c. If the two-dimensional curl of a vector field is positive throughout a region (on which the conditions of Green’s Theorem are met), then the circulation on the boundary of that region is positive (assuming counterclockwise orientation).
asked 2021-02-09
A particle moves along line segments from the origin to the points(3,0,0),(3,3,1),(0,3,1), and back to the origin under the influence of the force field \(\displaystyle{F}{\left({x},{y},{z}\right)}={z}^{{2}}{i}+{3}{x}{y}{j}+{4}{y}^{{2}}{k}\).
Use Stokes' Theorem to find the work done.
asked 2020-12-16
Let \(\displaystyle{f}={\left[{x}^{{2}}{y}^{{2}},-\frac{{x}}{{y}^{{2}}}\right]}\) and \(\displaystyle{R}:{1}\le{x}^{{2}}+{y}^{{2}},+{4},{x}\ge{0},{y}\ge{x}\). Evaluate \(\displaystyle\int_{{C}}{F}{\left({r}\right)}\cdot{d}{r}\) counterclockwise around the boundary C of the region R by Green's theorem.
asked 2020-11-02
Suppose that the plane region D, its boundary curve C, and the functions P and Q satisfy the hypothesis of Green's Theorem. Considering the vector field F = Pi+Qj, prove the vector form of Green's Theorem \(\displaystyle\oint_{{C}}{F}\cdot{n}{d}{s}=\int\int_{{D}}\div{F}{\left({x},{y}\right)}{d}{A}\)
where n(t) is the outward unit normal vector to C.
...