(IV) Find the work done by F in moving a particle along any closed path C. (V) Consider two paths Cl and C2. Suppose the work done by the particle in moving through F along path Cl from r(O) to r(2) is 10. Find the work done by F in moving a particle along path C2 from r(O) to r(2). (VI) Consider two paths C3 and C4, described by the parametric equations Lost sint [²x] =²²₁ e-+ [ cont] +₂e="[t] sint cost [₁] = α [ 201 For simplicity, set 0₁ = 0 and α₂ = 1 to define the equations for C3 and Cy. cos (24) -2sin (24) +22 sin (24) [2cos (24 USE MATLAB to plot C3 and Cy from r(0) to r() indicating the direction of increasing (may manually add direction to plot)
(IV) Find the work done by F in moving a particle along any closed path C. (V) Consider two paths Cl and C2. Suppose the work done by the particle in moving through F along path Cl from r(O) to r(2) is 10. Find the work done by F in moving a particle along path C2 from r(O) to r(2). (VI) Consider two paths C3 and C4, described by the parametric equations Lost sint [²x] =²²₁ e-+ [ cont] +₂e="[t] sint cost [₁] = α [ 201 For simplicity, set 0₁ = 0 and α₂ = 1 to define the equations for C3 and Cy. cos (24) -2sin (24) +22 sin (24) [2cos (24 USE MATLAB to plot C3 and Cy from r(0) to r() indicating the direction of increasing (may manually add direction to plot)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please answer d,e,f
![(1) Find the real and positive constants and such that the following velocity field V is conservative
V(x, y, z) = [21x sin (772)]; +[√T ₂²e-Y]; + [x² cos(172)_2₂€¯Y] K
V.
(II) Consider a force field F(x,7,₁²)=√(x₁4₁²) where is the conservative form of from part (1). Find & such that. F-
(III) Can the divergence of F be zero on the plane z-O? Justify your answer using the divergence of F on this plane. Classify the
points on 2-0 as source, sink or neither.
No structure operates in perfect isolation. Structures always interact with their environment and these interactions entail energy losses
which need to be minimised (e.g., recall Figure 4 middle-right). Accordingly, an important consideration in the design and modelling of
components in renewable technologies concerns measures of the energy expended in the flows (e.g., air) around them. The work done
is one such measure.
(IV) Find the work done by F in moving a particle along any closed path C.
(V) Consider two paths Cl and C2. Suppose the work done by the particle in moving through F along path Cl from r(O) to r(2) is 10.
Find the work done by F in moving a particle along path C2 from r(O) to r(2).
(VI) Consider two paths C3 and C4, described by the parametric equations
#]
[²²] = α²₁ e-t [cont].
sint
't [sint.
cost
+aze-t
For simplicity, set a₁ = 0 and α²₂=1
(24)
[+] = α₁ [cat].
-2 sin (2t)
to define the equations for C3 and cy.
(g) Find the equation for C4, in cartesian coordinates (x,y). [Check that your equation
makes sense by comparing your result to the plot in part (f).]
+α2
+
[
USE MATLAB to plot (3 and (y from r(0) to r() indicating the direction of increasing (may manually add direction to plot)
sin (24)
(h) Find the work done by the force field & in moving a particle along path C4, from r(0) tor (Given
Gie) - Ezcas (2) E-toin234 (12sinh (tt)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12fdac56-4b52-4434-8b00-5772085cd8ad%2Fa38b83e8-1b85-4c1f-ba61-e275c281f96c%2F1mv48aj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(1) Find the real and positive constants and such that the following velocity field V is conservative
V(x, y, z) = [21x sin (772)]; +[√T ₂²e-Y]; + [x² cos(172)_2₂€¯Y] K
V.
(II) Consider a force field F(x,7,₁²)=√(x₁4₁²) where is the conservative form of from part (1). Find & such that. F-
(III) Can the divergence of F be zero on the plane z-O? Justify your answer using the divergence of F on this plane. Classify the
points on 2-0 as source, sink or neither.
No structure operates in perfect isolation. Structures always interact with their environment and these interactions entail energy losses
which need to be minimised (e.g., recall Figure 4 middle-right). Accordingly, an important consideration in the design and modelling of
components in renewable technologies concerns measures of the energy expended in the flows (e.g., air) around them. The work done
is one such measure.
(IV) Find the work done by F in moving a particle along any closed path C.
(V) Consider two paths Cl and C2. Suppose the work done by the particle in moving through F along path Cl from r(O) to r(2) is 10.
Find the work done by F in moving a particle along path C2 from r(O) to r(2).
(VI) Consider two paths C3 and C4, described by the parametric equations
#]
[²²] = α²₁ e-t [cont].
sint
't [sint.
cost
+aze-t
For simplicity, set a₁ = 0 and α²₂=1
(24)
[+] = α₁ [cat].
-2 sin (2t)
to define the equations for C3 and cy.
(g) Find the equation for C4, in cartesian coordinates (x,y). [Check that your equation
makes sense by comparing your result to the plot in part (f).]
+α2
+
[
USE MATLAB to plot (3 and (y from r(0) to r() indicating the direction of increasing (may manually add direction to plot)
sin (24)
(h) Find the work done by the force field & in moving a particle along path C4, from r(0) tor (Given
Gie) - Ezcas (2) E-toin234 (12sinh (tt)
Transcribed Image Text:The solutions of (a) (b) (c)are given as references, please solve for (d)
and (e)
We have already proved that:
n = 1 and y = 1
(a)
(b)
(c)
can let the velocity field V is conservative.
F = V¢ where $(x, y, z) = sin(xz) - z² e¯y
div(F) = −2e- < 0
therefore the points on the plane z = 0 are all sink.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 1 images
Follow-up Questions
Read through expert solutions to related follow-up questions below.
Follow-up Question
With the MATLAB code is r1 meant to be r1=exp(-t).*[sin(t); cos(t)], with the negative sign??
and what direction is t increasing in on the plot?
Can you also find the equation for C4, in cartesian coordinates (x,y). [Check that your equation makes sense by comparing your result to the plot in part (f).]
Solution
by Bartleby ExpertRecommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
