A parametric position vector has the following definitions: r(t) = (a+sin(kt)) êp + hêz with (t) = wt, a, ẞ, k > 0, and a > ẞ. (1) The final condition ensures a positive radius. The following sequence of plots shows the space curve traced out by the position vector (black) for the specific values a = 3, ß = 1, k = 2, h = 4, and w = 1. At the selected snapshots for t, the velocity vector (blue) and acceleration vector (red) is also given. t= 2π 11 6 л t= 11 10πT t= 11 Note: in all questions except (v) do not use the specific values for the plots. Using the general definitions in equation (1) and the cylindrical vector definitions in the lecture notes, (i) determine all first and second order derivatives of the cylindrical coordinates p(t), (t), z(t); (ii) determine the velocity vector in cylindrical coordinates; (iii) determine the acceleration vector in cylindrical coordinates; and (iv) determine when the velocity vector is perpendicular to the position vector. (v) Simplify the answer in (iv) for the parameter values used to construct the space curve in the figure. Is the result consistent with the space curve? (vi) Rewrite the position vector in cartesian form - you may use whichever conversion technique you wish. (The plots are drawn using the cartesian form of the vectors).
A parametric position vector has the following definitions: r(t) = (a+sin(kt)) êp + hêz with (t) = wt, a, ẞ, k > 0, and a > ẞ. (1) The final condition ensures a positive radius. The following sequence of plots shows the space curve traced out by the position vector (black) for the specific values a = 3, ß = 1, k = 2, h = 4, and w = 1. At the selected snapshots for t, the velocity vector (blue) and acceleration vector (red) is also given. t= 2π 11 6 л t= 11 10πT t= 11 Note: in all questions except (v) do not use the specific values for the plots. Using the general definitions in equation (1) and the cylindrical vector definitions in the lecture notes, (i) determine all first and second order derivatives of the cylindrical coordinates p(t), (t), z(t); (ii) determine the velocity vector in cylindrical coordinates; (iii) determine the acceleration vector in cylindrical coordinates; and (iv) determine when the velocity vector is perpendicular to the position vector. (v) Simplify the answer in (iv) for the parameter values used to construct the space curve in the figure. Is the result consistent with the space curve? (vi) Rewrite the position vector in cartesian form - you may use whichever conversion technique you wish. (The plots are drawn using the cartesian form of the vectors).
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
Transcribed Image Text:A parametric position vector has the following definitions:
r(t) = (a+sin(kt)) êp + hêz
with
(t) = wt, a, ẞ, k > 0, and a > ẞ. (1)
The final condition ensures a positive radius.
The following sequence of plots shows the space curve traced out by the position vector (black) for
the specific values a = 3, ß = 1, k = 2, h = 4, and w = 1. At the selected snapshots for t, the velocity
vector (blue) and acceleration vector (red) is also given.
t=
2π
11
6 л
t=
11
10πT
t=
11
Note: in all questions except (v) do not use the specific values for the plots.
Using the general definitions in equation (1) and the cylindrical vector definitions in the lecture notes,
(i) determine all first and second order derivatives of the cylindrical coordinates p(t), (t), z(t);
(ii) determine the velocity vector in cylindrical coordinates;
(iii) determine the acceleration vector in cylindrical coordinates; and
(iv) determine when the velocity vector is perpendicular to the position vector.
(v) Simplify the answer in (iv) for the parameter values used to construct the space curve in the
figure. Is the result consistent with the space curve?
(vi) Rewrite the position vector in cartesian form - you may use whichever conversion technique you
wish. (The plots are drawn using the cartesian form of the vectors).
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 2 steps
Recommended 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,
