Consider the Archimedean spiral given by: for t = [0,00). a) Plot the curve. x(t) = t cost, 1 y(t) = -t sint, 8 (10a) (10b) b) What are the slopes of the curve at t = 57/6 and t = 4π? c) Draw arrows on your plot to indicate the direction of motion of the path traced by a particle whose trajectory is described by Eqs. (10a)-(10b).
Consider the Archimedean spiral given by: for t = [0,00). a) Plot the curve. x(t) = t cost, 1 y(t) = -t sint, 8 (10a) (10b) b) What are the slopes of the curve at t = 57/6 and t = 4π? c) Draw arrows on your plot to indicate the direction of motion of the path traced by a particle whose trajectory is described by Eqs. (10a)-(10b).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Consider the Archimedean spiral given by:
x(t) =
for t = [0, ∞0).
a) Plot the curve.
= -t cost,
1
y(t) = t si
t sint,
(10a)
(10b)
b) What are the slopes of the curve at t = 57/6 and t = 4T?
c) Draw arrows on your plot to indicate the direction of motion of the path traced by a
particle whose trajectory is described by Eqs. (10a)-(10b).
d) Calculate the area enclosed between the curve and the y-axis, from its first point of
intersection with the y-axis until its first point of intersection with the x-axis. Do not
count the starting point t = 0 as an intersection point. Once you arrive at a definite
integral whose integrand depends only on t, compute the integral numerically.
Hint: First calculate the values of t for the intersection points. Then, find the area following the usual
x=b(t)
formula A = f(ty dr, but make a change of variables (just like you do in regular single-variable
x=a(t)
integration) following the parametric equation x = t cost/8.
e) Determine the length of the spiral after one revolution. You will need the following
integral:
√ √₁ +²²&t=1/√ [t√₁ + 1² + m (t+ √1 +²)] + C
In](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77ccc228-61ae-45bc-bff3-36e89ce5abe9%2Fae757236-8988-4ae3-a1df-a96f5a506e71%2Fhv8glok_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the Archimedean spiral given by:
x(t) =
for t = [0, ∞0).
a) Plot the curve.
= -t cost,
1
y(t) = t si
t sint,
(10a)
(10b)
b) What are the slopes of the curve at t = 57/6 and t = 4T?
c) Draw arrows on your plot to indicate the direction of motion of the path traced by a
particle whose trajectory is described by Eqs. (10a)-(10b).
d) Calculate the area enclosed between the curve and the y-axis, from its first point of
intersection with the y-axis until its first point of intersection with the x-axis. Do not
count the starting point t = 0 as an intersection point. Once you arrive at a definite
integral whose integrand depends only on t, compute the integral numerically.
Hint: First calculate the values of t for the intersection points. Then, find the area following the usual
x=b(t)
formula A = f(ty dr, but make a change of variables (just like you do in regular single-variable
x=a(t)
integration) following the parametric equation x = t cost/8.
e) Determine the length of the spiral after one revolution. You will need the following
integral:
√ √₁ +²²&t=1/√ [t√₁ + 1² + m (t+ √1 +²)] + C
In
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