A classical problem in the calculus of variations is to find the shape of a curve C such that a bead, under the influence of gravity, will slide from point A(0, 0) to point B(x,. Y,) in the least time, as in the figure below. It can be shown that a nonlinear differential equation for the shape y(x) of the path is y[1 + (y')?] = k, where k is a constant. A(0, 0) bead mg B(x1, y1) Find an expression for dx in terms of y and dy. dx = Use the substitution y = k sin?(e) to obtain a parametric form of the solution. The curve Cturns out to be a cycloid. x(0) =
A classical problem in the calculus of variations is to find the shape of a curve C such that a bead, under the influence of gravity, will slide from point A(0, 0) to point B(x,. Y,) in the least time, as in the figure below. It can be shown that a nonlinear differential equation for the shape y(x) of the path is y[1 + (y')?] = k, where k is a constant. A(0, 0) bead mg B(x1, y1) Find an expression for dx in terms of y and dy. dx = Use the substitution y = k sin?(e) to obtain a parametric form of the solution. The curve Cturns out to be a cycloid. x(0) =
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
![A classical problem in the calculus of variations is to find the shape of a curve \( \mathcal{C} \) such that a bead, under the influence of gravity, will slide from point \( A(0, 0) \) to point \( B(x_1, y_1) \) in the least time, as in the figure below. It can be shown that a nonlinear differential equation for the shape \( y(x) \) of the path is \( y(1 + (y')^2) = k \), where \( k \) is a constant.
**Diagram Description:**
- The graph shows a curve \( \mathcal{C} \) between two points: \( A(0, 0) \) at the origin and \( B(x_1, y_1) \).
- A bead is positioned on the curve, with a force \( mg \) acting vertically downward.
**Task:**
Find an expression for \( dx \) in terms of \( y \) and \( dy \).
\[ dx = \_\_\_\_\_ \]
Use the substitution \( y = k \sin^2(\theta) \) to obtain a parametric form of the solution. The curve \( \mathcal{C} \) turns out to be a cycloid.
\[ x(\theta) = \_\_\_\_\_ \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd608d232-a827-4526-b5b4-a763d0855f8e%2F7fb7e1c0-bdb1-4ff3-8c97-5e464fbb19b9%2Fhe9d5u_processed.png&w=3840&q=75)
Transcribed Image Text:A classical problem in the calculus of variations is to find the shape of a curve \( \mathcal{C} \) such that a bead, under the influence of gravity, will slide from point \( A(0, 0) \) to point \( B(x_1, y_1) \) in the least time, as in the figure below. It can be shown that a nonlinear differential equation for the shape \( y(x) \) of the path is \( y(1 + (y')^2) = k \), where \( k \) is a constant.
**Diagram Description:**
- The graph shows a curve \( \mathcal{C} \) between two points: \( A(0, 0) \) at the origin and \( B(x_1, y_1) \).
- A bead is positioned on the curve, with a force \( mg \) acting vertically downward.
**Task:**
Find an expression for \( dx \) in terms of \( y \) and \( dy \).
\[ dx = \_\_\_\_\_ \]
Use the substitution \( y = k \sin^2(\theta) \) to obtain a parametric form of the solution. The curve \( \mathcal{C} \) turns out to be a cycloid.
\[ x(\theta) = \_\_\_\_\_ \]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
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,
