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Q3-3.4-17E

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Fundamentals Of Differential Equations And Boundary Value Problems
Found in: Page 116
Fundamentals Of Differential Equations And Boundary Value Problems

Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

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Short Answer

In Problem 16, let I = 50 kg-m2 and the retarding torque be N-mIf the motor is turned off with the angular velocity at 225 rad/sec, determine how long it will take for the flywheel to come to rest.

The flywheel takes to come to rest in 300 sec.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Find the value of time

The equation is

Idωdt=-T150dωdt=-5ω-10∫dωω=∫dt \begingathered-20ω=t+cω(t)=(-t20+c)2\endgathered-20ω=t+cω(t)=(-t20+c)2

Step 2: Apply the given conditions

When ω0 = 225 , c = 15

ω(t)=(-t20+15)2t=20(15-ω(t))

at the moment when flywheel stopes rotating (t)=0,

so, t = 300 sec

Hence, the flywheel takes to come to rest in 300 sec.

Most popular questions for Math Textbooks

Use the fourth-order Runge–Kutta subroutine with h = 0.25 to approximate the solution to the initial value problem y'=1-y,y(0)=0 , at x = 1. Compare this approximation with the one obtained in Problem 6 using the Taylor method of order 4.

Use the improved Euler’s method subroutine with step size h = 0.2 to approximate the solution to the initial value problem y'=1x(y2+y),y(1)=1 at the points x = 1.2, 1.4, 1.6, and 1.8. (Thus, input N = 4.) Compare these approximations with those obtained using Euler’s method (see Exercises 1.4, Problem 6, page 28).

Use the fourth-order Runge–Kutta subroutine with h = 0.25 to approximate the solution to the initial value problem y'=2y-6,y(0)=1, at x = 1. (Thus, input N = 4.) Compare this approximation to the actual solution y=3-2e2x evaluated at x = 1.

Use the improved Euler’s method subroutine with step size h = 0.2 to approximate the solution to at the points x = 0, 0.2, 0.4, …., 2.0. Use your answers to make a rough sketch of the solution on [0, 2].

Local versus Global Error. In deriving formula (4) for Euler’s method, a rectangle was used to approximate the area under a curve (see Figure 3.14). With

\({\bf{g(t) = f(t,f(t))}}\) , this approximation can be written as \(\int\limits_{{{\bf{x}}_{\bf{n}}}}^{{{\bf{x}}_{{\bf{n + 1}}}}} {{\bf{g(t)dt}} \approx {\bf{hg(}}{{\bf{x}}_{\bf{n}}}{\bf{)}}} \)where \({\bf{h = }}{{\bf{x}}_{{\bf{n + 1}}}}{\bf{ - }}{{\bf{x}}_{\bf{n}}}\) .

  1. Show that if g has a continuous derivative that is bounded in absolute value by B, then the rectangle approximation has error\(\left( {\bf{O}} \right){{\bf{h}}^{\bf{2}}}\); that is, for some constant M, \(\left| {\int\limits_{{{\bf{x}}_{\bf{n}}}}^{{{\bf{x}}_{{\bf{n + 1}}}}} {{\bf{g(t)dt - hg(}}{{\bf{x}}_{\bf{n}}}{\bf{)}}} } \right| \le {\bf{M}}{{\bf{h}}^{\bf{2}}}\).This is called the local truncation error of the scheme. [Hint: Write \(\int\limits_{{{\bf{x}}_{\bf{n}}}}^{{{\bf{x}}_{{\bf{n + 1}}}}} {{\bf{g(t)dt - hg(}}{{\bf{x}}_{\bf{n}}}{\bf{)}}} {\bf{ = }}\int\limits_{{{\bf{x}}_{\bf{n}}}}^{{{\bf{x}}_{{\bf{n + 1}}}}} {\left[ {{\bf{g(t)dt - g(}}{{\bf{x}}_{\bf{n}}}{\bf{)}}} \right]{\bf{dt}}} \). Next, using the mean value theorem, show that\(\left| {{\bf{g(t)dt - g(}}{{\bf{x}}_{\bf{n}}}{\bf{)}}} \right| \le {\bf{B}}\left| {{\bf{t - }}{{\bf{x}}_{\bf{n}}}} \right|\) . Then integrate to obtain the error bound\(\left( {\frac{{\bf{B}}}{{\bf{2}}}} \right){{\bf{h}}^{\bf{2}}}\).]
  2. In applying Euler’s method, local truncation errors occur in each step of the process and are propagated throughout the further computations. Show that the sum of the local truncation errors in part (a) that arise after n steps is (O)h. This is the global error, which is the same as the [ss1] [m2] convergence rate of Euler’s method.

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