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

Spring Pendulum. Let a mass be attached to one end of a spring with spring constant k and the other end attached to the ceiling. Let $l_{o}$ be the natural length of the spring, and let l(t) be its length at time t. If $θ (t)$ is the angle between the pendulum and the vertical, then the motion of the spring pendulum is governed by the system
$l'' (t) - l (t) θ' (t) - gcosθ (t) + \frac{k}{m} (l - l_{o}) = 0 l^{2} (t) θ'' (t) + 2 l (t) l' (t) θ' (t) + gl (t) sinθ (t) = 0$
Assume g = 1, k = m = 1, and $l_{o}$ = 4. When the system is at rest, $l = l_{o} + \frac{mg}{k} = 5$ .
a. Describe the motion of the pendulum when $l (0) = 5 . 5, l' (0) = 0, θ (0) = 0, θ' (0) = 0$ .
b. When the pendulum is both stretched and given an angular displacement, the motion of the pendulum is more complicated. Using the Runge–Kutta algorithm for systems with h = 0.1 to approximate the solution, sketch the graphs of the length l and the angular displacement u on the interval [0,10] if $l (0) = 5 . 5, l' (0) = 0, θ (0) = 0 . 5, θ' (0) = 0$ .

a. The pendulum would just move up and down in a periodic way.

b. See the table.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Transform the equation

Here, the equation is:

$l'' (t) - l (t) θ' (t) - gcosθ (t) + \frac{k}{m} (l - l_{o}) = 0 l^{2} (t) θ'' (t) + 2 l (t) l' (t) θ' (t) + gl (t) sinθ (t) = 0$

The system can be written as;

$x_{1} = l x_{2} = l' = x'_{1} x_{3} = θ x_{4} = θ' = x'_{3}$

The transform equation is;

$x'_{1} = x_{2} x'_{2} = l'' = x_{1} x_{4} + \cos (x_{3}) - x_{1} + 4 x'_{3} = x_{4} x'_{4} = \frac{- 2 x_{2} x_{4} - \sin (x_{3})}{x_{1}}$

The initial conditions are;

$x_{1} (0) = l (0) = 5 . 5 x_{2} (0) = l' (0) = 0 x_{3} (0) = θ (0) = 0 . 5 x_{4} (0) = θ' (0) = 0$

Apply Runge-Kutte method

The values are;

T	L	$θ$
0	5.5	0.5
0.1	5.49	0.499
0.5	5.41	0.488
1	5.13	0.454
2	4.105	0.289
2.5	3.47	0.131
3	2.89	-0.105
4	2.11	-0.835
5	2.13	-1.511
6	3.344	-1.633
9.9	7.6868	-0.696

Graphs

Graph for l(t)

Graph for $θ (t)$ .

This is the required result.

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