Let us consider a pendulum swinging under the influence of gravitational force. Denote by y the angle of the pendulum with respect to the vertical, so that y = 0 corresponds to straight down. (Thus, y and y+27 are considered the same angle.) The (frictionless) motion of the pendulum is governed by the ODE mly" = -mg sin y, where l is the length of the pendulum rod, m is the mass, g is the gravitational constant. (a) Convert this second-order ODE into a system of first-order ODES. (b) For the system obtained in (a), let {w;}o and {z;}Lo denote the Euler approx- imations of the first and second components, respectively, with equal step size h. Write down the recurrence formulas for {w;}o and {z;}"-o· (c) Suppose the initial position of the pendulum is y(0) = 0 and the initial velocity is y(0) = 1. Using the recurrence formulas obtained in (b), find wo, w1, w2 and Z0, 21, %2.
Let us consider a pendulum swinging under the influence of gravitational force. Denote by y the angle of the pendulum with respect to the vertical, so that y = 0 corresponds to straight down. (Thus, y and y+27 are considered the same angle.) The (frictionless) motion of the pendulum is governed by the ODE mly" = -mg sin y, where l is the length of the pendulum rod, m is the mass, g is the gravitational constant. (a) Convert this second-order ODE into a system of first-order ODES. (b) For the system obtained in (a), let {w;}o and {z;}Lo denote the Euler approx- imations of the first and second components, respectively, with equal step size h. Write down the recurrence formulas for {w;}o and {z;}"-o· (c) Suppose the initial position of the pendulum is y(0) = 0 and the initial velocity is y(0) = 1. Using the recurrence formulas obtained in (b), find wo, w1, w2 and Z0, 21, %2.
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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Transcribed Image Text:Let us consider a pendulum swinging under the influence of gravitational force. Denote
by y the angle of the pendulum with respect to the vertical, so that y = 0 corresponds
to straight down. (Thus, y and y+2n are considered the same angle.) The (frictionless)
motion of the pendulum is governed by the ODE
mly" = -mg sin y,
where l is the length of the pendulum rod, m is the mass, g is the gravitational constant.
(a) Convert this second-order ODE into a system of first-order ODES.
(b) For the system obtained in (a), let {w;}o and {z;};o denote the Euler approx-
imations of the first and second components, respectively, with equal step size h.
Write down the recurrence formulas for {w;}o and {z:}-o-
(c) Suppose the initial position of the pendulum is y(0) = 0 and the initial velocity
is y (0) = 1. Using the recurrence formulas obtained in (b), find wo, wi, wz and
i=0-
20, 21, 2.
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