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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.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.5: Applications
Problem 19EQ
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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.
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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