A simple pendulum is idealized by considering a bob mass swinging on an inextensible, massless rod (negligible mass relative to the bob) about a frictionless pivot (Figure 1). 0 L Figure 1 Simple pendulum The pendulum oscillation is described by the initial-value problem consisting of the equation of motion with initial conditions: 8 += sin(0) = 0, e(to) = 0o, (to) = wo Here, L is the length of the pendulum, g acceleration due to gravity, the angle the pendulum makes with the vertical, 90 the initial angular displacement, and coo the initial angular velocity. a) Transform the second-order initial-value problem into a system of first-order initial-value problems. b) Compute the angular displacement (0.2) and angular velocity (0.2) using the forward Euler method with step size h = 0.1. Use L = 0.5 m, g = 9.81 m/s², to=0, 00=rad, and wo = 0 rad/s.
A simple pendulum is idealized by considering a bob mass swinging on an inextensible, massless rod (negligible mass relative to the bob) about a frictionless pivot (Figure 1). 0 L Figure 1 Simple pendulum The pendulum oscillation is described by the initial-value problem consisting of the equation of motion with initial conditions: 8 += sin(0) = 0, e(to) = 0o, (to) = wo Here, L is the length of the pendulum, g acceleration due to gravity, the angle the pendulum makes with the vertical, 90 the initial angular displacement, and coo the initial angular velocity. a) Transform the second-order initial-value problem into a system of first-order initial-value problems. b) Compute the angular displacement (0.2) and angular velocity (0.2) using the forward Euler method with step size h = 0.1. Use L = 0.5 m, g = 9.81 m/s², to=0, 00=rad, and wo = 0 rad/s.
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:A simple pendulum is idealized by considering a bob mass swinging on an inextensible, massless rod
(negligible mass relative to the bob) about a frictionless pivot (Figure 1).
¤ + = sin(0) =
3
0
Figure 1 Simple pendulum
The pendulum oscillation is described by the initial-value problem consisting of the equation of motion
with initial conditions:
L
= 0,
0 (to) = 0o, (to) =
= wo
Here, L is the length of the pendulum, g acceleration due to gravity, the angle the pendulum makes
with the vertical, 90 the initial angular displacement, and coo the initial angular velocity.
a) Transform the second-order initial-value problem into a system of first-order initial-value
problems.
b) Compute the angular displacement (0.2) and angular velocity (0.2) using the forward
Euler method with step size h = 0.1. Use L = 0.5 m, g = 9.81 m/s², to = 0, 0,= rad, and wo
= 0 rad/s.
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