Explanation Procedure used: Use the method of characteristics to solve the initial value problem describing the motion in terms of displacement θ ( t ) of a mass m attached to a pendulum of length L is a determined by the initial value problem, d 2 θ d t 2 + g L θ = 0 with initial conditions, θ ( 0 ) = θ 0 d θ d t ( 0 ) = v 0 Here, v 0 is the initial velocity of the mass. Substitute θ ( t ) = 0 to find the time at which the pendulum is vertical. Calculation: The initial value problem is, d 2 θ d t 2 + g L θ = 0 ( 1 ) Let ω 2 = g L then the equation ( 1 ) becomes, d 2 θ d t 2 + ω 2 θ = 0 The corresponding characteristics-equation for the above equation is as follows. r 2 + ω 2 = 0 The solutions of the above equation are determined as follows. r 2 = − ω 2 r = ± − ω 2 = ± ω i So, the roots of characteristics equation are r 1 = − ω i and r 2 = ω i . Thus, a general solution is given by, θ ( t ) = c 1 cos ω t + c 2 sin ω t … ( 2 ) Now, differentiate both sides of equation ( 2 ) with respect to t θ ′ ( t ) = − c 1 ω sin ω t + c 2 ω cos ω t … ( 3 ) Use initial condition θ ( 0 ) = θ 0 in equation ( 2 ) θ ( 0 ) = c 1 cos ( 0 ) + c 2 sin ( 0 ) θ 0 = c 1 Use initial condition d θ d t ( 0 ) = v 0 in equation ( 3

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ISBN: 9780128149485

Author: Abell, Martha L. L.

Publisher: Elsevier Science

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Chapter 5.5, Problem 16E

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To find: The values of t for which the pendulum is vertical for the undamped problem given by the initial value problem d2θdt2+gLθ=0, θ(0)=θ0, dθdt(0)=v0.

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Chapter 5.5, Problem 15E

Chapter 5.5, Problem 17E

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Chapter 5 Solutions

Introductory Differential Equations

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The values of t for which the pendulum is vertical for the undamped problem given by the initial value problem d 2 θ d t 2 + g L θ = 0 , θ ( 0 ) = θ 0 , d θ d t ( 0 ) = v 0 .

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To find: The values of t for which the pendulum is vertical for the undamped problem given by the initial value problem d2θdt2+gLθ=0, θ(0)=θ0, dθdt(0)=v0.

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Chapter 5 Solutions