Chapter 17.3, Problem 18E

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

The figure shows a pendulum with length I, and the angle θ from the vertical to the pendulum. It can be shown that θ, as a function of time, satisfies the nonlinear differential equation d 2 θ d t 2 + g L sin θ = 0 where g is the acceleration due to gravity. For small values of θ we can use the linear approximation sin θ = θ and then the differential equation becomes linear.(a) Find  the equation of motion of a pendulum with length 1 m. if θ is initially 0.2 rad and the initial angular velocity is d θ/dt =1 rad/s.(b) What is the maximum angle from the vertical?(c) What is the period of the pendulum (that is, the time to complete one back-and-forth swing)?(d) When will the pendulum first be vertical?(e) What is the angular velocity when the pendulum is vertical?

(a)

To determine

To find: The equation of motion of a pendulum with length.

Explanation

Given data:

Formula used:

Write the expression for general solution complex roots.

θ(t)=eαt[c1cos(βt)+c2sin(βt)] (1)

Write the expression for r .

r=α+βi (2)

Write the expression for auxiliary equation.

ar2+br+c=0 (3)

Write the expression for differential equation.

ay+by+cy=0 (4)

Consider the expression for differential equation as follows.

d2θdt2+gLsinθ=0 (5)

Consider the value of g as follows.

g=9.8

Substitute 9.8 for g , 1 for L and θ for sinθ in equation (5),

d2θdt2+9.81θ=0

d2θdt2+9.8θ=0 (6)

Modify equation (6) as follows.

θ+9.8θ=0 (7)

Modify equation (4) as follows.

aθ+bθ+cθ=0 (8)

Compare equation (7) and (8).

a=1b=0c=9.8

Substitute 1 for a , 0 for b and 9.8 for c in equation (3),

(1)r2+(0)r+9.8=0r2+9.8=0r2=9.8

r=±9.8i (9)

Compare equation (2) and (9).

α=0β=9.8

Substitute 0 for α and 9.8 for β in equation (1),

θ(t)=e(0)t[c1cos(9

(b)

To determine

To find: The maximum angle from the vertical.

(c)

To determine

To find: The period of the pendulum.

(d)

To determine

To find: The time at which pendulum should be vertical for first time.

(e)

To determine

To find: The angular velocity when the pendulum is vertical.

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