Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550



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?



To determine

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


Given data:

sinθθ , L=1m , θ(0)=0.2rad , dθ(0)dt=1rads , d2θdt2+gLsinθ=0 .

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.


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


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).


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


r=±9.8i (9)

Compare equation (2) and (9).


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



To determine

To find: The maximum angle from the vertical.


To determine

To find: The period of the pendulum.


To determine

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


To determine

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

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Chapter 17 Solutions

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Sect-17.1 P-11ESect-17.1 P-12ESect-17.1 P-13ESect-17.1 P-14ESect-17.1 P-15ESect-17.1 P-16ESect-17.1 P-17ESect-17.1 P-18ESect-17.1 P-19ESect-17.1 P-20ESect-17.1 P-21ESect-17.1 P-22ESect-17.1 P-23ESect-17.1 P-24ESect-17.1 P-25ESect-17.1 P-26ESect-17.1 P-27ESect-17.1 P-28ESect-17.1 P-29ESect-17.1 P-30ESect-17.1 P-31ESect-17.1 P-32ESect-17.1 P-33ESect-17.1 P-34ESect-17.2 P-1ESect-17.2 P-2ESect-17.2 P-3ESect-17.2 P-4ESect-17.2 P-5ESect-17.2 P-6ESect-17.2 P-7ESect-17.2 P-8ESect-17.2 P-9ESect-17.2 P-10ESect-17.2 P-11ESect-17.2 P-12ESect-17.2 P-13ESect-17.2 P-14ESect-17.2 P-15ESect-17.2 P-16ESect-17.2 P-17ESect-17.2 P-18ESect-17.2 P-19ESect-17.2 P-20ESect-17.2 P-21ESect-17.2 P-22ESect-17.2 P-23ESect-17.2 P-24ESect-17.2 P-25ESect-17.2 P-26ESect-17.2 P-27ESect-17.2 P-28ESect-17.3 P-1ESect-17.3 P-2ESect-17.3 P-3ESect-17.3 P-4ESect-17.3 P-5ESect-17.3 P-6ESect-17.3 P-7ESect-17.3 P-8ESect-17.3 P-9ESect-17.3 P-10ESect-17.3 P-11ESect-17.3 P-12ESect-17.3 P-13ESect-17.3 P-14ESect-17.3 P-15ESect-17.3 P-16ESect-17.3 P-17ESect-17.3 P-18ESect-17.4 P-1ESect-17.4 P-2ESect-17.4 P-3ESect-17.4 P-4ESect-17.4 P-5ESect-17.4 P-6ESect-17.4 P-7ESect-17.4 P-8ESect-17.4 P-9ESect-17.4 P-10ESect-17.4 P-11ESect-17.4 P-12ECh-17 P-1RCCCh-17 P-2RCCCh-17 P-3RCCCh-17 P-4RCCCh-17 P-5RCCCh-17 P-1RQCh-17 P-2RQCh-17 P-3RQCh-17 P-4RQCh-17 P-1RECh-17 P-2RECh-17 P-3RECh-17 P-4RECh-17 P-5RECh-17 P-6RECh-17 P-7RECh-17 P-8RECh-17 P-9RECh-17 P-10RECh-17 P-11RECh-17 P-12RECh-17 P-13RECh-17 P-14RECh-17 P-15RECh-17 P-16RECh-17 P-17RECh-17 P-18RECh-17 P-19RECh-17 P-20RECh-17 P-21RE

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