Chapter 17.3, Problem 18E

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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

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.

Write the expression for

Write the expression for auxiliary equation.

Write the expression for differential equation.

Consider the expression for differential equation as follows.

Consider the value of

Substitute

Modify equation (6) as follows.

Modify equation (4) as follows.

Compare equation (7) and (8).

Substitute

Compare equation (2) and (9).

Substitute

(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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