Chapter 17.3, Problem 4E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

A force of 13 N is needed to keep a spring with a 2-kg mass stretched 0.25 m beyond its natural length. The damping constant of the spring is *c* = 8.

(a) If the mass starts at the equilibrium position with a velocity of 0.5 m/s, find its position at time *t*.

(b) Graph the position function of the mass.

(a)

To determine

**To find:** The position of the mass at any time

Explanation

**Given data:**

The spring is stretched beyond its natural length, so

**Formula used:**

Write the expression for Hookeâ€™s Law.

Here,

Write the expression for damping force.

Here,

Write the expression for Newtonâ€™s Second Law.

Write the expression for auxiliary equation.

Write the expression for the roots.

Here,

Write the expression for general solution of under damping case.

Substitute

Substitute

Find the auxiliary equation using equation (4).

Find the value of

Substitute

Simplify

Find the value of

Substitute

Since, the spring starts at equilibrium the value of

(b)

To determine

**To graph:** The position function of the mass.

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