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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Suppose that 2 J of work is needed to stretch a spring from its natural length of 30 cm to a length of 42 cm.

(a) How much work is needed to stretch the spring from 35 cm to 40 cm?

(b) How far beyond its natural length will a force of 30 N keep the spring stretched?

To determine

(a)

To find:

The work needed to stretch the spring from 35 cm to 40 cm

Explanation

1) Concept:

Use Hook’s law and formula of work done

2) Law and Formula:

Hook’s law:

The force required to maintain a spring stretched x units beyond its natural length is proportional to x, fx=kx Where k is a positive constant called the spring constant

Work done:

W=abf(x)dx

3) Calculation:

a) According to Hook’s law, the force required to hold the spring stretched x meters beyond its natural length is fx=kx

From the given information, 2 J of work is needed to stretch a spring from its natural length of 30 cm to length of 42 cm

Therefore, the amount of stretched is 42-30=12 cm=0.12 m

Thus,

2=00.12kxdx

Simplify,

2=kx2200.12

2=k(0.12)22-k(0)22

2=0.0144k2-0

2=0.0072k

k=20

To determine

b)

To find:

How far beyond its natural length will a force of 30 N keep the spring stretched?

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