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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

If 6 J of work is needed to stretch a spring from 10 cm to 12 cm and another 10 J is needed to stretch it from 12 cm to 14 cm, what is the natural length of the spring?

To determine

To find:

The natural length of the spring L

Explanation

1) Concept:

i) Use Hook’s law and formula of work done for finding the natural length of the spring.

ii) 100cm=1m

2) Law and Formula:

Hook’s law:

The force required to stretch a spring 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=abfx dx

3) Calculation:

Let L be the natural length of the spring in meters.

6 J of work is needed to stretch a spring from 10 cm to 12 cm

That is 6 J of work is needed to stretch a spring from 0.1 m to 0.12 m

Means for W=6 J the length of spring L is stretched from 0.10-Lm  to 0.12-Lm

By using the formula for work done

6=0.10-L0.12-Lkx dx=12kx20.10-L0.12-L=12k0.12-L2-0.10-L2

6=12k0.12-L2-0.10-L2

6=k20.0144-20.12L+L2-0.0100+20.10L-L2

6=k20.0144-0.24L-0.0100+0.20L

6=k20.0044-0.04L

12=k0.0044-0.04L (1)

Similarly,

10 J of work is needed to stretch a spring from 12 cm to 14 cm

That is 10 J of work is needed to stretch a spring from 0.12 m to 0.14 m

Means for W=10 J the length of spring L is stretched from 0

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