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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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BuyFindarrow_forward

Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it.

A chain lying on the ground is 10 m long and its mass is 80 kg. How much work is required to raise one end of the chain to a height of 6 m?

To determine

To find:

Work required to raise one end of the chain to a height of 6 m

Explanation

1) Concept:

Approximate the required work by using the concept of Riemann sum.Then express the work as an integral and evaluate it.

2) Given:

Length of chain 10 m

Mass of chain 80 kg

3) Definition 4:

W=limnt=1nfxi* x=abfx dx

4) Calculation:

Chain lying on the ground is 10m long

Out of the 10m long wire, one end of the chain is raised to a height of 6m, 4 m of the chain lying along the ground.

Total length of the chain if one end of the chain is raised to a height of 6m is 6m

Total mass of chain 80 kg

Total length of chain 10 m

Mass per unit length of chain is (80kg)/(10m)=8 kg/m

The force acting on the chain is F=mg

Where m is mass of chain and g is acceleration due to gravity

Force per unit length acting on the chain is F=(8 kg/m)(9.8 m/s2)=78.4N/m

The work done acting on the chain is W=F·d

Where F is force and d is distance

To find the work required to raise one end of the chain to a height of 6 m

Use an argument similar to the one that leads to Definition 4

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