Chapter 16.2, Problem 45E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

A 160-lb man carries a 25-lb can of paint up a helical staircase that encircles a silo with a radius of 20 ft. If the silo is 90 ft high and the man makes exactly three complete revolutions climbing to the top, how much work is done by the man against gravity?

To determine

**To find:** The work done by the man against the gravity.

Explanation

**Given data:**

The weight of the man is 165-lb.

Weight of the paint can is 25-lb.

The staircase encircles a silo, the radius of the silo is 20 ft, and height of the silo is 90 ft.

The man needs to make three complete revolutions to reach the top the staircase.

**Formula used:**

Write the expression to find the work done on the object with the force vector

Here,

Write the expression to find the vector

Here,

When the man moves to up, the gravity of the earth tries to pull the combined weight of man and the paint towards downside. Therefore, the force exerted by man is equal and opposite to the force exerted by the gravity.

The combined weight of the man and paint is 185-lb. As the direction of force exerted by the man is up side (*z*-direction), the force is written as follows.

As the radius of silo is 20 ft, height is 90 ft, and man needs 3 revolutions to reach top, parameterize the staircase as follows.

Consider the parameter as
*x*, *y*, and *z* parameters.

One revolution means

Write the position vector

**Calculation of**
**:**

Substitute

Sect-16.1 P-1ESect-16.1 P-2ESect-16.1 P-3ESect-16.1 P-4ESect-16.1 P-5ESect-16.1 P-6ESect-16.1 P-7ESect-16.1 P-8ESect-16.1 P-9ESect-16.1 P-10E

Sect-16.1 P-11ESect-16.1 P-12ESect-16.1 P-13ESect-16.1 P-14ESect-16.1 P-15ESect-16.1 P-16ESect-16.1 P-17ESect-16.1 P-18ESect-16.1 P-21ESect-16.1 P-22ESect-16.1 P-23ESect-16.1 P-24ESect-16.1 P-25ESect-16.1 P-26ESect-16.1 P-29ESect-16.1 P-30ESect-16.1 P-31ESect-16.1 P-32ESect-16.1 P-33ESect-16.1 P-34ESect-16.1 P-35ESect-16.1 P-36ESect-16.2 P-1ESect-16.2 P-2ESect-16.2 P-3ESect-16.2 P-4ESect-16.2 P-5ESect-16.2 P-6ESect-16.2 P-7ESect-16.2 P-8ESect-16.2 P-9ESect-16.2 P-10ESect-16.2 P-11ESect-16.2 P-12ESect-16.2 P-13ESect-16.2 P-14ESect-16.2 P-15ESect-16.2 P-16ESect-16.2 P-17ESect-16.2 P-18ESect-16.2 P-19ESect-16.2 P-20ESect-16.2 P-21ESect-16.2 P-22ESect-16.2 P-23ESect-16.2 P-24ESect-16.2 P-25ESect-16.2 P-26ESect-16.2 P-31ESect-16.2 P-32ESect-16.2 P-33ESect-16.2 P-34ESect-16.2 P-35ESect-16.2 P-36ESect-16.2 P-37ESect-16.2 P-38ESect-16.2 P-39ESect-16.2 P-40ESect-16.2 P-41ESect-16.2 P-42ESect-16.2 P-43ESect-16.2 P-44ESect-16.2 P-45ESect-16.2 P-46ESect-16.2 P-47ESect-16.2 P-48ESect-16.2 P-49ESect-16.2 P-50ESect-16.2 P-51ESect-16.2 P-52ESect-16.3 P-1ESect-16.3 P-2ESect-16.3 P-3ESect-16.3 P-4ESect-16.3 P-5ESect-16.3 P-6ESect-16.3 P-7ESect-16.3 P-8ESect-16.3 P-9ESect-16.3 P-10ESect-16.3 P-11ESect-16.3 P-12ESect-16.3 P-13ESect-16.3 P-14ESect-16.3 P-15ESect-16.3 P-16ESect-16.3 P-17ESect-16.3 P-18ESect-16.3 P-19ESect-16.3 P-20ESect-16.3 P-21ESect-16.3 P-22ESect-16.3 P-23ESect-16.3 P-24ESect-16.3 P-25ESect-16.3 P-26ESect-16.3 P-28ESect-16.3 P-29ESect-16.3 P-30ESect-16.3 P-31ESect-16.3 P-32ESect-16.3 P-33ESect-16.3 P-34ESect-16.3 P-35ESect-16.4 P-1ESect-16.4 P-2ESect-16.4 P-3ESect-16.4 P-4ESect-16.4 P-5ESect-16.4 P-6ESect-16.4 P-7ESect-16.4 P-8ESect-16.4 P-9ESect-16.4 P-10ESect-16.4 P-11ESect-16.4 P-12ESect-16.4 P-13ESect-16.4 P-14ESect-16.4 P-17ESect-16.4 P-18ESect-16.4 P-19ESect-16.4 P-20ESect-16.4 P-21ESect-16.4 P-22ESect-16.4 P-23ESect-16.4 P-24ESect-16.4 P-25ESect-16.4 P-26ESect-16.4 P-27ESect-16.4 P-28ESect-16.4 P-29ESect-16.4 P-30ESect-16.4 P-31ESect-16.5 P-1ESect-16.5 P-2ESect-16.5 P-3ESect-16.5 P-4ESect-16.5 P-5ESect-16.5 P-6ESect-16.5 P-7ESect-16.5 P-8ESect-16.5 P-9ESect-16.5 P-10ESect-16.5 P-11ESect-16.5 P-12ESect-16.5 P-13ESect-16.5 P-14ESect-16.5 P-15ESect-16.5 P-16ESect-16.5 P-17ESect-16.5 P-18ESect-16.5 P-19ESect-16.5 P-20ESect-16.5 P-21ESect-16.5 P-22ESect-16.5 P-23ESect-16.5 P-24ESect-16.5 P-25ESect-16.5 P-26ESect-16.5 P-27ESect-16.5 P-28ESect-16.5 P-29ESect-16.5 P-30ESect-16.5 P-31ESect-16.5 P-32ESect-16.5 P-33ESect-16.5 P-34ESect-16.5 P-35ESect-16.5 P-36ESect-16.5 P-37ESect-16.5 P-38ESect-16.5 P-39ESect-16.6 P-1ESect-16.6 P-2ESect-16.6 P-3ESect-16.6 P-4ESect-16.6 P-5ESect-16.6 P-6ESect-16.6 P-13ESect-16.6 P-14ESect-16.6 P-15ESect-16.6 P-16ESect-16.6 P-17ESect-16.6 P-18ESect-16.6 P-19ESect-16.6 P-20ESect-16.6 P-21ESect-16.6 P-22ESect-16.6 P-23ESect-16.6 P-24ESect-16.6 P-25ESect-16.6 P-26ESect-16.6 P-29ESect-16.6 P-30ESect-16.6 P-33ESect-16.6 P-34ESect-16.6 P-35ESect-16.6 P-36ESect-16.6 P-37ESect-16.6 P-38ESect-16.6 P-39ESect-16.6 P-40ESect-16.6 P-41ESect-16.6 P-42ESect-16.6 P-43ESect-16.6 P-44ESect-16.6 P-45ESect-16.6 P-46ESect-16.6 P-47ESect-16.6 P-48ESect-16.6 P-49ESect-16.6 P-50ESect-16.6 P-51ESect-16.6 P-52ESect-16.6 P-53ESect-16.6 P-54ESect-16.6 P-56ESect-16.6 P-59ESect-16.6 P-60ESect-16.6 P-61ESect-16.6 P-62ESect-16.6 P-63ESect-16.7 P-1ESect-16.7 P-2ESect-16.7 P-3ESect-16.7 P-4ESect-16.7 P-5ESect-16.7 P-6ESect-16.7 P-7ESect-16.7 P-8ESect-16.7 P-9ESect-16.7 P-10ESect-16.7 P-11ESect-16.7 P-12ESect-16.7 P-13ESect-16.7 P-14ESect-16.7 P-15ESect-16.7 P-16ESect-16.7 P-17ESect-16.7 P-18ESect-16.7 P-19ESect-16.7 P-20ESect-16.7 P-21ESect-16.7 P-22ESect-16.7 P-23ESect-16.7 P-24ESect-16.7 P-25ESect-16.7 P-26ESect-16.7 P-28ESect-16.7 P-37ESect-16.7 P-38ESect-16.7 P-39ESect-16.7 P-40ESect-16.7 P-41ESect-16.7 P-42ESect-16.7 P-43ESect-16.7 P-44ESect-16.7 P-45ESect-16.7 P-46ESect-16.7 P-47ESect-16.7 P-48ESect-16.7 P-49ESect-16.8 P-1ESect-16.8 P-2ESect-16.8 P-3ESect-16.8 P-4ESect-16.8 P-5ESect-16.8 P-6ESect-16.8 P-7ESect-16.8 P-8ESect-16.8 P-9ESect-16.8 P-10ESect-16.8 P-11ESect-16.8 P-12ESect-16.8 P-13ESect-16.8 P-14ESect-16.8 P-15ESect-16.8 P-17ESect-16.8 P-18ESect-16.8 P-19ESect-16.8 P-20ESect-16.9 P-1ESect-16.9 P-2ESect-16.9 P-3ESect-16.9 P-4ESect-16.9 P-5ESect-16.9 P-6ESect-16.9 P-7ESect-16.9 P-8ESect-16.9 P-9ESect-16.9 P-10ESect-16.9 P-11ESect-16.9 P-12ESect-16.9 P-13ESect-16.9 P-14ESect-16.9 P-17ESect-16.9 P-18ESect-16.9 P-19ESect-16.9 P-20ESect-16.9 P-23ESect-16.9 P-24ESect-16.9 P-25ESect-16.9 P-26ESect-16.9 P-27ESect-16.9 P-28ESect-16.9 P-29ESect-16.9 P-30ESect-16.9 P-31ESect-16.9 P-32ECh-16 P-1RCCCh-16 P-2RCCCh-16 P-3RCCCh-16 P-4RCCCh-16 P-5RCCCh-16 P-6RCCCh-16 P-7RCCCh-16 P-8RCCCh-16 P-9RCCCh-16 P-10RCCCh-16 P-11RCCCh-16 P-12RCCCh-16 P-13RCCCh-16 P-14RCCCh-16 P-15RCCCh-16 P-16RCCCh-16 P-1RQCh-16 P-2RQCh-16 P-3RQCh-16 P-4RQCh-16 P-5RQCh-16 P-6RQCh-16 P-7RQCh-16 P-8RQCh-16 P-9RQCh-16 P-10RQCh-16 P-11RQCh-16 P-12RQCh-16 P-13RQCh-16 P-1RECh-16 P-2RECh-16 P-3RECh-16 P-4RECh-16 P-5RECh-16 P-6RECh-16 P-7RECh-16 P-8RECh-16 P-9RECh-16 P-10RECh-16 P-11RECh-16 P-12RECh-16 P-13RECh-16 P-14RECh-16 P-15RECh-16 P-16RECh-16 P-17RECh-16 P-18RECh-16 P-19RECh-16 P-20RECh-16 P-21RECh-16 P-22RECh-16 P-23RECh-16 P-24RECh-16 P-25RECh-16 P-27RECh-16 P-28RECh-16 P-29RECh-16 P-30RECh-16 P-31RECh-16 P-32RECh-16 P-33RECh-16 P-34RECh-16 P-35RECh-16 P-36RECh-16 P-37RECh-16 P-38RECh-16 P-39RECh-16 P-40RECh-16 P-41RECh-16 P-1PCh-16 P-2PCh-16 P-3PCh-16 P-5PCh-16 P-6P

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