   Chapter 16.2, Problem 46E

Chapter
Section
Textbook Problem

Suppose there is a hole in the can of paint in Exercise 45 and 9 lb of paint leaks steadily out of the can during the man’s ascent. How much work is done?

To determine

To find: The work done by the man against the gravity when the can leaks the paint.

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.

The can has a hole and it leaks the paint of 9-lb steadily when the man moves to the top side of the staircase.

Formula used:

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

W=abFr(t)dt (1)

Here,

F(t) is the force vector exerted by the man,

r(t) is the velocity vector of the object, which is the derivative of position vector r(t) ,

a is the lower limit of the scalar parameter and

b is the upper limit of the scalar parameter.

Write the expression to find the vector r(t) .

r(t)=ddt[r(t)] (2)

Here,

r(t) is the position vector.

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, the force of the man is towards the z-direction.

As the can leaks 9-lb of the paint steadily, the force is a function of time t and it is written as follows.

F=(0)i+(0)j+[1859-lb3-rev(2π)t]k=(0)i+(0)j+(18532πt)k

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 t and write the x, y, and z parameters.

One revolution means 2π and 3 complete revolutions means 6π .

x=20cost,y=20sint,z=906πt,0t6πx=20cost,y=20sint,z=15πt,0t6π

Write the position vector r(t) from the parametric equations as follows.

r(t)=20costi+20sintj+15πtk

Calculation of r(t) :

Substitute 20costi+20sintj+15πtk for r(t) in equation (2),

r(t)=ddt[20costi+20sintj+15πtk]=ddt(20cost)i+ddt(20sint)j+ddt(15πt)k=(20sint)i+(20cost)j+(15π)k

Calculation of work done by the man:

Substitute [(0)i+(0)j+(18532πt)k] for F , [(20sint)i+(20cost)j+(15π)k] for r(t)

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