Examples

Pumping water from spherical tank

Calculate the work done pumping water out of a spherical tank of radius .

Solution

(1) Slice the tank of water into horizontal layers:

Coordinate is at the center of the sphere, increasing upwards.

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(2) Calculate weight of single slice:

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(3) Work to lift out single slice:

Distance to raise a slice:

Then:

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(4) Total work by integrating over all slices:

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Note A: The integration runs over all slices, which start at (bottom of tank), and end at (top of tank).

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Question: Extra question: what if the spigot sits above the tank?

Question: Extra question: what if the tank starts at just of water depth?

Water pumped from a frustum

Find the work required to pump water out of the frustum in the figure. Assume the weight of water is .

Solution

(1) Find weight of a horizontal slice.

Coordinate at top, increasing downwards.

Use for radius of cross-section circle.

Linear decrease in from to :

Area is :

:

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(2) Find work to pump out a horizontal layer.

Layer at is raised a distance of .

Work to raise layer at :

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(3) Integrate over all layers.

Integrate from top to bottom of frustum:

Final answer is .

Raising a building

Find the work done to raise a cement columnar building of height and square base per side. Cement has a density of .

Solution

(1) Weight of each layer:

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(2) Work to lift layer into place:

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(3) Find total work as integral over the layers:

Raising a chain

An chain is suspended from the top of a building. Suppose the chain has weight density . What is the total work required to reel in the chain?

Solution

(1) Compute weight of a ‘link’ (vertical slice of the chain):

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(2) Work to raise link to top:

Each link (slice) is raised from height to height 80:

Then:

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(3) Integrate over the chain for total work:

Raising a leaky bucket

Suppose a bucket is hoisted by a cable up an tower. The bucket is lifted at a constant rate of and is leaking water weight at a constant rate of . The initial weight of water is . What is the total work performed against gravity in lifting the water? (Ignore the bucket itself and the cable.)

Solution

(1) Compute total force from water :

Choose coordinate at base, at top.

Rate of water weight loss per unit height:

Total water weight at height :

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(2) Work to raise bucket by :

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(3) Total work by integrating :

Change of method and integral formula!

For this example, we use the formula rather than the formula used in the earlier examples.

• This integral sums over the work to lift macroscopic material through each microscopic as if in sequence, and thus represents distance lifted.
• Earlier examples summed over the work to lift microscopic material through the macroscope (all the way up).