W05 Regular

Due date: Sunday 2/15, 11:59pm

Hydrostatic pressure

01

02

Fluid force on a parabolic plate

A parabolic plate is submerged vertically in water as in the figure:

The shape of the plate is bounded below by and above by the line . (Note that increases going up in this coordinate system.)

Compute the total fluid force on this plate.

(Hint: your integrand should contain as a factor.)

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Solution

Solutions - 0100-02

(1) Integral formula:

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(2) Integrand components:

So we have:

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(3) Integrate:

(Assuming and .)

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02

03

Fluid force on triangular plates

For diagrams 1, 2, 3 (L to R) below, set up an integral to compute the hydrostatic force on the plate.

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Solution

Solutions - 0100-03

(1) Integral formula:

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Option 1:

(2) Using at water line, increasing downwards: (a) Left:

(b) Center:

(c) Right:

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Option 2:

(2) Using at top of shape, increasing downwards: (a) Left:

(b) Center:

(c) Right:

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03

05

Fluid force on circular plates

For diagrams 1, 2, 3 (L to R) below, set up an integral to compute the hydrostatic force on the plate.

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Solution

Solutions - 0100-05

(1) Integral formula:

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Option 1:

(2) Using at water line, increasing downwards: (a) Left:

(b) Center:

(c) Right:

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Option 2:

(2) Using at center of shape, increasing downwards: (a) Left:

(b) Center:

(c) Right:

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Work performed

04

03

Work to raise a leaky bucket

A bucket of water is raised by a chain to the top of a -foot building. The water is leaking out, and the chain is getting lighter.

The bucket weighs , the initial water weighs , and the chain weighs , and the water is leaking at a rate of as the bucket is lifted at a constant rate of .

What is the total work required to raise the bucket of water?

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Solution

Solutions - 0120-03

(1) Integral formula:

Let at the ground and increase going up.

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(2) Compute force:

The force on the rope (at the top) when the bucket is at height is:

We know .

Water is leaking at . Therefore:

The weight of chain remaining is:

Put together:

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(3) Integrate:

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05

04

Work to pump water from cylindrical tank

A cylindrical tank is full of water and the water is pumped out the top. (See figure.) The length of the tank is and the radius is .

(a) Set up an integral for the total work performed assuming the tank is initially completely full.

(b) Set up an integral for the total work performed assuming the tank is initially full to and the water is pumped out of a spigot extending above the top of the tank.

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Solution

Solutions - 0120-04

(1) Integral formula:

Set at the center of the tank.

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(a) (2) Geometry:

So:

Tank length is so a horizontal slice is a rectangle with area .

Depth is:

Therefore:

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(b) (2) Change bounds and height:

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06

05

Work to build a pyramid

The Great Pyramid of Giza is tall and has a square base with on each side. It is built of stone with mass density .

Set up an integral that expresses the work (against gravity) required to build the pyramid.

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Solution

Solutions - 0120-05

(1) Integral formula:

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(2) Integrand components:

Option 1: Set at the base, going up.

Take a cross-sectional slice with a vertical plane. This intersects the surface of the pyramid in a triangle whose width is the side length of the square (the horizontal cross section) at height .

Note that . So we have:

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Option 2:

Set at the vertex, going down.

And:

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