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 $y=x^2$ and above by the line $y=1$. (Note that $y$ increases going up in this coordinate system.)

Compute the total fluid force on this plate.

(Hint: your integrand should contain $(1-y)$ as a factor.)

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Solution

Solutions---0100-02

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

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

Work performed

04 $★$

03

Work to raise a leaky bucket

A bucket of water is raised by a chain to the top of a $50$-foot building. The water is leaking out, and the chain is getting lighter.

The bucket weighs $4\mathrm{lbs}$, the initial water weighs $30\mathrm{lbs}$, and the chain weighs $0.25\mathrm{l}\mathrm{b}/\mathrm{f}\mathrm{t}$, and the water is leaking at a rate of $0.3\mathrm{l}\mathrm{b}\mathrm{s}/\mathrm{s}\mathrm{e}\mathrm{c}$ as the bucket is lifted at a constant rate of $3\mathrm{f}\mathrm{t}/\mathrm{s}\mathrm{e}\mathrm{c}$.

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

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Solution

Solutions---0120-03

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 $7\mathrm{m}$ and the radius is $5\mathrm{m}$.

(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 $3\mathrm{m}$ and the water is pumped out of a spigot extending $1\mathrm{m}$ above the top of the tank.

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Solution

Solutions---0120-04

06 $★$

05

Work to build a pyramid

The Great Pyramid of Giza is $140\mathrm{m}$ tall and has a square base with $230\mathrm{m}$ on each side. It is built of stone with mass density $2000\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$.

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

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Solution

Solutions---0120-05