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W05 - HW Solutions

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02 - Center of mass of a house

A “house” is a region bounded by the (non-regular) pentagon with vertex points at , (2, 3), (-2, 0), (-2, 3). Find the CoM of the house using additivity of moments by computing the moment of the rectangular base and the moment of the triangular roof separately.

Calculate the mass of the figure by calculating the areas of the rectangle and triangle separately.

By symmetry, note that , and .

Find and by the moment relation.

Find and by averaging out the and coordinates, respectively.

Find and by the moment relation.

Additivity of moments to find and .

Compute .

03 - CoM of region between curves

Find the CoM of the region between the graph of and the graph over .

Solution

Compute mass of the region

Write formula for .

Compute .

Write formula for .

Compute .

Compute .

05 - Fluid force on various plates

For each of the plates shown, (a)-(i), set up an integral to compute the fluid force on one face of the plate which is submerged in water.

Solution (a)

Write down the formula to find the fluid force on a submerged surface.

Identify the line .

In this case, we can set to the top of the plate, which is the same as the water line.

Identify .

At , .

At .

Using these points, the function is the line .

Identify bounds.

Shallowest point: .

Deepest point: .

.

Set up integral.

(b) Identify line .

Set as the top of the plate, so the water line is at .

To adjust for our formula, the becomes .

Identify .

At , .

At .

Using these points, the function is the line .

Identify bounds.

Shallowest point: .

Deepest point: .

.

Set up integral.

(c) Identify line .

Set as the top of the plate, so the water line is at .

To adjust for our formula, the becomes .

Identify .

At , .

At , .

Using these points, the function is the line .

Identify bounds.

Shallowest point: .

Deepest point: .

.

Set up integral.

(d) Identify line .

Set as the top of the plate, so the water line is at .

To adjust for our formula, the becomes .

Identify .

At , .

At , .

Using these points, the function is the line .

Identify bounds.

Shallowest point: .

Deepest point: .

Set up integral.

(e) Identify line .

Set as the top of the plate, so the water line is at .

To adjust for our formula, the becomes .

Identify .

At , ,

At , .

Using these points, the function is the line .

Identify bounds.

Shallowest point: .

Deepest point: .

.

Set up integral.

(f) Identify line .

Set as the water level.

Identify .

The submerged portion of the plate can be split into two portions: a square and a triangle.

The square has a constant width .

The triangle starts at .

At , .

At , .

Using these two points, the function is the line .

Identify bounds.

Square:

Shallowest point: .

Deepest point: .

Triangle:

Shallowest point: .

Deepest point: .

.

Set up integral.

(g) Identify line .

Set as the center of the circle, so the water line is at .

To adjust for our formula, the becomes .

Identify .

We know the formula for a circle of radius 2 is , so .

Note that at , , which is half of what we want.

So, .

Identify bounds:

Shallowest point: .

Deepest point: .

.

Set up integral.

(h) Identify line .

Set as the center of the circle, which is the water line.

Identify .

We know the formula for a circle of radius is .

So, .

Identify bounds:

Shallowest point: .

Deepest point: .

Set up integral.

(i) Identify line .

Set as the center of the circle, so the water line is at .

To adjust for our formula, our becomes

Identify .

We know the formula for a circle of radius 4 is .

So, .

Identify bounds:

Shallowest point: .

Deepest point: .

.

Set up integral.

07 - CoM from Simpson’s

Use Simpson’s rule (with 6 subintervals) to estimate the CoM of the region. You can use a calculator for your arithmetic. Write down formula for mass.

Write down formula for Simpson’s Rule.

Evaluate to approximate .

.

Thus, .

Write down formula for .

Evaluate to approximate and .

Write down formula for .

Evaluate to approximate

Compute .