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W05 - Examples

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Hydrostatic force

15 - Fluid force on a triangular plate

Find the total force on the submerged vertical plate with the following shape: Equilateral triangle, sides , top vertex at the surface, liquid is oil with density .

200

Solution Establish coordinate system: height increases going down.

Compute width function .

Drop a perpendicular from top vertex to the base.

Pythagorean Theorem: vertical height is .

Similar triangles: ratio must equal ratio .

Solve for :

Write integral using width function.

Bounds: shallowest: ; deepest: .

Integral formula:

Compute integral.

Simplify constants:

Compute integral without constants:

Combine for the final answer:

16 - Weight of water on a dam

Find the total hydrostatic force on an angled dam with the following geometric description: Tilted trapezoid. Base , Top , and vertical height . The base is tilted at an angle of .

200

Solution Establish coordinate system: height increases going down.

Find similar triangles to describe width function.

Draw line from lower corner to upper edge, perpendicular to upper and lower edges.

Identify as the distance from drawn line to outer edge of the trapezoid.

So ranges from at the bottom () to at the top .

Similar triangles yields equal ratios similarity equation:

Width function is

Simplify width function.

Solve similarity equation

Plug result into width function:

Simplify:

Incorporate angle of incline in strip thickness.

Infinitesimal thickness element is $dz=\csc 55^\circ,dy.

Calculate total force using integral formula.

Plug data into formula:

Simplify:

Plug in constants and if the final number is desired.

17 - CoM of a parabolic plate

Find the CoM of the region depicted: 200

Solution Compute the total mass.

Area under the curve with density factor :

Compute .

Formula:

Interpret and calculate:

Compute .

Formula:

Width of horizontal strips between the curves:

Interpret :

Plug data into integral:

Calculate integral:

Compute CoM coordinates from moments.

CoM formulas:

Insert data:

Therefore CoM is located at .

18 - Computing CoM using only vertical strips

Find the CoM of the region: 300

Solution Compute the total mass .

Area under the curve times density :

Compute using vertical strips.

Plug into formula:

Integration by parts.

Set , ; then , .

IBP formula:

Plug in data:

Evaluate:

Compute also using vertical strips.

Plug and into formula:

Integration by ‘power to frequency conversion’:

Use :

Integrate:

Compute CoM.

CoM via moment formulas:

Plug in data:

Plug in data:

CoM is given by .

19 - CoM of region between line and parabola

Compute the CoM of the region below and above with .

Solution Name the functions: (lower) and (upper) over .

Compute the mass .

Area between curves times density:

Compute using vertical strips.

Plug into formula:

Compute also using vertical strips.

Plug into formula:

Compute CoM.

Using CoM via moment formulas:

CoM is given by .

20 - Center of mass using moments and symmetry

Find the center of mass of the region below.

200

Solution Symmetry: CoM on Because the region is symmetric in the -axis, the CoM must lie on that Therefore .

Additivity of moments.

Write for the total -moment (distance measured to the -axis from above).

Write and for the -moments of the triangle and circle.

Additivity of moments equation:

Find moment of the circle .

By symmetry we know .

By symmetry we know .

Area of circle with is , so total mass is .

Centroid-from-moments equation:

Solve the equation for .

Solve:

Find moment of the triangle using integral formula

Similar triangles:

200

Similarity equation:

Integral formula:

Perform integral:

Conclude:

Apply additivity.

Additivity formula:

Total mass of region.

Area of circle is .

Area of triangle is .

Thus .

Compute center of mass from total and total .

We have and .

Plug into formula:

Final answer is .

21 - Center of mass - two part region

Find the center of mass of the region which combines a rectangle and triangle (as in the figure) by computing separate moments. What are those separate moments? Assume the mass density is .

200

Solution

By symmetry, the center of mass of the rectangle is located at .

Thus and .

Find moments of the rectangle.

Total mass of rectangle .

Apply moment relation:

Find moments of the triangle.

Area of vertical slice .

Distance from -axis .

Total integral:

Total integral:

Add up total moments.

General formulas: and

Plug in data: and

Find center of mass from moments.

Total mass of triangle .

Total combined mass .

Apply moment relation:

Therefore, center of mass is .