Due date: Thursday 2/12, 11:59pm
Hydrostatic pressure
01
01
Link to originalFluid force on a triangular plate
Find the total force on the submerged vertical plate that is an isosceles triangle with (bottom) base
and height , and assume it is submerged with the upper vertex below the surface. Liquid is oil with density .
Solution
Solutions - 0100-01
(1) Integral formula:
(2) Integrand components:
Width function:
Depth function:
(3) Integrate:
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02
04
Link to originalFluid force on trapezoidal plates
For diagrams 1, 2, 3 (L to R) below, set up an integral to compute the hydrostatic force on the plate.
Solution
Solutions - 0100-04
(a) Left:
Set
at the water line, increasing downwards.
Alternative: Set
at the top of the trapezoid. Obtain:
(b) Center:
Set
at the water line, increasing downwards.
Alternative: Set
at the top of the trapezoid. Obtain:
(c) Right:
Set
at the water line, increasing downwards. Link to original
Work performed
03
01
Link to originalPumping water from hemispherical tank
A hemispherical tank (radius
) is full of water. A pipe allows water to be pumped out, but requires pumping up above the top of the tank.
(a) Set up an integral that expresses the total work required to pump all the water out of the tank, assuming it is completely full.
(b) Now assume the tank start out full just to
. What does the integral become?
Solution
Solutions - 0120-01
(a) (1) Integral formula:
(2) Setup:
Coordinate system: set
at the top of the tank, increasing downwards. Horizontal slice of the tank: disk of radius
at depth , satisfies: Distance pumped up (add
for the spigot): Thus:
(b) (1) Change upper bound, top of water at
: Link to original
04
02
Link to originalBuilding a conical tower
Set up an integral that expresses the work done (against gravity) to build a circular cone-shaped tower of height
and base radius out of a material with mass density .
Solution
Solutions - 0120-02
(1) Integral formula:
Option 1: (2) Setup:
Set
at the bottom, increasing upwards. Radius of the cone with a QLIF:
Horizontal slice of the cone tower: disk of radius
at height , satisfies: The slice at
is raised a distance of . Thus:
Option 2: (2) Setup:
Set
at the top of the cone, increasing downwards. Now
is the distance from the ground up to the height of a slice indexed by . Radius function:
Thus:
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