MAT 21B Lecture 15: MAT 21B – Lecture 15 – Application of Definite Integrals on Fluid Forces
MAT 21B – Lecture 15 – Application of Definite Integrals on Fluid Forces
• The pressure,
where F is force and A is area. If force has and area has
, then the pressure P will have
as its units. We can find the force, F using
the formula for pressure to obtain
.
• In fluid, pressure is omni-dimensional. Pressure exits in every direction.
• The density of water,
and the fluid force on the base of the
container, with = length, = width. The pressure of the
container at depth, D is
(since )
• Problem: Find the force of the liquid inside the container with depth, D and a
constant density, .
The force of the liquid is
The pressure,
If the density, is a constant,
then
In other words,
.
• Example: What is the force of the water on this dam?
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Document Summary
Mat 21b lecture 15 application of definite integrals on fluid forces: the pressure, = where f is force and a is area. If force has and area has (cid:2870), then the pressure p will have (cid:3118) as its units. We can find the force, f using the formula for pressure to obtain =(cid:3118) (cid:2870)= (cid:4666) (cid:4667). The pressure of the container at depth, d is = = . (since = (cid:1875)) constant density, . The force of the liquid is = = (cid:4666)(cid:1877)(cid:4667)= (cid:4666)(cid:1877)(cid:4667) (cid:1875) (cid:1877)= (cid:2868) (cid:1875) (cid:4666)(cid:1877)(cid:4667)(cid:1877). In other words, = = . then (cid:1877)= (cid:2868: example: what is the force of the water on this dam? (cid:2868) The force, = = = (cid:4666)(cid:1877)(cid:4667)(cid:1875)(cid:4666)(cid:1877)(cid:4667)(cid:1877)= (cid:2869)(cid:2868) (cid:2868) (cid:4666)(cid:883)(cid:882) (cid:1877)(cid:4667) (cid:1877)= (cid:888)(cid:884) (cid:3119)(cid:4666)(cid:883)(cid:882) (cid:1877)(cid:4667) (cid:883)(cid:882)(cid:882) (cid:1877)= (cid:888)(cid:884) (cid:3119) (cid:883)(cid:882)(cid:882) (cid:2869)(cid:2868) (cid:2869)(cid:2868) (cid:2868) (cid:888)(cid:884)(cid:882)(cid:882) (cid:3118) [(cid:883)(cid:882)(cid:1877) (cid:2869)(cid:2870)(cid:1877)(cid:2870)](cid:2868)(cid:2869)(cid:2868)=(cid:888)(cid:884)(cid:882)(cid:882) (cid:3118)[(cid:4666)(cid:883)(cid:882)(cid:4667)(cid:2870) (cid:2869)(cid:2870)(cid:4666)(cid:883)(cid:882)(cid:4667)(cid:2870)]= (cid:2868) (cid:888)(cid:884)(cid:882)(cid:882) (cid:3118) (cid:887)(cid:882)(cid:2870)= (cid:885)(cid:883)(cid:882),(cid:882)(cid:882)(cid:882) . This time, we will be using a different coordinate system.
