Work pumping fluid from a tank (with many examples), Single Variable Calculus

In this video, we compute the work done pumping fluid (water, oil) from a tank. First diagram the situation with a slice of the fluid, then do (1) slice volume, (2) slice mass, (3) force on the slice, (4) work done lifting the slice, (5) total work. Examples include a cylindrical tank, an inverted cone, a hemisphere, a parabolic tank, and a triangular prism. (Error: the first numerical answer should be 3.9 x 10⁶ J.)

00:00 Intro
02:54 First Example: Pumping Fluid Out of a Cylindrical Tank [first solution should be 3.9*10^6 J.]
17:05 Second Example: Cone Tank
26:56 Third Example: Hemisphere Tank
37:28 Third Example revisited (different orientation)
41:15 Fourth Example: Parabolic Tank
47:55 Final Example: Triangular prism

Key steps outlined include:

1. Sketching the Problem: A critical first step is to draw a picture of the tank and the situation to visualize the problem better.
2. Slice Volume Calculation: Determine the volume of a fluid slice, considering the shape of the tank (cylindrical, conical, hemispherical, or triangular prism) and its dimensions.
3. Mass Calculation: Convert volume to mass by multiplying by the fluid's density, with examples given in both standard international units (meters, kilograms) and American units (feet, pounds) as applicable.
4. Force Calculation: Calculate the force required to move the fluid slice using the formula force = mass × acceleration, with acceleration due to gravity.
5. Work Calculation for a Slice: Determine the work required to move a single slice of fluid by considering force and displacement.
6. Total Work Calculation: Integrate the work for a single slice over the entire fluid volume to find the total work required to pump all the fluid out of the tank.

We work through several examples, including tanks that are fully or partially filled with fluid, and demonstrate how to apply this methodology to tanks of different shapes, including adjustments for non-cylindrical tanks and considerations for specific weights in different unit systems.

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