Setting up work and fluid-force integrals

Work with a changing force

For constant force, work is force times distance. When force varies, partition the motion, approximate force on each small interval, and integrate the products.

Springs use Hooke’s law F = kx, while lifting chains or liquids also requires tracking how much material moves how far.

Fluid force varies with depth

Pressure equals weight density times depth. A horizontal strip at one depth has nearly constant pressure, so its force is pressure times strip area.

The depth is measured from the fluid surface, not automatically from the coordinate origin.

Draw and label one representative slice

The slice is the model. Label its width, thickness, depth, and movement distance before writing the integral. Most setup errors are visible immediately on that diagram.

Worked example

Common mistakes

• Using the spring’s total length instead of displacement from equilibrium.
• Treating fluid pressure as constant across changing depth.
• Mixing centimeters and meters inside one setup.

Keep these ideas

• Build one differential contribution first.
• Depth and distance must be measured from the correct reference.
• Units are a powerful setup check.