Comprehensive Hydraulics Problem Solutions

I have five graduate-level hydraulics questions that need fully worked, textbook-quality solutions. You will show every step, state each assumption you make, and keep the units consistent from start to finish. Hand-drawn, clearly labeled diagrams are mandatory for every problem; scan them neatly so the geometry and force directions are unmistakable.

The set covers:
• Fluid mechanics – with special emphasis on viscous shear and torque (treat these two aspects as equally important, as this section is my top priority).
• Hydrostatics – complete force and centroid calculations on submerged surfaces.
• Momentum & energy – a water-jet striking a moving cart.
• Pipe flow – friction losses via Darcy-Weisbach, friction factor from the Colebrook/Moody approach, plus all minor losses.
• Open-channel flow – weir analysis, critical depth, and specific energy checks.

For every question you must:
1. Begin with the governing equations, then carry the algebra through to the numeric result.
2. Cite and apply the correct formula (Darcy-Weisbach, Colebrook, hydrostatic pressure prism, control-volume momentum, etc.).
3. Highlight where you insert numerical values, showing intermediate answers so I can trace your work.
4. Finish with a boxed final answer in proper SI units.

Deliver a single PDF that compiles the handwritten sketches and typed calculations (Word, LaTeX, or clearly written scans are fine) so the flow of each solution is easy to follow.