Work and Energy Calculator

Calculus 3 · Axiom Academy

One relationship runs every push, pull, and lift — work = force · displacement = |F| |d| cosθ. Put it in your hands. You drag a crate 8 m across the floor with a 50 N force. Do the full 50 N count? Only the part of the force pointing along the motion does work — angle it up and you waste effort. Three moves with one dot product. Only the part along the motion works Same 50 N force, same 8 m slide — just tilt it. Watch the force split into the green part along the displacement (that's the part doing work) and the grey part pushing uselessly into the floor. Straight-on is all work; tilt it away and the work drains off. A real number this time. You push a box with a 100 N force angled up from the handle and it slides 5 m along the floor. Only the horizontal share of your push, 100 cosφ , travels with the box — multiply that by the 5 m and you have the work. Same dot product, put to work. Straight up does nothing at all Here's the surprise. Keep the 50 N and the 8 m, but sweep the angle all the way to 90° — force straight up, motion straight across. You're pushing just as hard, yet the work drops to zero . The curve is cosθ: every degree off-line costs you, and perpendicular costs everything. One dot product, three moves: split it , compute it , weigh it . Work only counts the force along the motion — W = |F| |d| cosθ , zero when they're perpendicular. When the force varies along a curved path you just add up the pieces: W = ∫ C F · dr — same idea, one dot product at a time.

This is the written version of the interactive lesson above. See the full Calculus 3 course.