Dimensional Analysis
Drag-and-drop unit checker. Builds on Powers of Ten & Units.
A physics student derives that the time for a ball to fall from height h is t = √(2mh/g), where m is the ball's mass and g is gravitational acceleration (9.8 m/s²). Their friend says "I can tell that's wrong WITHOUT doing any experiment."
How can the friend tell it's wrong just by looking at it?
Check the units: m·h/g has dimensions [kg·m / (m/s²)] = [kg·s²]. Take the square root: [kg^½·s]. That's NOT seconds! The mass doesn't belong.
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Toggle between equations and watch the dimension colors balance — or clash.
[LHS] = [RHS]
You can DERIVE that pendulum period T ∝ √(L/g) purely from dimensions — without solving any differential equation!
In 1999, Mars Climate Orbiter was lost because one team used pounds, another used Newtons. Dimensional mismatch destroyed the spacecraft.
Medication dosages are calculated per kilogram of body weight. A unit error can be fatal — dimensional analysis saves lives.
Before running expensive simulations, engineers use dimensional analysis to identify which variables actually matter.
“Dimensional analysis is the physicist's cheat code — it catches errors, derives formulas, and costs nothing but a moment's thought.”
Unit Builder
Combine physical quantities to match the target dimensions. It's a unit puzzle!
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Try to build the dimensions of Energy [ML²T⁻²] from mass, velocity, and height.
There are often multiple ways to combine quantities to get the same dimensions — but physics picks the one that matches reality.