Dimensional Formulae Summary
| Quantity | Formula | Dimensions | SI Unit |
|---|---|---|---|
| Work/Energy | W = Fd | [M^{1}$$L^{2}$$T^{-2}] | J |
| Kinetic Energy | ½ | [M^{1}$$L^{2}$$T^{-2}] | J |
| Gravitational PE | mgh | [M^{1}$$L^{2}$$T^{-2}] | J |
| Spring PE | ½ | [M^{1}$$L^{2}$$T^{-2}] | J |
| Power | W/t | [M^{1}$$L^{2}$$T^{-3}] | W |
| Spring constant | F/x | [M^{1}$$L^{0}$$T^{-2}] | N/m |
| Momentum | mv | [M^{1}$$L^{1}$$T^{-1}] | kg·m/s |
| Impulse | F | [M^{1}$$L^{1}$$T^{-1}] | N·s |
| Coefficient of restitution | v_sep/v_app | Dimensionless | — |
| Efficiency | P_out/P_in | Dimensionless | — |
Dimensional Analysis Checks
- P = Fv: [F][v] = ML × L = ML^{2}$$T^{-3} ✓
- KE = /2m: [p]^{2}/[m] = (ML)^{2}/M = M^{2}$$L^{2}$$T^{-2}/M = ML^{2}$$T^{-2} ✓
- v = √(gR): [g][R] = L × L = L^{2}$$T^{-2}, so √(gR) = L = m/s ✓
- W_friction = −μmgd: μ dimensionless, [mgd] = M × L × L = ML^{2}$$T^{-2} ✓