Uses:

1. Checking equations: Every term in an equation must have the same dimensions. If [LHS] ≠ [RHS], the equation is certainly wrong. But dimensional correctness does NOT guarantee physical correctness.

2. Deriving relations: If a quantity Q depends on variables x, y, z, write Q = k*$x^a$$y^b$$z^c$. Equate dimensions of both sides to get a, b, c. This gives the functional form but NOT the constant k.

3. Unit conversion: Use the formula $n_2$ = $n_1$$\frac{M_1}{M_2}$^a$\frac{L_1}{L_2}$^b*$\frac{T_1}{T_2}$^c.

Limitations:

1. Cannot find dimensionless constants (2*pi, 1/2, e, etc.)
2. Cannot distinguish quantities with the same dimensions (work vs torque)
3. Cannot handle log, exp, sin, cos functions
4. Limited to 3 unknowns in mechanics (M, L, T provide 3 equations)
5. Fails for dimensionless combinations (e.g., angle dependence)

Classic example: T = ksqrt$\frac{L}{g}$ for a simple pendulum. Dimensional analysis finds the $L^{1/2}$$g^{-1/2}$ dependence but not k = 2*pi.