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 = kx^ay^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*(M_1/M_2)^a*(L_1/L_2)^b*(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 = k*sqrt(L/g) for a simple pendulum. Dimensional analysis finds the L^{1/2}g^{-1/2} dependence but not k = 2pi.