Chapter 1 - Introduction to Physics - Problems and Conceptual Exercises - Page 17: 51

$p=\frac{1}{2}$ and $q=-\frac{1}{2}$

Dimension of T: [T]. Dimension of L: [L]. Dimension of g: $\frac{[L]}{[T]^{2}}$. $2π$ has no dimensions. $T=2πL^{p}g^{q}$ $[T]=[L]^{p}(\frac{[L]}{[T]^{2}})^{q}$ $[T]=\frac{[L]^{p+q}}{[T]^{2q}}$ $[T]^{1}\times [L]^{0}=[L]^{p+q}\times [T]^{-2q}$ $-2q=1$ and $p+q=0$ $-2q=1$ $q=-\frac{1}{2}$ $p+q=0$ $p-\frac{1}{2}=0$ $p=\frac{1}{2}$