Numerical 1: Convert G = $6.67 \times 10^{-11}$N$m^{2}$$kg^{-2}$ from SI to CGS

Step 1 — Dimensional formula of G:

G \text{ has dimensions } [$M^{-1}$ L^3 $T^{-2}$]

So a = −1, b = 3, c = −2.

Step 2 — Set up conversion:

$G_\text{CGS} = G_\text{SI} \times \left(\frac{M_\text{SI}}{M_\text{CGS}}\right)^{-1} \times \left(\frac{L_\text{SI}}{L_\text{CGS}}\right)^{3} \times \left(\frac{T_\text{SI}}{T_\text{CGS}}\right)^{-2}$

Step 3 — Substitute known conversions:

$= 6.67 \times 10^{-11} \times \left(\frac{1 \text{ kg}}{1 \text{ g}}\right)^{-1} \times \left(\frac{1 \text{ m}}{1 \text{ cm}}\right)^{3} \times \left(\frac{1 \text{ s}}{1 \text{ s}}\right)^{-2}$

$= 6.67 \times 10^{-11} \times (10^3)^{-1} \times (10^2)^{3} \times 1$

Step 4 — Simplify:

$$= 6.67 \times 10^{-11} \times 10^{-3} \times 10^{6} = 6.67 \times 10^{-8} \text{ dyne cm}^2 \text{ g}^{-2}$$$