Unit Conversions

NEET regularly asks to convert force from newton to dyne, energy from joule to erg, pressure from pascal to bar, or larger astronomical distances using light-years. The trick is to track each base unit independently and use powers of ten cleanly.

Concept

Choose a factor of 1 written as a ratio of equivalent units, e.g. $1\,m / 100\,cm$ or $1\,kg / 1000\,g$. Multiply your quantity by these factors so unwanted units cancel and the desired ones remain.

Useful equivalences:

• $1\,N = 10^{5}\,dyne$
• $1\,J = 10^{7}\,erg$
• $1\,Pa = 10\,dyne/cm^{2}$
• $1\,atm = 1.013 \times 10^{5}\,Pa$
• $1\,eV = 1.6 \times 10^{-19}\,J$
• $1\,ly \approx 9.46 \times 10^{15}\,m$

Formula Derivation

For a derived quantity with dimensional formula $M^{a} L^{b} T^{c}$, conversion from system 1 to system 2:

$n_2 = n_1 \left(\frac{M_1}{M_2}\right)^{a} \left(\frac{L_1}{L_2}\right)^{b} \left(\frac{T_1}{T_2}\right)^{c}$

Example: Force has $a=1, b=1, c=-2$. Converting newton to dyne:

$n_2 = 1 \times \left(\frac{1\,kg}{1\,g}\right)^{1} \left(\frac{1\,m}{1\,cm}\right)^{1} \left(\frac{1\,s}{1\,s}\right)^{-2} = 1000 \times 100 \times 1 = 10^{5}$

So $1\,N = 10^{5}\,dyne$.

NEET-style Worked Example

Express the gravitational constant $G = 6.67 \times 10^{-11}\,N\,m^{2}/kg^{2}$ in CGS units (dyne $cm^{2}/g^{2}$).

Dimensional formula of $G$ is $M^{-1} L^{3} T^{-2}$, so

$n_2 = 6.67 \times 10^{-11} \times \left(\frac{1\,kg}{1\,g}\right)^{-1} \left(\frac{1\,m}{1\,cm}\right)^{3} \left(\frac{1\,s}{1\,s}\right)^{-2}$

$= 6.67 \times 10^{-11} \times (10^{3})^{-1} \times (10^{2})^{3} = 6.67 \times 10^{-11} \times 10^{-3} \times 10^{6}$

$= 6.67 \times 10^{-8}\,dyne\,cm^{2}/g^{2}$

Common Confusions

• Track the exponents of each base unit. Sign errors are common when exponents are negative.
• $1\,L = 10^{-3}\,m^{3}$, not $10^{-2}$.
• Temperature conversions are not multiplicative for Celsius (offset by 273.15).
• $1\,year \approx 3.15 \times 10^{7}\,s$ — useful for astronomical problems.

Key Takeaways

• Use $n_2 = n_1 (M_1/M_2)^{a}(L_1/L_2)^{b}(T_1/T_2)^{c}$.
• $1\,N = 10^{5}\,dyne$, $1\,J = 10^{7}\,erg$.
• Cancel units as if they were algebraic symbols.