Units & Measurements — Solved Examples

UNITS & MEASUREMENTS — SOLVED EXAMPLES (JEE/NEET style) EXAMPLE 1 — Dimensional check Q: Is $v^{2}=u^{2}+2as$ dimensionally correct? Sol: $[v^{2}]=[L\,T^{-1}]^{2}=[L^{2}\,T^{-2}]$, $[u^{2}]=[L^{2}\,T^{-2}]$, $[as]=[L\,T^{-2}][L]=[L^{2}\,T^{-2}]$. All terms $[L^{2}\,T^{-2}]$ ⇒ consistent. ✔ EXAMPLE 2 — Same dimensions Q: Show impulse and momentum have the same dimensions. Sol: Impulse $=F\,t=[M\,L\,T^{-2}][T]=[M\,L\,T^{-1}]$; momentum $=mv=[M][L\,T^{-1}]=[M\,L\,T^{-1}]$. Equal ⇒ impulse $=$ change in momentum. ✔ EXAMPLE 3 — Error propagation Q: $Z=\dfrac{A^{2}B}{\sqrt{C}}$ with percentage errors $A:1\%,\ B:2\%,\ C:4\%$. Find % error in $Z$. Sol: $\dfrac{\Delta Z}{Z}=2\dfrac{\Delta A}{A}+\dfrac{\Delta B}{B}+\tfrac12\dfrac{\Delta C}{C}=2(1\%)+1(2\%)+\tfrac12(4\%)=6\%$. ✔ EXAMPLE 4 — Significant figures (addition) Q: Add $24.36+0.0623+256.2$ to correct significant figures. Sol: Raw sum $=280.6223$. Keep the least decimal places; $256.2$ has 1 ⇒ answer $=280.6$. ✔ EXAMPLE 5 — Least count (screw gauge) Q: Pitch $=1\ \text{mm}$, 100 circular-scale divisions. Find the least count. Sol: $\text{LC}=\dfrac{\text{pitch}}{\text{divisions}}=\dfrac{1\ \text{mm}}{100}=0.01\ \text{mm}=10\ \mu\text{m}$. ✔