Units & Measurements — Solved Examples

UNITS & MEASUREMENTS — SOLVED EXAMPLES (JEE/NEET style) EXAMPLE 1 — Dimensional check Q: Check whether v² = u² + 2as is dimensionally correct. Sol: [v²] = [L T^-1]² = [L² T^-2]. [u²] = [L² T^-2]. [as] = [L T^-2][L] = [L² T^-2]. All three terms are [L² T^-2] → dimensionally consistent. ✔ EXAMPLE 2 — Same dimensions Q: Show that impulse and momentum have the same dimensions. Sol: Impulse = Force × time = [M L T^-2][T] = [M L T^-1]. Momentum = mass × velocity = [M][L T^-1] = [M L T^-1]. Equal → that is why impulse = change in momentum. ✔ EXAMPLE 3 — Error propagation Q: A quantity is Z = A²B / √C. Percentage errors are A: 1%, B: 2%, C: 4%. Find % error in Z. Sol: ΔZ/Z = 2(ΔA/A) + 1(ΔB/B) + ½(ΔC/C) = 2(1%) + 1(2%) + 0.5(4%) = 2% + 2% + 2% = 6%. ✔ Note the ½ comes from √C = C^(1/2). EXAMPLE 4 — Significant figures Q: Add 24.36 + 0.0623 + 256.2 and give the answer to correct significant figures. Sol: Raw sum = 280.6223. In addition we keep the least decimal places. 256.2 has only 1 decimal place, so the answer = 280.6. ✔ EXAMPLE 5 — Least count / screw gauge Q: A screw gauge has a pitch of 1 mm and 100 divisions on the circular scale. Find its least count. Sol: LC = pitch / divisions = 1 mm / 100 = 0.01 mm = 10 µm. ✔ So it can measure to the nearest 0.01 mm.