The density $\rho$ of a uniform cylinder is determined by measuring its mass m, length $l$ and diameter d. The measured values of $\mathrm{m}, l$ and d are $97.42 \pm 0.02 \mathrm{~g}, 8.35 \pm 0.05 \mathrm{~mm}$ and $20.20 \pm 0.02 \mathrm{~mm}$, respectively. Calculated percentage fractional error in $\rho$ is ______.

JEE Main · 2026 · 06 Apr 2026 · Shift 1 · Units & Measurement

The density

ρ

of a uniform cylinder is determined by measuring its mass m, length

l

and diameter d. The measured values of

m, l

and d are

97.42 \pm 0.02 g, 8.35 \pm 0.05 mm

and

20.20 \pm 0.02 mm

, respectively. Calculated percentage fractional error in

ρ

is ______.

Answer: (B)

With

ρ = \frac{m}{\frac{π}{4} d ^{2} ℓ}

, the fractional errors of a product/quotient add:

\frac{Δ ρ}{ρ} = \frac{Δ m}{m} + 2 \frac{Δ d}{d} + \frac{Δ ℓ}{ℓ}

. Substituting,

\frac{0.02}{97.42} + 2 \times \frac{0.02}{20.20} + \frac{0.05}{8.35} = 0.00021 + 0.00198 + 0.00599 = 0.00818

. The percentage error in density is

0.82%

JEE Main · 2026 · 06 Apr 2026 · Shift 1 · Single correct option