A metal wire has mass g, radius mm and length cm. The maximum possible percentage error in the measurement of density will nearly be [2023]

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Solution

Q.
A metal wire has mass $(0.4 \pm 0.002)$ g, radius $(0.3 \pm 0.001)$ mm and length $(5 \pm 0.02)$ cm. The maximum possible percentage error in the measurement of density will nearly be [2023]

1 1.6%

2 1.4%

3 1.2%

4 1.3%

Ans.
(1)

Volume of the wire, $V = π r^{2} L$

Density of the wire $ρ = \frac{m}{V} = \frac{m}{π r^{2} L}$

For percentage error,

$\frac{Δ ρ}{ρ} \times 100 = \frac{Δ m}{m} \times 100 + 2 \times \frac{Δ r}{r} \times 100 + \frac{Δ L}{L} \times 100$

$∴$ $\frac{Δ ρ}{ρ} \times 100 = \frac{0.002}{0.4} \times 100 + 2 \times \frac{0.001 \times 100}{0.3} + \frac{0.02}{5} \times 100$

$= 0.5 + 0.67 + 0.4 = 1.57 \approx 1.6 %$

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