Question

The density $\rho$ 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 $\rho$ is

• A. 0.63%
• B. 0.82%
• C. 0.72%
• D. 0.25%

Hint:

Density of a cylinder is $\rho = \frac{4m}{\pi d^2 l}$. Apply the formula for fractional error propagation for multiplication and division.

Answer 1

The Correct Option is B

To find the percentage fractional error in density $\rho$, we start with the formula for the density of a cylinder. Density is defined as mass divided by volume. For a cylinder with diameter $d$ and length $l$, the volume $V$ is given by $V = \pi \left(\frac{d}{2}\right)^2 l = \frac{\pi d^2 l}{4}$.
Substituting this into the density formula:
$$\rho = \frac{m}{V} = \frac{4m}{\pi d^2 l}$$
In error analysis, when a physical quantity is calculated from several measured variables using multiplication or division, the fractional error in the result is the sum of the fractional errors of the individual terms, multiplied by their respective powers. For $\rho = \frac{4m}{\pi d^2 l}$, the fractional error $\frac{\Delta \rho}{\rho}$ is:
$$\frac{\Delta \rho}{\rho} = \frac{\Delta m}{m} + 2\frac{\Delta d}{d} + \frac{\Delta l}{l}$$
Given measured values:
Mass $m = 97.42 \pm 0.02$ g $\implies \Delta m = 0.02, m = 97.42$
Length $l = 8.35 \pm 0.05$ mm $\implies \Delta l = 0.05, l = 8.35$
Diameter $d = 20.20 \pm 0.02$ mm $\implies \Delta d = 0.02, d = 20.20$
Now, calculate each fractional error term:
1. $\frac{\Delta m}{m} = \frac{0.02}{97.42} \approx 0.0002053$
2. $2\frac{\Delta d}{d} = 2 \times \frac{0.02}{20.20} = \frac{0.04}{20.20} \approx 0.0019802$
3. $\frac{\Delta l}{l} = \frac{0.05}{8.35} \approx 0.0059880$
Adding these values together:
$$\frac{\Delta \rho}{\rho} = 0.0002053 + 0.0019802 + 0.0059880 = 0.0081735$$
To find the percentage error, multiply by 100:
Percentage error $= 0.0081735 \times 100\% \approx 0.817\%$
Rounding to two decimal places, we get $0.82\%$.