Animated Solution for Physics - Physics and Measurement: In the density measurement of a cube, the mass and edge length are measured as $(10.00\pm 0.10)$ kg and $(0.10\pm 0.01)$ m, respectively. The error in the measurement of density is

The process of measuring physical quantities is never absolutely perfect. Every instrument has a limit to its precision, and as physicists, we must account for these tiny uncertainties. In this problem, we are tasked with finding the error in the density of a cube, given the measurements of its mass and edge length.

We are given the mass of the cube and its edge length, along with their respective absolute errors: $M=10.00\pm 0.10\text{ kg}$ $l=0.10\pm 0.01\text{ m}$

Here, the base value of the mass is $10.00\text{ kg}$, and its absolute error $\Delta M$ is $0.10\text{ kg}$. Similarly, the base value of the edge length is $0.10\text{ m}$, and its absolute error $\Delta l$ is $0.01\text{ m}$.

To find the error in density, we first need the formula that connects density, mass, and volume. For a cube, the volume is simply the cube of its edge length ($V=l^3$). Therefore, the density $\rho$ is given by:

When physical quantities are multiplied or divided, their relative errors add up to give the maximum permissible relative error in the final result. Furthermore, if a quantity is raised to a power, that power becomes a multiplier for its relative error. Applying these rules to our density formula, we get:

Notice how the error in the length is magnified by a factor of 3. This is a crucial insight: errors in quantities with higher powers have a much larger impact on the final result!

Now, let's substitute our given values into the error propagation equation:

First, we calculate the fractional error contributed by the mass:

Next, we calculate the fractional error contributed by the length:

Adding these together gives us the total relative error in the density:

We have found that the relative error is $0.31$. By definition, relative error is a ratio of two identical units, making it a unitless quantity.

However, if you look closely at the options provided in the question, they all have the unit ${\text{kg/m}}^3$. This unit corresponds to absolute error ($\Delta \rho$), not relative error. If we were to calculate the absolute error, it would be $\Delta \rho =0.31\times \rho =0.31\times 10000=3100{\text{ kg/m}}^3$, which is nowhere to be found in the options.

This indicates a typographical error in the design of the question. The examiners intended to ask for the relative error but mistakenly attached the units of absolute error to the options. In a competitive exam scenario, the most logical step is to identify the numerical match, which is $0.31$, corresponding to option (d). Officially, this question was marked as a bonus due to this anomaly.