Animated Solution for Physics - Physics and Measurement: A wire has a mass $(0.3\pm 0.003)\text{ g}$, radius $(0.5\pm 0.005)\text{ mm}$ and length $(6\pm 0.06)\text{ cm}$. The maximum percentage error in the measurement of its density is

The journey to finding the maximum percentage error in the density of a wire is a classic exploration of how uncertainties propagate in physical measurements. Imagine you are in a laboratory, carefully measuring the dimensions of a cylindrical wire. You have its mass, its radius, and its length, but no measurement is perfect. Each comes with a tiny margin of error. Our mission is to determine how these tiny errors combine to affect the final calculated density.

Let's start by laying out the facts. We are given the mass of the wire as $m=0.3\pm 0.003\text{ g}$. The radius is measured as $r=0.5\pm 0.005\text{ mm}$, and the length is $L=6\pm 0.06\text{ cm}$.

Notice that the units are all different—grams, millimeters, and centimeters. You might be tempted to convert them all to a standard unit like SI. But here is a beautiful shortcut: when we calculate percentage errors, we are dealing with ratios ($\frac{\Delta x}{x}$). The units in the numerator and denominator perfectly cancel each other out! This means we can proceed directly without any tedious unit conversions.

To find the error in density, we first need the formula for density itself. Density ($\rho$) is defined as mass divided by volume:

For a cylindrical wire, the volume $V$ is the cross-sectional area multiplied by the length:

Substituting this into our density formula gives us our master equation:

Now, we apply the principle of error propagation. When quantities are multiplied or divided, their relative errors add up to give the maximum relative error of the result. Furthermore, if a quantity is raised to a power, that power becomes a multiplier for its relative error. The constant $\pi$ is an exact mathematical value, so it has zero uncertainty and drops out of our error equation.

This leads us to the expression for the maximum percentage error in density:

Notice the factor of 2 in front of the radius term. Because the radius is squared in the volume formula, any error in measuring the radius is magnified twofold in the final density calculation. This is a crucial insight for experimental physics: always measure the squared or cubed quantities with the highest precision!

With our error equation ready, it is time to plug in the numbers. Let's calculate the percentage error contributed by each measurement individually.

For the mass:

For the radius (remembering the multiplier):

For the length:

Finally, we sum these individual contributions to find the total maximum percentage error in the density:

The maximum percentage error in the measurement of its density is $4\%$. This elegant problem reminds us that in the world of physics, errors don't just exist; they compound, and understanding how they do is the key to mastering experimental science.