Animated Solution for Physics - Physics and Measurement: A physical quantity $z$ depends on four observables $a,b,c$ and $d$, as $z=\frac{a^2{b}^{2/3}}{\sqrt{c}{d}^3}$. The percentages of error in the measurement of $a,b,c$ and $d$ are $2\%$, $1.5\%$, $4\%$ and $2.5\%$ respectively. The percentage of error in $z$ is

In the world of experimental physics, no measurement is perfectly precise. Every instrument has its limitations, and every reading carries a tiny shadow of doubt known as an 'error'. But what happens when we take these slightly imperfect measurements and plug them into a complex formula? Do the errors cancel out, or do they conspire to create a bigger mess?

Let's explore this through a classic problem where a physical quantity $z$ depends on four observables $a,b,c$, and $d$ according to the relation:

When calculating the error in a derived quantity, we must always prepare for the worst-case scenario. This means we want to find the maximum possible percentage error.

The mathematical tool we use here is derived from calculus (specifically, logarithmic differentiation). If we take the natural logarithm of both sides of our equation and differentiate, we get a beautiful linear relationship for the relative errors.

The golden rule is simple: When quantities are multiplied or divided, their relative errors add up. Furthermore, any power to which a quantity is raised simply becomes a multiplier for its relative error.

Crucially, even if a term is in the denominator (like $c$ and $d$ in our formula), its error still adds to the total error. Why? Because an error is an uncertainty. A positive error in the denominator decreases the final value, while a negative error increases it. To find the maximum possible deviation, we assume all errors align to push the final result in the same direction.

Let's apply this rule to our specific equation. We can rewrite the square root of $c$ as $c^{1/2}$. The maximum percentage error in $z$ is given by:

Notice how the powers $2$, $2/3$, $1/2$, and $3$ have gracefully stepped down to become coefficients.

Now, it's just a matter of substituting the given percentage errors for each observable: - Error in $a=2\%$ - Error in $b=1.5\%$ - Error in $c=4\%$ - Error in $d=2.5\%$

Plugging these in:

Let's compute each term carefully: - The contribution from $a$ is $2\times 2\%=4\%$ - The contribution from $b$ is $\frac{2}{3}\times 1.5\%=1\%$ - The contribution from $c$ is $\frac{1}{2}\times 4\%=2\%$ - The contribution from $d$ is $3\times 2.5\%=7.5\%$

Finally, we sum them all up to find the total maximum percentage error:

And there we have it! The maximum percentage error in the measurement of $z$ is $14.5\%$.