Question

The errors in the measurement of radius of a circle is $2\;$. Find the error in the area of the circle.

Answer 1

We know that error in physics refers to the inaccuracy which arises due to the nature of the experiment or due to humans. Using error analysis, as discussed below, we can calculate the percentage error which might occur in the given question as follows.

Formula used:
$A=\pi r^2$

Complete step-by-step solution:
Given that $\dfrac{\delta r}{r}=2$ or we can rewrite the same as $\dfrac{\delta r}{r}\times 100=2$, this is called the percentage error in radius $r$.
We also know that $A=\pi r^2$where $A$ is the area of circle whose radius is $r$. we need to find the percentage error in $A$ which is nothing but $\dfrac{\delta A}{A}\times 100$
To begin with let us differentiate$A=\pi r^2$ with respect to on both sides, then we have the following
$\implies \dfrac{\ dA}{dr}=2\pi r$
$\implies \dfrac{\ dA}{dr}\delta r=2\pi r\delta r$
Since , $\dfrac{dA}{dr}\delta r=\delta A$, substituting we have
$\implies \delta A=2\pi r\delta r$
Multiplying and dividing by radius $r$ on RHS , we have
$\implies \delta A=2\pi r^2\dfrac{\delta r}{r}$
Since, $A=\pi r^2$, replacing we have
$\implies \delta A=2A\dfrac{\delta r}{r}$
Now dividing both sides by area $A$, we have
$\implies \dfrac{\delta A}{A}=2\dfrac{\delta r}{r}$
Since we are calculating the percentage errors in $A$, we can multiply $\;100$ on both sides of the equation, then rewriting the above as follows
$\implies \dfrac{\delta A}{A}\times 100=2\dfrac{\delta r}{r}\times 100$
Substituting the given value, we have
$\implies \dfrac{\delta A}{A}\times 100=2\times 2$
$\therefore \dfrac{\delta A}{A}\times 100=4$
Thus the error in the area of the circle is $4\;$ is the required answer.

Additional information:
Errors can be classified as the following on the basis of how they arise in the experiment, for example gross error, systematic error, instrumental error, environmental error, observational error, random error.

Note: Here, to calculate the required answer, we are differentiating the formula for the area of the circle. This is a very easy question and can be calculated easily. However note that due to differentiation, we are multiplying 2 to the percentage error in radius, and not squaring the terms.