Question

The radius of a circle is measured to be 11.6 0.4 m. (a) Calculate the area of the circle including its uncertainty Enter a number. differs from the correct answer by more than 10%. Double check your calculations. (b) Calculate the circumference of the circle including its uncertainty.

Answer 1

The error in radius is,

$\Delta r = 0.4$

(a) The area of the circle is,

$A = \pi r^2 \Rightarrow A = \pi (11.6)^2 = 422.73 \, m^2$

The uncertainity in area will be given by,

(Multiplication with constant, makes the error also to be get multiplied by the constant; and x^n error is given by nx )

$\Delta A = \pi \left( 2 \frac{\Delta r}{r} A \right)$

$\Rightarrow \Delta A = \pi \left( 2 \frac{0.4}{11.6} 422.73 \right) = 91.59 \, m^2$

Therefore the area is,

$\Rightarrow A \pm \Delta A = 422.73 \pm 91.59 \, m^2$

(b) The circumference of the circle is given by,

$C = 2 \pi r \Rightarrow C = 2 \pi (11.5 \pm 0.4)$

$\Rightarrow C = 72.26 \pm 2.51 \, m$

(Since here only constant is multiplied, we can just multiply the error with a constant)