Chapter 1, Problem 1CQ

Can dimensional analysis determine whether the area of a circle is πr² or 2πr²? Explain.

To determine

Whether the dimensional analysis determine the area of circle is $\pi r^2$ or $2\pi r^2$.

Answer to Problem 1CQ

The dimensional analysis cannot determine whether the area of a circle is $\pi r^2$ or $2\pi r^2$.

Explanation of Solution

Write the dimensional formula for the area of the circle $\pi r^2$.

$\begin{array}{c}\pi r^2=\left[L^0\right]\left[L^2\right]\\ =\left[L^2\right]\end{array}$ (1)

Here,

$L$ is the dimension of the length.

Write the dimensional formula for the area of the circle $2\pi r^2$.

$\begin{array}{c}2\pi r^2=\left[L^0\right]\left[L^0\right]\left[L^2\right]\\ =\left[L^2\right]\end{array}$ (2)

Here,

$L$ is the dimension of the length.

From the equation (1) and (2) the dimension of the area is same in both cases. So that the dimensional analysis cannot be determine whether the area of a circle is $\pi r^2$ or $2\pi r^2$.

The dimensions of the variable do not depend on the numeric number.

Conclusion:

Therefore, the dimensional analysis cannot determine whether the area of a circle is $\pi r^2$ or $2\pi r^2$.