Radius of Curvature Formula

The radius of the approximate circle at a particular point is the radius of curvature. The curvature vector length is the radius of curvature. The radius changes as the curve moves. Denoted by R, the radius of curvature is found out by the following formula.

Formula for Radius of Curvature

\(\large R=\frac{(1+(\frac{dy}{dx})^{2})^{3/2}}{|\frac{d^{2}y}{dx}|}\)

In polar coordinates r=r(Θ), the radius of curvature is given by

\(\large R=\frac{r^{2}+r_{\theta}^{r}}{r^{2}+2r^{2}_{\theta}-rr_{\theta\theta}}\)

Here, rθ=dr ⁄ dθ and rθθ=d2r ⁄ dθ2

Solved Examples Using Curvature Radius Formula

Question: Find the radius of curvature for the cubic at the point x = 2?

Solution: 

\(Y = 5x^{3}-x+1$\)

x=2

\(\frac{dy}{dx}=10^{2}+1$\)\) \(\left(\frac{dy}{dx}\right)^{2}=\left(10^{2}-1\right)^{2}$\) \(=100x^{4}-20x^{2}+1\) \(=\frac{d^{2}y}{dx}=20x^{2}\)

Using the formula

\(R=\frac{(1+(\frac{dy}{dx})^{2})^{3}}{|\frac{d^{2}y}{dx}|}\) \(R=\frac{\left(1+100x^{4}-20x^{2}+1\right)^{\frac{3}{2}}}{\left|20x\right|}\; at\;x=2\) \(R=1481.51\)

Practise This Question

Morula is a developmental stage___________