Question

Find the curvature and radius of curvature of the plane curve at the given value of \(x\). $$ y=\sqrt{a^{2}-x^{2}}, \quad x=0 $$

Short answer

After evaluating the derivations and getting expressions for \(y'\), \(y''\), the curvature \(k\) can be found, and subsequently the radius of curvature \(R\) is calculated.

Step-by-step solution

Derive y

Differentiate \(y=\sqrt{a^2 - x^2}\) with respect to x to find \(y'\). This involves using the chain rule of differentiation.

Derive y'

Differentiate \(y'\) with respect to x to find \(y''\). This also involves the chain rule of differentiation.

Find the curvature (k)

Substitute \(y'\) and \(y''\) values into the curvature formula \(k = |y''| /(1 + (y')^2)^{3/2}\) to find k.

Find the radius of curvature (R)

Find the radius of curvature using the formula \(R = 1 / k\).