Question

The curvature at a point P of a parametric curve x = x(t), y = y(t) is given below, where the dots indicate derivatives with respect to t, so ẋ = dx/dt. Use the formula to find the curvature of the cycloid x = θ − sin(θ), y = 1 − cos(θ) at the top of one of its arches. $ \kappa = \dfrac {\left| \dot{x} \ddot{y} - \ddot{x} \dot{y} \right|} {[\dot{x}^2 + \dot{y}^2]^{3/2}} $

Answer 1

Answer:

$\kappa = \frac{\cos \theta -1}{\left[2\cdot (1 - \cos \theta) \right]^{\frac{3}{2} }}$

Step-by-step explanation:

The first and second derivatives of the components of the parametric curve are:

$\dot x = 1 - \cos \theta$ and $\ddot x = \sin \theta$

$\dot y = \sin \theta$ and $\ddot y = \cos \theta$

After substitution, the equation of curvature is:

$\kappa = \frac{(1-\cos\theta)\cdot (\cos \theta)-\sin^{2} \theta}{\left[(1-\cos\theta)^{2}+\sin^{2}\theta \right]^{\frac{3}{2} }}$

$\kappa = \frac{\cos \theta -1}{\left[2\cdot (1 - \cos \theta) \right]^{\frac{3}{2} }}$