Question

Prove that the radius of curvature for the catenary y = c cos h x/c is equal to the portion of the normal intercepted between the curve and the X-axis and that it varies as the square of the ordinate. 

Answer 1

Equation of the curve is

y = c cos h x/c                                                                              ......(1)

∴  dy/dx = y₁ = sin h x/c                                                               .......(2)

and dy1/dx = y₂ = (1/c) cos h (x/c)                                                    ......(3)

thus p = 

                                                            .....(4)

Now portion of the normal intercepted between the curve and the X-axis is

                                                                       ......(5)

Clearly from eqn (4) and (5) we see that ρ (radius of curvature) = n (length of the normal)

= c cos h² x/c = y²/c                            using (1)

∴ ρ varies as square of the ordinate