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