Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/184

Illustrative Example 1. Find the radius of curvature at any point of the catenary $y={\tfrac {a}{2}}(e^{\tfrac {x}{a}}+e^{-{\tfrac {x}{a}}})$.

Solution.	${\frac {dy}{dx}}$	$={\frac {1}{2}}(e^{\frac {x}{a}}-e^{-{\frac {x}{a}}});{\frac {d^{2}y}{dx^{2}}}={\frac {1}{2a}}(e^{\frac {x}{a}}-e^{-{\frac {x}{a}}})$.
Substituting in (42),
$R={\frac {\left[1+\left({\frac {e^{\frac {x}{a}}-e^{-{\frac {x}{a}}}}{2}}\right)^{2}\right]^{\frac {3}{2}}}{\frac {e^{\frac {x}{a}}-e^{-{\frac {x}{a}}}}{2a}}}$		$={\frac {\left({\frac {e^{\frac {x}{a}}-e^{-{\frac {x}{a}}}}{2}}\right)^{3}}{\frac {e^{\frac {x}{a}}-e^{-{\frac {x}{a}}}}{2a}}}={\frac {a(e^{\frac {x}{a}}-e^{-{\frac {x}{a}}})^{2}}{4}}={\frac {y^{2}}{a}}$. Ans.
If the equation of the curve is given in parametric form, find the first and second derivatives of y with respect to x from (A) and (B), §97, namely:
(G)	${\frac {dy}{dx}}$	$={\frac {\frac {dy}{dt}}{\frac {dx}{dt}}}$, and
(H)	${\frac {d^{2}y}{dx^{2}}}$	$={\frac {{\frac {dx}{dt}}{\frac {d^{2}y}{dt^{2}}}-{\frac {dy}{dt}}{\frac {d^{2}x}{dt^{2}}}}{\left({\frac {dx}{dt}}\right)^{3}}}$;

and then substitute the results in (42).^[1]

Illustrative Example 2. Find the radius of curvature of the cycloid

$x=a(t-\sin t)$

$y=a(t-\cos t)$

Solution.	${\frac {dx}{dt}}=a(1-\cos t)$,	${\frac {dy}{dt}}=a\sin t$;
	${\frac {d^{2}x}{dt^{2}}}=a\sin t$,	${\frac {d^{2}y}{dt^{2}}}=a\cos t$,

Substituting in (G) and (H), and then in (42), we get

${\frac {dy}{dx}}={\frac {\sin t}{1-\cos t}},{\frac {d^{2}y}{dx^{2}}}={\frac {a(1-\cos t)a\cos t-a\sin ta\sin t}{a^{3}(1-\cos t)^{3}}}={\frac {1}{a(1-\cos t)^{2}}}$.

$R={\frac {\left[1+\left({\frac {\sin t}{1-\cos t}}\right)^{2}\right]^{\frac {3}{2}}}{-{\frac {1}{a(1-\cos t)^{2}}}}}=-2a{\sqrt {2-2\cos t}}.$ Ans.

1. Substituting (G) and (H), and then in (42) gives $R={\tfrac {\left[\left({\tfrac {dx}{dt}}\right)^{2}+\left({\tfrac {dy}{dt}}\right)^{2}\right]}{{\tfrac {dx}{dt}}{\tfrac {d^{2}y}{dt^{2}}}-{\tfrac {dy}{dt}}{\tfrac {d^{2}x}{dt^{2}}}}}$.