Question

A point ${}^{\prime }{O}^{\prime }$ is situated on a circle of radius $R$ and with centre $O$, another circle of radius $3R/2$ is described. Inside the smaller crescent shaped area intercepted between these circles, a circle of radius $R/8$ is placed. If the same circle moves in contact with the original circle of radius $R$, then find the length of the arc described by its centre in moving from one extreme position to the other.

Answer 1

Consider the problem

$AB=R$

$AC=\frac{3R}{2}+\frac{R}{8}=\frac{13R}{8}$

$BC=R-\frac{R}{8}=\frac{7R}{8}$

In triangle $ABC$

$\cos \theta =\frac{A{B}^2+B{C}^2-A{C}^2}{2AB.BC}$

$\begin{array}{c}=\frac{R^2+\frac{45R^2}{64}-\frac{169R^2}{64}}{2\times \frac{7R^2}{8}}\\ \\ =\frac{56R^2}{64}\times \frac{8}{14R^2}=-\frac{1}{2}\end{array}$

$\cos \theta =-\frac{1}{2}$

$\theta ={120}^{\circ }$

For symmetry

$\angle ABD={120}^{\circ }$

$\angle BDC={120}^{\circ }=\frac{2\pi }{3}$

$CD=BC\times \frac{2\pi }{3}=\frac{7R}{8}\times \frac{2\pi }{3}=\frac{7\pi R}{12}$