Problem source

\begin{center}
\begin{tikzpicture}
    % Main lines
    \draw (1,1) -- (5,7);
    \draw (5,7) -- (5,4);
    
    % Arc at point O
    \draw (5,7) ++ (-123.7:1.19) arc (-123.7:-90:1.19);
    
    % Labels
    \node at (5.72,6.56) {$\sin^{-1}\tfrac{3}{5}$};
    \node at (5.16,7.36) {$O$};
    \node at (4.88,3.69) {$N$};
    \node at (0.64,0.82) {$A$};
    \node at (2.52,4.27) {$B$};
    
    % Zigzag path
    \draw (3,4) -- (3.09,3.8) -- (2.64,3.78) -- (2.86,3.51) -- (2.41,3.45) -- 
          (2.67,3.2) -- (2.25,3.15) -- (2.52,2.88) -- (2.03,2.85) -- 
          (2.27,2.5) -- (1.77,2.49) -- (1.96,2.12) -- (1.5,2.09) -- 
          (1.68,1.76) -- (1.28,1.71) -- (1.47,1.42) -- (1.04,1.41) -- 
          (1.32,1.07) -- (1,1);
    
    % Dots
    \filldraw[black] (1,1) circle (2pt);
    \filldraw[black] (3,4) circle (2pt);
\end{tikzpicture}
\end{center}
The end $O$ of a smooth light rod $OA$ of length $2a$ is a fixed point. The rod $OA$ makes a fixed angle $\sin^{-1}\frac{3}{5}$ with the downward vertical $ON,$ but is free to rotate about $ON.$
A particle of mass $m$ is attached to the rod at $A$ and a small ring $B$ of mass $m$ is free to slide on the rod but is joined to a spring of natural length $a$ and modulus of elasticity $kmg$. The vertical plane containing the rod $OA$ rotates about $ON$ with constant angular velocity $\sqrt{5g/2a}$ and $B$ is at rest relative to the rod. Show that the length of $OB$ is 
\[
\frac{(10k+8)a}{10k-9}.
\]
Given that the reaction of the rod on the particle at $A$ makes an angle $\tan^{-1}\frac{13}{21}$ with the horizontal, find the value of $k$. Find also the magnitude of the reaction between the rod and the ring $B$.