25.12 seconds
As we have been given ion the question,
Tangential speed of her brother at the Wooden Horse = 6 m/s
Tangential speed of Sheila = 4 m/s
Radius of merry go round, r = 8 m
Now,
We know that,
v = r.ω
So,
Angular velocity of brother is given by,
[tex]v=r.\omega\\\omega = \frac{v}{r}=\frac{6}{8}\\\omega =0.75\,rad/sec.[/tex]
And,
Angular velocity of Sheila is given by,
[tex]v=r.\omega\\\omega = \frac{v}{r}=\frac{4}{8}\\\omega =0.5\,rad/sec.[/tex]
Relative Angular Velocity = 0.75 - 0.5 = 0.25 rad/s.
Therefore, the time taken by Sheila to encounter her brother again is given by,
[tex]Time=\frac{Total\,Distance}{Relative\,Speed}\\Time=\frac{2\pi}{0.75-0.5}\\Time,t=\frac{2\pi}{0.25}\\t=8\pi\\Time,t=25.12\,s[/tex]
Therefore, Time Taken to encounter again is 25.12 seconds.