$($a$)$ A Francis turbine has runner diameter $1$ m at the entry and $0\mathrm{.}5$ m at the exit. The

vane angle is radial at entry and the fixed vane angle is $15.$ Water at the exit

leaves the vanes axially. The working head is $30$ m and radial component of flow

is constant. What will be the rpm of the wheel and vane angle at exit? Also

calculate the synchronous speed to couple the turbine with an alternator of $50$

cycles $\frac{?}{s},$ assuming $12$ pairs of poles.