Problem $9$

A trapezoidal channel has a bottom width $B$ of $5$ m $,$ side slopes of $2$:$1,$ and $n=0\mathrm{.}025.$

a$.$ Determine the normal slope $S_0$ at a normal depth $y_n$ of $1$ m when the discharge $Q$ is $10\frac{m^3}{s}.$

$($Hint: using Manning's formula$)$

b$.$ Determine the critical slope $S_c$ and the corresponding normal depth $y_n$ when the discharge $Q$ is $10\frac{m^3}{s}.$

$($Hint: the critical slope $S_c$ is the normal slope $S_0$ when the flow condition is at critical: $S_c=S_0$ and $y_n=y_c)$

c$.$ Determine the critical slope $S_c$ at the normal depth $y_n$ of $1$ m $,$ and compute the corresponding discharge $Q.$

$($Hint: the critical slope $S_c$ is the normal slope $S_0$ when the flow condition is at critical: $S_c=S_0$ and the Froude number $Fr=\frac{V}{V}gD=1,$ where $D$ is the hydraulic depth $=\frac{A}{T}$