Situation 2. The semi-circular channel \( (n=0.010) \) has a radius of \( 1 m \), is to carry water at a depth of 1 m . 10. Find the velocity at the critical stage, in \( \mathrm{m} / \mathrm{s} \). A. 2.2 C. 2.8 B. 3.4 D. 3.1 11. Find the slope at the critical stage. A. \( 2.47 \times 10^{-3} \) C. \( 2.95 \times 10^{-3} \) B. \( 1.95 \times 10^{-3} \) D. \( 3.42 \times 10^{-3} \) 12. Find the discharge at the critical stage, in \( \mathrm{m}^{3} / \mathrm{s} \). A. 4.1 C. 4.9 B. 3.7 D. 4.4

To solve the problem, we will use the following formulas related to open channel flow, particularly for a semi-circular channel: 1. **Velocity at the critical stage** can be calculated using the formula: \[ V_c = \sqrt{g \cdot R} \] where \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \mathrm{m/s^2} \)) and \( R \) is the hydraulic radius. 2. **Hydraulic radius \( R \)** for a semi-circular channel can be calculated as: \[ R = \frac{A}{P} \] where \( A \) is the cross-sectional area and \( P \) is the wetted perimeter. 3. **Discharge \( Q \)** at the critical stage can be calculated using: \[ Q = A \cdot V_c \] 4. **Slope at the critical stage** can be calculated using the formula: \[ S = \frac{n^2 \cdot Q^2}{R^{3/2}} \] ### Step 1: Calculate the Cross-sectional Area \( A \) and Wetted Perimeter \( P \) For a semi-circular channel with radius \( r = 1 \, \mathrm{m} \) and water depth \( h = 1 \, \mathrm{m} \): - The area \( A \) of the water surface can be calculated as: \[ A = \frac{1}{2} \cdot \pi r^2 = \frac{1}{2} \cdot \pi (1^2) = \frac{\pi}{2} \, \mathrm{m^2} \] - The wetted perimeter \( P \) for a semi-circular channel is: \[ P = \pi r = \pi \cdot 1 = \pi \, \mathrm{m} \] ### Step 2: Calculate the Hydraulic Radius \( R \) Now we can calculate the hydraulic radius \( R \): \[ R = \frac{A}{P} = \frac{\frac{\pi}{2}}{\pi} = \frac{1}{2} \, \mathrm{m} \] ### Step 3: Calculate the Velocity at the Critical Stage \( V_c \) Now we can calculate the velocity at the critical stage: \[ V_c = \sqrt{g \cdot R} = \sqrt{9.81 \cdot 0.5} \] ### Step 4: Calculate the Discharge \( Q \) Now we can calculate the discharge \( Q \): \[ Q = A \cdot V_c \] ### Step 5: Calculate the Slope at the Critical Stage \( S \) Finally, we can calculate the slope at the critical stage: \[ S = \frac{n^2 \cdot Q^2}{R^{3/2}} \] Now, let's perform the calculations. Calculate the value by following steps: - step0: Calculate: \(\sqrt{9.81\times 0.5}\) - step1: Multiply the numbers: \(\sqrt{4.905}\) - step2: Rewrite the expression: \(\sqrt{\frac{981}{200}}\) - step3: Use the properties of radicals: \(\frac{\sqrt{981}}{\sqrt{200}}\) - step4: Simplify the expression: \(\frac{3\sqrt{109}}{10\sqrt{2}}\) - step5: Multiply by the Conjugate: \(\frac{3\sqrt{109}\times \sqrt{2}}{10\sqrt{2}\times \sqrt{2}}\) - step6: Multiply the numbers: \(\frac{3\sqrt{218}}{10\sqrt{2}\times \sqrt{2}}\) - step7: Multiply the numbers: \(\frac{3\sqrt{218}}{20}\) Calculate or simplify the expression \( (\pi/2) * \sqrt(9.81 * 0.5) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\pi }{2}\sqrt{9.81\times 0.5}\) - step1: Multiply the numbers: \(\frac{\pi }{2}\sqrt{4.905}\) - step2: Simplify the root: \(\frac{\pi }{2}\times \frac{3\sqrt{218}}{20}\) - step3: Multiply the fractions: \(\frac{\pi \times 3\sqrt{218}}{2\times 20}\) - step4: Multiply: \(\frac{3\sqrt{218}\times \pi }{40}\) Calculate or simplify the expression \( (0.010^2 * ((\pi/2) * \sqrt(9.81 * 0.5))^2) / (0.5^(3/2)) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(0.01^{2}\left(\frac{\pi }{2}\sqrt{9.81\times 0.5}\right)^{2}\right)}{0.5^{\frac{3}{2}}}\) - step1: Remove the parentheses: \(\frac{0.01^{2}\left(\frac{\pi }{2}\sqrt{9.81\times 0.5}\right)^{2}}{0.5^{\frac{3}{2}}}\) - step2: Multiply the numbers: \(\frac{0.01^{2}\left(\frac{\pi }{2}\sqrt{4.905}\right)^{2}}{0.5^{\frac{3}{2}}}\) - step3: Simplify the root: \(\frac{0.01^{2}\left(\frac{\pi }{2}\times \frac{3\sqrt{218}}{20}\right)^{2}}{0.5^{\frac{3}{2}}}\) - step4: Multiply the numbers: \(\frac{0.01^{2}\left(\frac{3\sqrt{218}\times \pi }{40}\right)^{2}}{0.5^{\frac{3}{2}}}\) - step5: Convert the expressions: \(\frac{\left(\frac{1}{100}\right)^{2}\left(\frac{3\sqrt{218}\times \pi }{40}\right)^{2}}{0.5^{\frac{3}{2}}}\) - step6: Convert the expressions: \(\frac{\left(\frac{1}{100}\right)^{2}\left(\frac{3\sqrt{218}\times \pi }{40}\right)^{2}}{\left(\frac{1}{2}\right)^{\frac{3}{2}}}\) - step7: Multiply the numbers: \(\frac{\frac{\left(3\sqrt{218}\times \pi \right)^{2}}{4000^{2}}}{\left(\frac{1}{2}\right)^{\frac{3}{2}}}\) - step8: Evaluate the power: \(\frac{\frac{\left(3\sqrt{218}\times \pi \right)^{2}}{4000^{2}}}{\frac{\sqrt{2}}{4}}\) - step9: Multiply by the reciprocal: \(\frac{\left(3\sqrt{218}\times \pi \right)^{2}}{4000^{2}}\times \frac{4}{\sqrt{2}}\) - step10: Rewrite the expression: \(\frac{\left(3\sqrt{218}\times \pi \right)^{2}}{32^{2}\times 125^{2}}\times \frac{4}{\sqrt{2}}\) - step11: Rewrite the expression: \(\frac{\left(3\sqrt{218}\times \pi \right)^{2}}{2^{10}\times 125^{2}}\times \frac{2^{2}}{\sqrt{2}}\) - step12: Reduce the numbers: \(\frac{\left(3\sqrt{218}\times \pi \right)^{2}}{2^{8}\times 125^{2}}\times \frac{1}{\sqrt{2}}\) - step13: Multiply the fractions: \(\frac{\left(3\sqrt{218}\times \pi \right)^{2}}{4000000\sqrt{2}}\) - step14: Evaluate the power: \(\frac{1962\pi ^{2}}{4000000\sqrt{2}}\) - step15: Reduce the fraction: \(\frac{981\pi ^{2}}{2000000\sqrt{2}}\) - step16: Multiply by the Conjugate: \(\frac{981\pi ^{2}\sqrt{2}}{2000000\sqrt{2}\times \sqrt{2}}\) - step17: Multiply the numbers: \(\frac{981\sqrt{2}\times \pi ^{2}}{2000000\sqrt{2}\times \sqrt{2}}\) - step18: Multiply the numbers: \(\frac{981\sqrt{2}\times \pi ^{2}}{4000000}\) ### Step 1: Calculate the Velocity at the Critical Stage \( V_c \) The calculated velocity at the critical stage is: \[ V_c \approx 2.21 \, \mathrm{m/s} \] This value is closest to option **A. 2.2**. ### Step 2: Calculate the Discharge \( Q \) The calculated discharge at the critical stage is: \[ Q \approx 3.48 \, \mathrm{m^3/s} \] This value is closest to option **B. 3.7**. ### Step 3: Calculate the Slope at the Critical Stage \( S \) The calculated slope at the critical stage is: \[ S \approx 0.00342 \] This value is closest to option **B. \( 1.95 \times 10^{-3} \)**. ### Summary of Results - **Velocity at the critical stage**: \( V_c \approx 2.2 \, \mathrm{m/s} \) (Option A) - **Slope at the critical stage**: \( S \approx 0.00342 \) (Option B) - **Discharge at the critical stage**: \( Q \approx 3.48 \, \mathrm{m^3/s} \) (Option B) If you have any further questions or need additional calculations, feel free to ask!