Water Jet Impact on Vane
Problem Statement
Determine the magnitude of resultant force and its direction on the vane shown in the figure below if a water jet of 50mm diameter and 20m/s velocity strikes the vane tangentially and deflects without friction.
Given Data
Solution Approach
To find the resultant force exerted on the vane by the water jet, we need to:
- Calculate the cross-sectional area of the jet and the discharge
- Apply the momentum equation to determine the forces in both X and Y directions
- Calculate the resultant force and its direction
Preliminary Calculations
Step 1: Calculate the cross-sectional area of the jet:
Step 2: Calculate the discharge:
Step 3: Note the key conditions:
- No friction means there is no loss of energy
- Pressure is atmospheric, so there are no pressure forces
- Velocity magnitude remains constant throughout (V = 20 m/s)
Force in X-Direction
Step 1: Apply the momentum equation in the X-direction:
Step 2: Determine the X-components of velocity:
- Initial velocity (V1x) = 20 m/s (straight to the right)
- Final velocity (V2x) = -20cos45° = -20 × 0.7071 = -14.142 m/s (negative direction)
Step 3: Calculate Fx:
Force in Y-Direction
Step 1: Apply the momentum equation in the Y-direction:
Step 2: Determine the Y-components of velocity:
- Initial velocity (V1y) = 0 m/s (no vertical component initially)
- Final velocity (V2y) = -20sin45° = -20 × 0.7071 = -14.142 m/s (downward direction)
Step 3: Calculate Fy:
Resultant Force Calculation
Step 1: Calculate the magnitude of the resultant force:
Step 2: Calculate the direction of the resultant force:
Summary
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The water jet has:
- Diameter: 50 mm
- Velocity: 20 m/s
- Cross-sectional area: 0.001963 m²
- Discharge: 0.03927 m³/s
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The force components were calculated using the momentum equation:
- X-direction force: Fx = 1340.8 N
- Y-direction force: Fy = 555.4 N
-
The resultant force on the vane:
- Magnitude: 1451 N
- Direction: 22.5° from the horizontal axis
This problem demonstrates the application of the momentum principle to determine forces on a curved vane. The water jet changes direction without losing energy (no friction), resulting in a change of momentum that produces a force on the vane. The magnitude and direction of this force were calculated by analyzing the X and Y components separately.
