A jet of water strikes and leaves a fixed, smooth curved vane tangentially at inlet and outlet. What is the magnitude of the force component exerted by the jet on the vane along the normal to the vane surface? (α = inlet jet angle with tangent, β = outlet jet angle with tangent; ρ = density, a = jet area, V = jet speed)

Introduction:
Hydraulic forces on curved vanes are evaluated by resolving momentum change along convenient axes. When a jet meets and leaves a vane tangentially, the tangential component relates to whirl, while the normal component depends on the normal projections of the jet velocity at entry and exit.

Given Data / Assumptions:

• Fixed, smooth curved vane.
• Jet speed magnitude V; area a; density ρ.
• Inlet and outlet are tangential to the vane surface.
• Angles α (inlet) and β (outlet) measured from the tangent to the jet line.

Concept / Approach:
By linear momentum, force on the vane equals mass flow rate times the vector change in jet velocity. Resolve along the normal to the vane. For tangential approach/exit, the normal components at inlet and outlet are V sin α and V sin β (both directed into the vane normal on their respective sides). Because the vane is fixed, the normal force component sums these magnitudes.

Step-by-Step Solution:
1) Mass flow rate: m_dot = ρ a V.2) Normal component of inlet velocity: V_n,in = V sin α.3) Normal component of outlet velocity: V_n,out = V sin β (same sense into the vane).4) Force along normal: F_n = m_dot * (V_n,in + V_n,out) = ρ a V^2 (sin α + sin β).

Verification / Alternative check:
Special case β = 0 (perfectly tangential exit) gives F_n = ρ a V^2 sin α, matching flat-plate normal result.

Why Other Options Are Wrong:
Cosine forms correspond to tangential components, not normal.

Difference forms imply opposing normal components, which is incorrect for tangential approach/exit on a fixed vane.

2 cos α is unrelated to normal summation.

Common Pitfalls:
Confusing angle definitions (with tangent vs with vane chord) and mixing tangential/normal projections.

Final Answer:
ρ * a * V^2 * (sin α + sin β)