Problem Statement
A jet of water of diameter 50 mm moving with a velocity of 20 m/s strikes a fixed plate in such a way that the angle between the jet and the plate is 60°. Find the force exerted by the jet on the plate (i) in the direction normal to the plate, and (ii) in the direction of the jet.
Given Data & Constants
- Diameter of jet, \(d = 50 \, \text{mm} = 0.05 \, \text{m}\)
- Velocity of jet, \(V = 20 \, \text{m/s}\)
- Angle between jet and plate, \(\theta = 60^\circ\)
- Density of water, \(\rho = 1000 \, \text{kg/m}^3\)
Solution
1. Calculate Area and Mass Flow Rate
(i) Force in the Direction Normal to the Plate (\(F_n\))
The force normal to the plate is the rate of change of momentum in that direction. The initial velocity component normal to the plate is \(V \sin(\theta)\), and the final velocity is 0.
(ii) Force in the Direction of the Jet (\(F_x\))
This force is the component of the normal force (\(F_n\)) acting back along the original direction of the jet.
(i) Force normal to the plate: \( \approx 680.6 \, \text{N} \)
(ii) Force in the direction of the jet: \( \approx 589.4 \, \text{N} \)
Explanation of the Forces
When a jet strikes an inclined plate, the force it exerts is not in the direction of the jet itself. The primary force is generated by the change in momentum perpendicular (normal) to the plate's surface.
- Normal Force (\(F_n\)): This is the total force exerted by the jet on the plate. It's calculated using the component of the jet's initial velocity that is normal to the plate (\(V \sin\theta\)), as this is the only component of momentum that is destroyed by the plate.
- Force in Jet Direction (\(F_x\)): This is the force that one would feel if they were holding the nozzle and trying to keep it stationary. It is not the full normal force, but rather the component of the normal force that acts back along the jet's original axis. This is found by resolving the normal force vector.
