Lecture - 6 ( jet strikes on inclined flat moving plate )


Jet strikes on inclined flat moving plate
Let us consider an plate which is free to move linearly and inclind at at an angle θ. The jet of water is striking at the centre of the plate horizontally. Mass flow rate of water Jet is denoted by 'm' in Kilogram per second. As we know the plate is moving with velocity U with respect to ground and velocity of water Jet at entry of plate is V1 with respect to ground. m1 is the mass flow rate exiting from the above end of the plate and m2 is the mass flow rate exiting from the below end of the plate. Point2 denote both the the above and below exit point of jet on the plate. Point1 is the entry point of jet on the centre of plate. U1' represent velocity of entry point of plate and 'U2' represent velocity of exit point of plate. When Jet strikes the plate, it will move in linear direction so here U = U1 = U2. Now in this case also concepts of relative velocity will be used.
V1 represents velocity of jet at entry with respect to ground. Remember one thing that any relative term which is represented with respect to ground are absolute term. So here V1 is also called as absolute velocity of jet at entry. Similarly U is also called as absolute velocity of plate because U is velocity of plate with respect to ground. V2 represents velocity of Jet at exit with respect to ground and it is also called as absolute velocity of Jet at exit.
Vr1 represent relative velocity of water jet at entry with respect to plate. Vr2 represent relative velocity of water Jet at exit with respect to plate.

Now we have to calculate forces in horizontal and vertical direction and also in normal and tangential direction of plate. We will do it by taking component of Forces. Let's draw the normal force component perpendicular to plate and tangential force component tangential to plate. Both the forces are shown in green colour.
m=ρ*A*Vr1 =ρ*A*(V1-U1); this is the mass flow rate which is reaching on the plate

In this scenario I am considering that our observer is at the blade so velocity of blade for our observer is zero and water jet striking the plate with relative velocity Vr1. The mass flow rate of water striking on the plate is 'ρ*A*(V1-U1)' because here our plate is also moving with velocity U1.
Fn = linear momentum of Jet at entry point 1 in normal direction - linear momentum of jet at exit point 2 in normal direction = [m*Vr1*cos(90-θ)] - [(m1)*0 + (m2)*0] = m*Vr1*sinθ
 Fn = ρ*a*(Vr1^2)*sinθ = ρ*a*((V1-U1)^2)*sinθ
Now let us calculate the forces in horizontal and vertical direction. If we consider the below figure we can easily calculate force in horizontal and vertical direction. We just consider component of 'FN ' in horizontal and vertical direction.

Fx = FN cos(90-θ) = ρ*a*(Vr1^2)*{(sinθ)^2}
Fy = FN sin (90-θ) = ρ*a*(Vr1^2)*sinθ*cosθ


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