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Friday, 8 February 2019

FORCE EXERTED BY JET ON HINGED PLATE

We have already seen the derivation of continuity equationBernoulli’s equationmomentum equationvelocity of sound in an isothermal processvelocity of sound in an adiabatic process, fundamentals of stagnation properties i.e. stagnation pressure, stagnation temperature and stagnation density for compressible fluid flow in our previous posts. 


Now we will see here the derivation of expression of force exerted by a jet on a hinged plate with the help of this post. Let us first brief here the basic concept of impact of jets and after that we will derive the expression of force exerted by a jet on hinged plate. 

Impact of jets  

Let us consider that we have one pipe through which liquid is flowing under pressure. Let us assume that a nozzle is fitted at outlet of pipe. Liquid which will come through the outlet of nozzle will be in the form of jet. 

If a plate, which may be moving or fixed, is placed in the path of jet, there will be one force which will be exerted by the jet over the surface of plate. The force which will be exerted by the jet over the surface of plate, which might be moving or fixed, will be termed as impact of jet. 

Force exerted by a jet on a hinged plate 

Let us consider a jet of water striking a vertical plate at the center which is hinged at point O as displayed here in following figure. 

When jet of water will strike the plate, the plate will swing through an angle about the hinge as displayed here in following figure. 

Let us consider the following terms from above figure.
x = Distance of center of jet from hinge O
θ = Angle of swing about hinge
W = Weight of plate acting at C.G of the plate 

Initial position of the vertical plate is displayed above in figure by dotted lines. When water jet will strike the plate, the plate will swing through an angle θ. After the jet strikes the plate, point A will be now at point A’ as displayed in above figure. 

Distance OA = Distance OA’ = x 

Weight of plate will be acted through the point A’ after water jet strikes the plate. 

There will be two forces acting on the plate as mentioned here. 
  1. Force due to jet of water normal to the plate, F
  2. Weight of the plate, W 

Force due to jet of water normal to the plate

Fn = ρ a V2 Sin (90 – θ)  

Weight of the plate, W 

Once the jet strikes the plate, for equilibrium condition we will have following equation as mentioned here. 

Moment of force Fn about hinge = Moment of weight W about hinge 

Moment of force Fn about hinge = Fn x OB = ρ a V2 Sin (90 – θ)   x OB
Moment of force Fn about hinge = ρ a V2 Cos θ   x OB
Moment of force Fn about hinge = ρ a V2 Cos θ   x (OA / Cos θ)
Moment of force Fn about hinge = ρ a V2 x OA
Moment of force Fn about hinge = ρ a V2

Now we will determine the moment of weight W about hinge 

Moment of weight W about the hinge = W x OA’ Sin θ = W x Sin θ 

Once the jet strikes the plate, for equilibrium condition, we will have 

Moment of force Fn about hinge = Moment of weight W about hinge
ρ a V2 x = W x Sin θ

Sin θ = ρ a V2 /W 


Above equation, derived here, provides the angle of swing of plate when water jet strikes the vertical plate which is fixed at hinge.  

Do you have any suggestions? Please write in comment box. 

Reference: 

Fluid mechanics, By R. K. Bansal 
Image courtesy: Google  

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