Hydraulic System
February 08, 2019

## FORCE EXERTED BY JET ON HINGED PLATE

We have already seen the derivation of

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Sin θ = ρ a V

*continuity equation**,**Bernoulli’s equation**,**momentum equation**,**velocity of sound in an isothermal process*,*velocity 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.
We have also discussed the fundamentals of impact of
jets,

*force exerted by a jet on vertical flat plate*,*force exerted by a jet on stationary inclined flat plate*and*force exerted by a jet on stationary curved plate*in our recent post.
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.

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**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.

- Force due to jet of water normal to the plate, F
_{n } - Weight of the plate, W

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*Force
due to jet of water normal to the plate*

*Force due to jet of water normal to the plate*

F

_{n}= ρ a V^{2}Sin (90 – θ)####
*Weight
of the plate, W *

*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 F

_{n}about hinge = Moment of weight W about hinge
Moment of force F

_{n}about hinge = F_{n}x OB = ρ a V^{2}Sin (90 – θ) x OB
Moment of force F

_{n}about hinge = ρ a V^{2}Cos θ x OB
Moment of force F

_{n}about hinge = ρ a V^{2}Cos θ x (OA / Cos θ)
Moment of force F

_{n}about hinge = ρ a V^{2}x OA
Moment of force F

_{n}about hinge = ρ a V^{2}x
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 F

_{n}about hinge = Moment of weight W about hinge
ρ a V

^{2}x = W x Sin θ####
Sin θ = ρ a V^{2} /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