# The curved plate moving of jet

## Force on the curved plate when the plate is moving in the direction of the jet

Let a jet of water strikes a curved plate at the centre of the plate which is moving with a uniform velocity in the direction of the jet as shown in Fig.

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Fig. Jet striking a curved moving plate

Let

v = Absolute velocity of jet,

a = Area of a jet,

u = Velocity of the plate in the direction of the jet.

The relative velocity of the jet of water or the velocity with which jet strikes the curved plate

**= (v - u**)

If the plate is smooth and the loss of energy due to impact of a jet is zero, then the velocity with which the jet will be leaving the curved vane

**= (v - u)**

This velocity can be resolved into two components, one in the direction of the jet and other perpendiculars to the direction of the jet.

(-ve sign is taken as at the outlet, the component is in the opposite direction of the jet).

Mass of the water striking the plate

= ρa x velocity with which jet strikes the plate

**= ρa (v –u)**

Force exerted by a jet of water on the curved plate in the direction of the jet, Fx

Fx = Mass striking per second x [Initial velocity with which jet strikes the plate in the direction of jet – Final velocity]

= ρa(v – u) [(v – u) – (– (v – u) cosθ)]

= ρa(v – u) [(v – u) + (v – u) cosθ]

**= ρa(v – u)^2 [ 1 + cosθ]**

Work is done by the jet on the plate per second

= Fx x Distance travelled per second in the direction of x

= Fx x u

= ρa(v – u)^2 [ 1 + cosθ] x u

**= ρa(v – u)^2 u [ 1 + cosθ]**

**Ex.**A jet of water of diameter 7.5 cm strikes a curved plate at its centre with a velocity of 20 m/s. This curved plate is moving by a velocity of 8 m/s in the direction from the jet. Each jet is deflected through an angle of 165°. Considering the plate smooth,

Find

- Force exerted at the plate into the direction from the jet,
- Power of the jet, and
- The efficiency of the jet.

**Given**

The diameter of the jet,

d = 7.5 cm

= 0.075 m

Area,

a = (π/4) (0.075)^2

= 0.004417

The velocity of the jet,

v = 20 m/s

The velocity of the plate,

u = 8 m/s

The angle of deflection of the jet

= 165°

**Solution**

Angle made by the relative velocity at the outlet of the plate,

θ = 180° - 165°

**= 15°**

Force exerted by the jet on the plate in the direction of the jet is given by,

Fx = ρa (v - u)^2 (1 + cosθ)

= 1000 x 0.004417 x (20 - 8)^2 [1 + cos15°]

**= 1250.38 N**

Work done to the jet on this plate is

**/sec**= Fx x u

= 1250.38 x 8

**= 10003.04 N m/s**

Power of the jet

= 10003.04/1000

**= 10 kW**

= Output / Input

= (Work done by jet/sec) / ( Kinetic energy of jet/sec )

= (1250.38 x 8) / 1/2(pav^3)

= (1250.38 X 8) / 1/2(1000 x 0.004417 x 20^3)

**= 56.4%**

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