# Some momentum proofs needed

1. Aug 9, 2006

### John O' Meara

A jet of liquid of cross-sectional area A and density p moves with speed vj in the positive x-direction and impinges against a perfectly smooth blade B, which deflects the stream at right angles but does not slow it down. (a) If the blade is stationary, prove that the rate of arrival of mass at the blade is dm/dt = pAvj. (b) If the impulse-momentum theorem is applied to a small mass dm, prove that the x-component of the force acting on this mass for the time interval dt is given by Fx = -(vj)dm/dt. (c) Prove that the steady force exerted on the blade in the x-direction is Fx = pA(vj)^2.
If the blade moves to the right with a seed vb ( vb < vj) derive the equations for (d) the rate of arrival of mass at the moving blade ( e) the force Fx on the blade, and (f) the power delivered to the blade.

Assuming that the deflected jet moves in the positive y-direction (a) p = density = mass/volume then Axvj= volume per unit time therefore pAvj =mass flow rate = dm/dt. (b) The jet of liquid exerts a force F on the blade now the x-component of the force acting on this mass over the time interval dt is the negative of this force =-Fx but Fxdt = dm(vj) therefore this force = -(vj)dm/dt. (c).... (d) pA(vj-vb). (e)... (f)...