Question 5.12: A vane, with a turning angle θ, is attached to a cart which ...

Choose the CV and coordinate systems as shown in Fig. 5.16(a). X Y is the inertial reference frame, while reference frame xy moves with the cart.

Applying the linear momentum conservation equation to the moving CV, we have

\sum \vec{F}_{C V}-0=0+\rho \int_{C S} \vec{V}_{x y z}\left(\vec{V}_{x y z} . \hat{n}\right) d A                                (5.36)

Note that the correction term on the left-hand side is zero, since the reference frame xyz is non-accelerating. The right-hand side of Eq. (5.36) may be further expanded by noting that because of the frictionless nature of the flow, the speed of water relative to the cart remains unaltered as it enters and leaves the CV (this may be ascertained by applying Bernoulli’s equation along a streamline connecting the inlet and the exit), although the flow direction gets altered, thereby giving rise to a rate of change of linear momentum. Mathematically, the scenario may be represented as

\sum \vec{F}_{C V}=\rho\left[-\left(V-V_{c}\right) \hat{i}\left(V-V_{c}\right) A\right]+\rho\left[\left(V-V_{c}\right)(\cos \theta \hat{i}+\sin \theta \hat{j})\left(V-V_{c}\right) A\right]                        (5.37)

The x component of the force is

F_{x}=\rho A\left(V-V_{c}\right)^{2}(\cos \theta-1)                     (5.37a)

The y component of the force is

F_{y}=\rho\left[\left(V-V_{c}\right) \sin \theta\left(V-V_{c}\right) A\right]=\rho A\left(V-V_{c}\right)^{2} \sin \theta                    (5.37b)

These are the x and y components of force exerted by the cart on the water jet. By Newton’s third law, the x and y components of force exerted by the water on the cart are \rho A\left(V-V_{c}\right)^{2}(1-\cos \theta) and -\rho A\left(V-V_{c}\right)^{2} \sin \theta , respectively.