1 answer

Fluid mechanics

Question:

Chapter 5, Problem 052
Water flows from a large tank into a dish as shown in Fig. P5.52. (a) If at the instant shown the tank and the water in it weigh W1 lb, what is the tension, T1, in the cable supporting the tank? (b) If at the instant shown the dish and the water in it weigh W2 lb, what is the force, F2, needed to support the dish?
image.pngFig. P5.52


Answers

Applying bernoullis principle to the tank, \(\left(\frac{P_{2}-P_{1}}{\rho}\right)+\left(\frac{V_{2}^{2}-V_{1}^{2}}{2 g}\right)+\left(z_{2}-z_{1}\right)=0\)

\(0+\frac{V_{2}^{2}}{2 g}+10=0 \Rightarrow V_{2}=25.4 \mathrm{ft} / \mathrm{s}\)

From conservation of linear momentum for tank

\(\rho_{w} A_{\text {out }} V_{\text {out }}^{2}=T_{1}-W_{1}\)

\(-1.94 \times\left(\frac{\pi}{4} \times 0.1^{2}\right) \times\left(25.4^{2}\right)=T_{1}-W_{1}\)

\(T_{1}=\left(W_{1}-9.83\right) \mathrm{lb}\)

Applying bernoullis principle to the dish,

\(\left(\frac{P_{3}-P_{2}}{\rho}\right)+\left(\frac{V_{3}^{2}-V_{2}^{2}}{2 g}\right)+\left(z_{3}-z_{2}\right)=0\)

\(0+\frac{V_{3}^{2}-25.4^{2}}{2 \times 32.2}+12=0 \Rightarrow V_{3}=37.66 \mathrm{ft} / \mathrm{s}\)

From conservation of linear momentum for dish.

\(m_{i n} V_{i n}=F_{2}-W_{2} \Rightarrow\left(\rho_{w} A_{\text {out }} V_{\text {out }}\right) V_{3}=F_{2}-W_{2}\)

\(\left[1.94 \times\left(\frac{\pi}{4} \times 0.1^{2}\right) \times 25.4\right] \times 37.66=F_{2}-W_{2}\)

\(F_{2}=\left(W_{2}+14.6\right) \mathrm{lb}\)

.

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