Water flows through a curved pipe AB that rotates with a constant clockwise angular velocity of 90 rpm. If the velocity of the water relative to the pipe is 8 m/s, determine the total acceleration of a particle of water at Point P.

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Accepted Answer

Let the curved pipe be a rotating frame of reference. Its angular velocity is [latex]90 \mathrm{\ rpm}\circlearrowright[/latex] or [latex]\omega=9.4248\mathrm{\ rad}/\mathrm{s}\circlearrowright[/latex].

Motion of the frame of reference at Point [latex]P^{\prime}[/latex].

[latex]\mathbf{v}_{P^{\prime}}=(A P)\omega\ ⦨ 45^{\circ}=(0.5{\sqrt{2}})(9.4248)\ ⦨45^{\circ}=6.6643\mathrm{~m/s~}\ ⦨ 45^{\circ}[/latex]

[latex]{{\mathbf{a}_{P^{\prime}}=(A P)\omega^{2}\ ⦪45^{\circ}=(0.5\sqrt{2})(9.4248)^{2}\ ⦪45^{\circ}=62.81~\mathrm{m}/s^2\ ⦪ 45^{\circ}}}[/latex]

Motion of water relative to the frame at Point P.

[latex]\mathbf{v}_{P/F}=8\ \mathrm{m}/\mathrm{s}\leftarrow [/latex]

[latex]{\bf a}_{P/F}=\frac{( u_{P/F})^{2}}{ ho}\downarrow [/latex]

[latex]={\frac{(8\mathrm{\ m}/s)^{2}}{0.5\mathrm{\ m}}}\downarrow [/latex]

[latex]=128 \ m/s^2\downarrow [/latex]

Coriolis acceleration

[latex]\begin{array}{c}{{a_{c}=2\omega u_{P/F}}}\ {{=(2)(9.4248)(8)}}\ {{=150.797\mathrm{~m/s^{2}}}}↑\end{array}[/latex]

Acceleration of water at Point P.

[latex]\begin{array}{l}{{\mathbf{a}_{P}=\mathbf{a}_{P^{\prime}}+\mathbf{a}_{P/F}+\mathbf{a}_{c}}}\ {{\mathbf{a}_{P}=[62.81\mathbf\ ⦪45^{\circ}]+[128\,\downarrow]+[150.797\uparrow ]}}\ \quad {{=[44.413\:\mathbf{m}/s^{2} o]+[21.616\:\mathbf{m}/s^{2}\downarrow]}}\end{array}[/latex]

[latex]\mathbf{a}_{P}=49.4~\mathrm{m}/\mathrm{s}^2\ ⦪26.0^{\circ}\blacktriangleleft[/latex]

Question 15.255: Water flows through a curved pipe AB that rotates with a con...

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