The boat in Fig. 7-37a is propelled by the vertical cylinder, which has a height of 3 m, a diameter of 0.5 m, and is rotating at 55 rad / s. If the boat is traveling forward at 3 m / s, and the wind speed against the side of the boat is 4 m / s, determine the force the cylinder exerts on the

Question

The boat in Fig. 7-37a is propelled by the vertical cylinder, which has a height of 3 m, a diameter of 0.5 m, and is rotating at 55 rad / s. If the boat is traveling forward at 3 m / s, and the wind speed against the side of the boat is 4 m / s, determine the force the cylinder exerts on the boat. Take [latex] \rho_a=1.22 kg / m ^3 [/latex]

Accepted Answer

Fluid Description. We have steady flow of an ideal fluid.

Analysis. The force per unit height acting on the rotating cylinder is defined by Eq. 7-72, F = ρUΓ. Since the boat (cylinder) is moving, we must find the velocity of the wind relative to the boat, Fig. 7-37b. It is

[latex] F_y=- ho U \Gamma [/latex] (7-72)

[latex] \begin{gathered} V _{w / b}= V _w- V _b \ V _{w / b}=\{-4 j -3 ext{i} \} m / s \end{gathered} [/latex]

The magnitude of the velocity is then

[latex] V_{w / b}=\sqrt{(-4 m / s )^2+(-3 m / s )^2}=5 m / s [/latex]

And the circulation of the flow around the cylinder is

[latex] \begin{aligned} \Gamma=\oint V \cdot d s & =(55 rad / s )(0.25 m )[2 \pi(0.25 m )] \ & =21.60 m ^2 / s \end{aligned} [/latex]

Since the cylinder has a height of 3 m, the force acting on the cylinder is, therefore,

[latex] \begin{aligned} & F= ho V_{w / b} \Gamma h=\left(1.22 kg / m ^3 ight)(5 m / s )\left(21.60 m ^2 / s ight)(3 m ) \ & F=395 N \end{aligned} [/latex]

This force acts perpendicular to [latex] V _{w / b} [/latex] as shown in Fig. 7-37c, and so it is the x component of this force that propels the boat forward.

NOTE: This idea for propulsion, based on the Magnus effect, was first employed by Anton Flettner in the 1920s. His "Flettner rotor ship" had two rotating cylinders, each 15 m high, having a diameter of 3 m, and driven by an electric motor.

Question 7.11: The boat in Fig. 7-37a is propelled by the vertical cylinde...

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