A mixing beater consists of three thin rods, each 11.0 cm long. The rods diverge

Question

A mixing beater consists of three thin rods, each 11.0 cm long. The rods diverge from a central hub, separated from each other by 120°, and all turn in the same plane. A ball is attached to the end of each rod. Each ball has cross-sectional area 3.70 cm2 and is so shaped that it has a drag coefficient of 0.560. The drag force on each ball is

R =

D A v2

where D is the drag coefficient,

the density of the fluid, A the cross-sectional area, and v the speed of the object moving through the fluid.

(a) Calculate the power input required to spin the beater at 1000 rev/min in water.
W
(b) The beater is taken out of the water and held in air. If the input power remains the same (it wouldn't, but if it did), what would be the new rotation speed?
rev/min

Explanation / Answer

a) r = 0.11 m, theta = 120 deg

A = 3.70 * 10-4 m2, D = 0.560

w = 1000 rev/min = 1000 * 2pi /60 = 104.72

torque = 3 r R

P = torque * w

drag coefficient R = 1/2 D p A v2

torque = 3 r * [1/2 D p A (rw)2]

P = 3/2 * r3 D A w3 p = 3/2 * 0.113 * 0.56 * 3.7 * 10-4 * 104.723 * 1000

input power = 475 W

b) w = [2 P / 3 r3 D A p]1/3

= [2 * 475 / 3 * 0.113 * 0.560 * 3.7 * 10-4  * 1.20]1/3

= 985 * [60 / 2 pi]

= 9410 rev/min

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