The mill
Water from a reservoir is used to rotate a mill of radius R with angular velocity ω constant through the jet flowing from the orifice of area S situated at a depth h of its level. With the jet perpendicularly incident on each shovel, with totally inelastic shock, calculate the torque of the friction forces on the mill axis, where ρ and g, respectively, are the mass, specific gravity and acceleration of gravity.
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1 solution
According to energy conservation, the speed at which water exits the reservoir is given by v = √2gh After the inelastic shock, a portion of the water with mass Δm will have velocity ωR. Being thus, the linear momentum variation of this water mass will be given by Δp = Δm ⋅ (v - ω R).
To determine the mass of water reaching one of the blades in a time interval Δt, we will consider a cylindrical portion that emerges from the hole with area of the base S and length L. The mass contained in this cylinder is given by Δm = ρ ⋅ S ⋅ L. Note that after a time interval Δt the blade shifts ωRΔt. Therefore, the length of the cylinder that reaches the blade is given by L = (v - ω R) Δt. Therefore the mass of the cylinder will be: Δm = ρS (v - ω R) Δt Thus, the average force applied by the water on a shovel of the mill is given by:
F = Δ t Δ p = ρS(v – ωR)²
F = = ρSv²(1- v ω R )²
The torque of the friction forces has the same modulus of the torques of the water portion that hits the shovel, then:
M = R ⋅ F = ρSv²R(1- v ω R )²
M = 2ρghRS(1- √ 2 g h ω R )²