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Problem 2.76 - Problem 2.76[Difficulty 5 Geometry of...

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Problem 2.76 [Difficulty: 5] Given: Geometry of rotating bearing Find: Expression for shear stress; Maximum shear stress; Expression for total torque; Total torque Solution: Basic equation τ μ du dy = dT r τ dA = Assumptions: Newtonian fluid, narrow clearance gap, laminar motion From the figure r R sin θ ( ) = u ω r = ω R sin θ ( ) = du dy u 0 h = u h = h a R 1 cos θ ( ) ( ) + = dA 2 π r dr = 2 π R sin θ ( ) R cos θ ( ) d θ = Then τ μ du dy = μ ω R sin θ ( ) a R 1 cos θ ( ) ( ) + = To find the maximum τ set θ μ ω R sin θ ( ) a R 1 cos θ ( ) ( ) + d d 0 = so R μ ω R cos θ ( ) R a cos θ ( ) + ( ) R a + R cos θ ( ) ( ) 2 0 = R cos θ ( ) R a cos θ ( ) + 0 = θ acos R R a + = acos 75 75 0.5 + = θ 6.6 deg = τ 12.5 poise 0.1 × kg m s poise
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