To determine The torque that must be applied to the shaft to maintain the motion Explanation Given: The radius of the tube ( r ) is 40 mm and 80 mm . The angular velocity of the tube ( ω ) is 4.5 rad/s . The thickness of thin film ( t ) is 1.5 mm. The viscosity of oil ( μ ) is 0.0586 N ⋅ s/m 2 . Calculation: The velocity profile within the oil is considered to be constant when the velocity distribution is linear. Calculate the velocity gradient ( d u d y ) . d u d y = U t U t = w r t d u d y = ( ω r t ) s − 1 Calculate the shear stress ( τ ) using the relation. τ = μ d u d y τ = ( μ × ω r t ) N / m 2 Calculate the cross sectional area of the tube ( d A ) . d A = ( 2 π r d r ) m 2 Determine the force ( d F ) for shaded element using the relation. F = τ A Differentiate the equation on both sides. d F = τ d A d F = μ × ω r t × 2 π r d r = ( 2 π μ ω r 2 d r t ) N Take moment about point D to obtain the torque ( T ). r d F − T = 0 Integrate r with the limits of 0 to R

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ISBN: 9780134649290

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 1, Problem 49P

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The torque that must be applied to the shaft to maintain the motion

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Chapter 1 Solutions

Fluid Mechanics (2nd Edition)

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